### 5.3.2 Confinement potential and edge states

Let us discuss in more detail the edge states and their chirality by considering a confinement potential that depends only on the y-component of the position, $V_{\text{conf}}(y)$. Within an (approximate) diagonalisation of the Hamiltonian, taking into account this potential, the Landau gauge $A_L = B(-y, 0, 0)$ imposes itself as the natural choice because it respects translation invariance in the x-direction in line with the symmetry imposed on the confinement potential. This means that the wave-vector component $k_x$ remains a good quantum number, as we have seen in section 5.2.5. The Hamiltonian is directly diagonalised if we use the approximation

$$V_{\text{conf}}(y) \approx V_{\text{conf}}(y_0 = k_x l_B^2) + \delta V(y_0; a, a^{\dagger}), \quad (5.78)$$

where we neglect the second term, as we have already implicitly done in equation (5.76) for the more general (disorder) potential. This approximation is fully justified if the electrostatic potential varies slowly in space,

$$|\nabla V| \ll \frac{\Delta_{n-1,n}}{R_C}, \quad (5.79)$$

where $\Delta_{n-1,n}$ is the spacing between adjacent Landau levels (we have $\Delta_{n-1,n} = \hbar\omega_C$ for a parabolic band) and $R_C = l_B\sqrt{2n+1}$ is the cyclotron radius.<fnref n="23" />

Since, within the approximation $V_{\text{conf}}(y) \approx V_{\text{conf}}(y_0)$, the Hamiltonian is already diagonal, we may immediately write down the spectrum, which reads

$$\delta V(y_0; a, a^{\dagger}) \approx \frac{\partial V}{\partial y_0} \eta_y = -\frac{\partial V}{\partial y_0} \frac{l_B}{\sqrt{2}} (a^{\dagger} + a),$$

where we have expressed $\eta_y = -\Pi_x/eB$ [see equation (5.54)] in terms of the ladder operators (5.28). If we treat this term in perturbation theory, we notice that the first-order term vanishes, $\langle n | \delta V(y_0) | n \rangle = 0$, since the ladder operators are not diagonal in the number basis, $\langle n | a^{(\dagger)} | n \rangle = 0$. One therefore needs to consider the second-order term, which yields a contribution

$$\langle \delta V(y_0) \rangle^{(2)} \sim \left| \frac{\partial V}{\partial y_0} \right|^2 \frac{R_C^2}{\Delta_{n-1,n}}. \quad (5.80)$$

This term is negligible as long as $|\langle \delta V(y_0) \rangle^{(2)}| \ll \Delta_{n-1,n}$, which is nothing other than the condition (5.79).

<footnote n="23">²³This approximation may be justified by a series expansion of the potential around $y_0$, using $y = y_0 + \eta_y$. At linear order, one finds</footnote>
(for an underlying parabolic band)

$$E_{n,k_x} = \hbar\omega_C \left(n + \frac{1}{2}\right) + V_{\text{conf}}(k_x l_B^2). \qquad (5.81)$$

We recognise, here, the two points mentioned already above: (i) the degeneracy of the Landau levels in the wave vector $k_x$ is lifted, as one may explicitly see from the second term; and (ii) we obtain again a family of 1D electronic bands where the dispersion in $k_x$ is given by the confinement potential. One may then also attribute a group velocity to the states $|n, k_x\rangle$, in the $x$-direction,

$$v_x = \frac{\partial E_{n,k_x}}{\hbar \partial k_x} = \frac{\partial V_{\text{conf}}}{\partial y} \frac{l_B^2}{\hbar} \text{sgn}(B) = \frac{1}{eB} \frac{\partial V_{\text{conf}}}{\partial y}, \qquad (5.82)$$

which is nothing other than the drift velocity, which we have already encountered in the form of equation (5.77) for a general potential: the sense of propagation, which is the chirality, is simply given by the sign of the group velocity. It is imposed by the gradient of the potential in the $y$-direction, which naturally changes from one edge to the other, as well as by the orientation of the magnetic field.<fnref n="24" />

Notice finally that, since all Landau levels are bent upwards at the sample edge, the $n$ completely filled Landau levels (for a filling factor $\nu = n$) successively cross the Fermi level there (see figure 5.7). These levels therefore form $n$ 1D metallic channels at the edge. Furthermore, as we have mentioned above, the gradient $\partial V_{\text{conf}}/\partial k_x$, and thus the group velocity, is the same for all $n$ levels so that these 1D metallic channels are chiral. These metallic channels need to be contrasted with the states one obtains when a Landau level crosses the Fermi level somewhere in the bulk of the sample. In this case, the crossing between the Landau and the Fermi levels forms a closed line and thus a localised non-current carrying state, as we have seen above. Now, what about an impurity – a scatterer – situated at the edge? Naturally there are then possible scattering mechanisms, but they couple only the $n$ edge channels among one another that all have the same chirality. Only forward-scattering from a channel $a$ to a channel $b$ is therefore possible, while backscattering does not exist if one considers a single edge. Indeed, the

<footnote n="24">²⁴More precisely, it is the product of the charge and the $B$-field that matters. To be exhaustive, we mention that the sense of propagation naturally also changes when we change the sign of the charge of the carriers.</footnote>
<figure><img src="image_53.png" /></figure>

Figure 5.9: Hall bar with six contacts. The current is driven through the system by the contacts 1 (source) and 4 (drain), which are also used in the measurement of the two-terminal resistance, while the remaining contacts are used to measure the different four-terminal resistances. In order to measure the longitudinal resistance, one uses two contacts at the same sample edge, that is two contacts that are not separated by the source or the drain. Here, we have chosen the contacts 2 and 3 to measure the longitudinal resistance $R_L$ (in blue). The transverse Hall resistance $R_H$ (in red) is measured between two contacts (e.g. 3 and 5) at opposite edges, i.e. separated by the source or the drain. The magnetic field is chosen to be oriented into the +z-direction so that a clock-wise chirality is imposed on the edge channels.

channels with opposite chirality are spatially separated from the former by a macroscopic distance since they are situated at the other edge, unless they are artificially brought together in local constriction. It is in this sense that the edge channels are ballistic: *backscattering at the same edge is forbidden by the chirality of the metallic edge channels.*

## 5.4 Understanding the quantum Hall effect within the Landauer-Büttiker formalism

With the former discussion of edge channels, we are now armed to understand the quantum Hall effect within the Landauer-Büttiker formalism introduced
in the last two chapters. We consider for the moment a GaAs heterostructure, where the Landau-level spectrum is formed from a single parabolic band to fix the overall picture. The specific case of graphene is then discussed later on, in section 5.4.4. We situate the Fermi level between the Landau levels $n-1$ and $n$. If we take into account the twofold spin degeneracy of the Landau levels, this situation corresponds to an integer filling factor $\nu = 2n$, and consequently to $M = 2n$ edge channels. In the spirit of the Landauer-Büttiker formalism, we make a clear distinction between the two-terminal and the different four-terminal resistances. We have already mentioned, in the introduction to this chapter, that the *Hall resistance* and *longitudinal resistance* are four-terminal resistances, which are mainly defined within the Hall-bar geometry with six terminals, sketched in figure 5.9. We use a convention where the contacts are enumerated in a clockwise manner starting with 1 for the source, and contact 4 is the drain.

In order to specify the different resistances, recall that the Landauer-Büttiker formula (4.5), $I_i = \sum_j G_{ij}V_j$, can be written with the help of the conductance matrix

$$ \mathcal{G} = (G_{ij}) = -\frac{e^2}{\hbar} (\mathcal{T}_{ij}), \qquad (5.83) $$

in terms of the quantum of conductance, $e^2/\hbar$.<fnref n="25" /> With this convention of the sign in the conductance matrix, its off-diagonal elements are directly given in terms of the transmission coefficients $\mathcal{T}_{ij}$ from contact $j$ to contact $i$, while the diagonal elements are formally defined as

$$ \mathcal{T}_{ii} = -(M - \mathcal{R}_{ii}). \qquad (5.84) $$

As a consequence of the chirality of the edge channels, there is no backscattering so that $R_{ii} = 0$, and the diagonal elements of the conductance matrix are thus directly given in terms of the number of edge modes $M$. What about the other elements, i.e. the transmission coefficients $\mathcal{T}_{ij}$ in the $j$-th line? Again due to the chirality of the edge channels, which we have chosen to be clockwise (a magnetic field oriented in the $+x$-direction) in figure 5.9, the contact $i$ only receives electrons transmitted from the contact $i-1$, so

<footnote n="25">²⁵In contrast to the general discussion of the Landauer-Büttiker formalism in chapter 4, we do not absorb the factor of 2 for the spin degeneracy into the quantum of conductance, but we take it into account in the number of channels $M = 2n$. This is more practical in the discussion of a possible spin-degeneracy lifting, e.g. due to the Zeeman effect, as well as in view of the study of the quantum anomalous and spin Hall effects in chapter 7.</footnote>
that the only non-zero off-diagonal elements of the conduction matrix are given by

$$ \mathcal{T}_{ij} = \mathcal{T}_{i,i-1}\delta_{j,i-1}. \qquad (5.85) $$

Remember from multi-channel scattering (see section 3.6) that this coefficient is given by the sum $\mathcal{T}_{i,i-1} = \sum_{a,b} \mathcal{T}_{i,i-1;ab}$ of all the transmission coefficients from the mode $b$ (in the terminal $i-1$) to mode $a$ (in the terminal $i$). Since there may be scatterers at the edge, this transmission matrix is not necessarily diagonal. However, since this transmission matrix must reflect the absence of backscattering, $\mathcal{R}_{ii} = 0$, we can use the same argument as in the discussion of the general constraints imposed on the conduction matrix in section 4.1 that the currents must be independent of the reference potential. The sum over all elements in the conduction matrix must therefore still vanish so that we find

$$ \mathcal{T}_{i,i-1} = \sum_{a,b} \mathcal{T}_{i,i-1;ab} = -M \qquad (5.86) $$

as for fully ballistic channels, even if we allow for forward-scattering events at the edge. The conduction matrix finally reads

$$ \mathcal{G}_{\text{QHE}}(B) = \frac{e^2}{h} \begin{pmatrix} M & 0 & 0 & 0 & 0 & -M \\ -M & M & 0 & 0 & 0 & 0 \\ 0 & -M & M & 0 & 0 & 0 \\ 0 & 0 & -M & M & 0 & 0 \\ 0 & 0 & 0 & -M & M & 0 \\ 0 & 0 & 0 & 0 & -M & M \end{pmatrix}, \qquad (5.87) $$

where we have explicitly put the magnetic field in the argument. Indeed, one can immediately verify the reciprocity relations: while the conduction matrix is apparently not symmetric, we obtain from equation (4.23), $\mathcal{T}_{ij}(B) = \mathcal{T}_{ji}(-B)$,

$$ \mathcal{G}_{\text{QHE}}(-B) = \mathcal{G}_{\text{QHE}}^{t}(B). \qquad (5.88) $$

This reciprocity relation has now a transparent physical interpretation in terms of the chirality of the edge channels. As mentioned above, we invert the chirality (the sense of propagation along the edge channels) if we change the orientation of the magnetic field $B \to -B$. In this case, the contact $i$ no longer receives electrons transmitted from contact $i-1$ but from contact $i+1$. This simply amounts to transposing the conduction matrix, in agreement with the Onsager reciprocity relation (4.23).
In order to obtain the different resistances $R_{14,ij} = (V_i - V_j)/I$, let us first fix the maximal possible number of voltages $V_i$ in the different contacts. In order to do so, we start investigating contacts with no current flow, i.e. the contacts different from the source and the drain. For the contacts $i = 2, 3, 5$ and 6, the Landauer-Büttiker thus yields

$$I_i = 0 = \frac{e^2}{h} M(V_i - V_{i-1}), \qquad (5.89)$$

that is the potential in the contact $i$ must be the same as in the preceding contact $i-1$,

$$V_i = V_{i-1}. \qquad (5.90)$$

This is a special form of Millman's theorem (4.27), which we have encountered in chapter 4. Consequently, all contacts (2 and 3) on the upper sample edge are at the same potential $V$ as the source (contact 1), and similarly the potentials in the contacts at the lower edge (5 and 6) coincide with that $V'$ in the drain (contact 4). This result may be understood on physical grounds: since the contacts 2 and 3 are only filled by electrons from the source without any electron loss in the contacts, they must be at thermodynamic equilibrium with the source and thus necessarily at the same chemical potential. For the same reason, the contacts 5 and 6 are at equilibrium with the drain.

The above results for the potentials already allow us to obtain the longitudinal resistances, which are measured by a voltage drop either between the contacts 2 and 3 or 5 and 6,

$$R_L = R_{14,23} = \frac{V_2 - V_3}{I} \quad \text{and} \quad R_L = R_{14,56} = \frac{V_5 - V_6}{I}. \qquad (5.91)$$

There is thus no potential drop, because $V_2 = V_3 = V$ and $V_5 = V_6 = V'$, as we have just mentioned, and the longitudinal resistance therefore vanishes

$$R_L = 0, \qquad (5.92)$$

as one expects for a ballistic four-terminal resistance and in agreement with experimental observation (see figure 5.1).

In order to obtain the Hall resistance, we may choose, as in figure 5.9, the contacts 3 and 5 to measure the voltage drop,<fnref n="26" /> $R_H = (V_3 - V_5)/I =$

<footnote n="26">²⁶As it is clear from the potentials in the different contacts, one obtains the same Hall resistance via a measurement between 2 and 5, 3 and 6, or 2 and 6.</footnote>
$(V - V')/I$. In order to obtain this resistance, we also require a relation between the current and the two different voltages.<fnref n="27" /> This is obtained by using the Landauer-Büttiker formula in one of the current contacts, e.g. the source where one finds

$$I_1 = I = \frac{e^2}{h} M(V_1 - V_6) = \frac{e^2}{h} M(V - V'). \qquad (5.93)$$

One may thus immediately read off the Hall resistance,

$$R_H = R_{14,35} = \frac{V_3 - V_5}{I} = \frac{V - V'}{I} = \frac{h}{e^2 M}, \qquad (5.94)$$

which agrees with the values of the plateaus measured in the Hall resistance (see again figure 5.1).

To complete this discussion of the resistances, let us finally calculate the two-terminal resistance $R_{2t} = R_{14,14}$ measured between the source and the drain. One readily finds

$$R_{2t} = R_{14,14} = \frac{V_1 - V_4}{I} = \frac{V - V'}{I} = \frac{h}{e^2 M}, \qquad (5.95)$$

i.e. one finds the same result as for the Hall resistance. This has actually led to some confusion among physicists, who would normally interpret $R_{2t}$ as a longitudinal resistance. However, there is no contradiction because one should not mix up two- and four-terminal resistances, as we know by now on the basis of the Landauer-Büttiker formalism. Indeed, the result simply reflects the fact that we are confronted with a ballistic conductor composed of $M$ channels, whose two-terminal conductance is precisely given by the inverse of the resistance in equation (5.95).

### 5.4.1 Appearance of plateaus in the Hall resistance and percolation picture

The above Landauer-Büttiker treatment of the edge conductance allows us to understand the signature of the quantum Hall effect: a quantised Hall resistance, $R_H = h/e^2 M$, accompanied by a vanishing longitudinal resistance,

<footnote n="27">²⁷Notice that there is only one voltage to determine because we can always choose a reference potential that sets one of the voltages to zero.</footnote>
$R_L = 0$. However, strictly speaking, this reasoning is valid only at precise values of the filling factor and does not yet indicate the evolution of the resistances if we sweep the filling factor around these precise values, $\nu = M = 2n$. Indeed, precisely at these values, we have $M = n_{el} \times 2\pi l_B^2 = hn_{el}/eB$, which, substituted into equation (5.94), yields

$$R_H = \frac{h}{Me^2} = \frac{h}{\nu e^2} = \frac{B}{en_{el}} \qquad (5.96)$$

that is nothing other than the classical value of the Hall resistance! This is a somewhat frustrating result in view of the length of this chapter until now: all this effort to get back to the starting point? Burn the author of these notes on stake! Fortunately (also for the author), this is not the case, and we have almost arrived at the end of our journey to a complete understanding of the quantum Hall effect.

In order to understand the quantisation of the Hall resistance over a larger range of magnetic field around its value corresponding to an integer filling $\nu = 2n$, we need to take into account the trapping (or localisation) of electrons or holes by impurities in the sample that we have already mentioned above in section 5.3. Let us consider figure 5.10 to illustrate the appearance of plateaus in the Hall resistance. There, the three different columns (a), (b) and (c) correspond to three different filling factors. The first column (a) refers to the situation, which we discussed in the preceding subsection and where the magnetic field corresponds precisely to the filling factor $\nu = 2n = M$. In this case, the Fermi level is situated precisely in the gap between the Landau levels $n-1$ and $n$, as it is shown in the first line, where we represent the density of states corresponding to eventually disorder-broadened Landau levels. The second line is a screenshot of the filling of the Landau level $n-1$, which is no longer completely flat but is given by a potential landscape with valleys and summits. Precisely at $\nu = 2n$, this level is naturally empty. The values of the longitudinal resistance $R_L = R_{xx}$ (in red) and of the Hall resistance $R_H = R_{xy}$ (in blue) are shown in the last line. These are the values $R_L = 0$ and $R_H = h/Me^2$, which we have just calculated within the Landauer-Büttiker formalism.

Let us now slightly vary the magnetic field. In our illustration in the second column of figure 5.10, we lower the magnetic field such as to increase a bit the filling factor $\nu + \delta\nu$. Indeed, the degeneracy of each Landau level is slightly lowered so that all electrons no longer fit into the lowest $n$ Landau levels, and some electrons are therefore promoted to the Landau level $n$,
<figure><img src="image_54.png" /><figcaption>Figure 5.10: Percolation picture of the quantum Hall effect. The columns (a), (b) and (c) correspond to different values of the filling factor, varying from $\nu = n$ (a) to $\nu = n + \delta\nu$ (b) and $\nu \approx n + 1$ (c, for a half-filled level $n$ if we take into account the spin degeneracy of each Landau level). The first line represents the filling in a diagram representing the density of states, where every Landau level is broadened by disorder. The second line shows the filling of the electrostatic potential landscape of the (originally unoccupied) Landau level $n$. The light blue areas represent electronic puddles, i.e. the potential minima (or valleys of the potential landscape) filled with electrons, and the dark blue contours represent the equipotential lines delimiting the puddles. The associated transport measurements for the Hall resistance $R_H = R_{xx}$ (red) and the longitudinal resistance $R_L = R_{xy}$ (blue) are sketched in the last line, in comparison with the classical Hall resistance (green).</figcaption></figure>
which was formerly unoccupied. This is represented in the first line of column (b) of figure 5.10, where the Fermi level is now situated in the level $n$ in its lower tail. If we take into account the potential landscape, the electrons occupy preferentially its minima (the valleys) for energetic reasons, where they form puddles. Since each electronic state occupies a minimal surface of $\Sigma = 2\pi l_B^2$, these puddles grow if we increase further the filling factor, by lowering the magnetic field. These puddles are shown in the second line of column (b).

What can we say about the transport properties? The edge states, which are responsible for the electronic transport precisely at $\nu = 2n$, are unaffected by this successive filling of the valleys in the potential landscape. Furthermore, even if the electrons at the equipotential lines delimiting the puddles (the cuts of the Fermi level and the potential landscape) acquire dynamical properties due to the Hall drift, they only turn in circles around the puddles, as we have seen above. Since these puddles are situated in the bulk and neither connected among each other nor to the sample edges, these electrons cannot contribute to the electronic transport, which remains therefore entirely controlled by the edge states. This means that the transport properties, represented by the longitudinal and Hall resistances, remain unaltered as compared to the original situation of an integer filling factor shown in column (a). The values of both the longitudinal and the Hall resistances thus remain pinned at their original value even if we have lowered the magnetic field, as it is shown in the last line of column (b). The same argument holds true for holes in the Landau level $n-1$ that we would have created had we lowered the filling factor by increasing the magnetic field. This is the origin of the plateau in the Hall resistance: its value remains pinned at its precisely quantised value $h/Me^2$ even if we sweep the magnetic field around its value corresponding to $\nu = 2n$.

If the filling factor is further increased, a value is reached where the Hall resistance jumps from the plateau at $R_H = h/Me^2$ to $R_H = h/M'e^2$, here with $M' = M + 2$. This can also be understood qualitatively from column (c) of figure 5.10. Indeed, the puddles become macroscopic when a substantial fraction of the potential landscape of the Landau level $n$ is filled. They may then connect the opposite edges. Via the equipotential lines delimiting these macroscopic puddles, an electron can now be transmitted from the upper edge, where it has been injected by the left contact, to the lower edge, via these extended states in the bulk of the sample as opposed to those at the sample edges. This is pictorially represented in the first
line of column (c), where the closed equipotential lines and the associated localised electronic states are represented by circles in the density of states (in the tails of the disorder-broadened Landau levels), while the extended states are represented by open lines at the centre of the Landau levels. Since the sense of propagation is opposite at the lower edge as compared to the upper one, the electron is therefore reflected to the left contact so that the transport is no longer ballistic. Indeed, these backscattering processes across the bulk of the sample yield a non-zero reflection coefficient. This leakage of electrons furthermore causes a voltage drop at the upper edge, *i.e.* the chemical potential is no longer constant all along this edge, and one therefore obtains a non-zero longitudinal resistance, as it is shown in the last line of column (c).

The picture of the quantum Hall effect, summarised in figure 5.10, is also called the *percolation picture* because the puddles percolate over macroscopic distances (bridging the sample edges) at the plateau transition. The percolation picture has found quite a spectacular experimental proof in scanning-tunneling-spectroscopy measurements the results of which are reproduced in figure 5.11. Scanning-tunneling spectroscopy is a special mode of the scanning-tunneling microscope (STM), where the mobile atomically sharp tip of the STM sweeps the sample, measuring the tunneling current locally at each position. In the spectroscopic mode, the tunneling current is measured (at each position) as a function of the voltage between the STM tip and the sample. If the quantum state at the associated energy $-eV$ above the Fermi level is unoccupied in the sample, the electron can tunnel easily from the tip into this state, and the tunneling current is proportional to the number of available states, which is nothing other than the (local) density of states. More precisely, one measures the differential tunneling conductance $dI/dV$ in order to increase the contrast.

This type of measurements turns out to be quite involved. In order to obtain a measurable current, a significant tunnel effect is required. However, the effect varies exponentially with the distance between the tip and the electronic system, but as we have seen in the introduction (chapter 1), high-mobility 2D electron systems are usually obtained at the interface in semiconductor heterostructures and thus situated at a substantial distance from the crystal surface. In order to circumvent this problem, experimentalists from Aachen, Germany, have fabricated a particular heterostructure based on (negatively doped) InSb, where the 2D electron gas is situated relatively close to the crystal surface so that the tunnel effect is substantial.
<figure><img src="image_55.png" /><figcaption>Figure 5.11: Experimental verification of the percolation picture within scanning-tunneling spectroscopic measurements on a 2D electron gas in negatively doped InSb. The differential conductance $dI/dV$ is locally measured, where $I$ is the tunneling current and $V$ the voltage between the STM tip and the sample. This allows one to probe the metallic states in the system (here the equipotential lines of the impurity potential) at different energies $eV$ above the Fermi level. The differential conductance $dI/dV(r)$ at the position $r$ is proportional to the *local density of states*, that is the density of states at this position, and sweeping the STM thus allows one to map out the potential landscape of a Landau level (here the lower spin branch of the $n=0$ level). Panels (a)-(g) show local maps (on a square of roughly 150 × 150 nm²) upon increase of the voltage $V$. Panels (c), (d) and (e) show maps close to the percolation threshold, where the equipotential lines (or the edges of the puddles if the Fermi level is at this energy) are extended over larger and larger distances. (h) Density of states integrated spatially (corresponding to the lower spin branch of the $n=0$ level). The letters indicate the voltages at which the maps (a)-(g) have been obtained. (i) Numerically calculated local density of states for a potential landscape sharing the same characteristics as the measured one. (j) Map of the measured density of states in the upper spin branch of $n=0$ close to the percolation threshold (on a larger scale). From Hashimoto *et al.*, Phys. Rev. Lett. **101**, 256802 (2008).</figcaption></figure>
Their measurements, which are reproduced in figure 5.11, allow one to measure the metallic states that delimit the electronic puddles in the bulk of the sample and thus to map out the equipotential lines of the potential landscape. Panels (a)-(g) display the tunneling conductance of the sample in a square of size ~150×150 nm², as a function of the voltage V between the tip and the sample. Panel (h) shows the (spatially) integrated tunneling conductance, which is proportional to the density of states of a (disorder-broadened) Landau level; here, it is in the lower spin branch of the Landau level $n = 0$, where the measurements have been performed. In figures 5.11(a) and (b), one clearly notices the closed equipotential lines around the potential minima. These equipotential lines grow in size when their energy approaches the value at the centre of the Landau level, where the density of states has its maximum, as shown in panel (h). The corresponding maps (c), (d) and (e) clearly show this increase in size, and one notices in figure 5.11(d) the formation of a network of extended states percolating through the bulk of the sample and thus connecting points over macroscopic distances. This is precisely the situation, which is required for a transition between plateaus in the Hall resistance, accompanied by a non-zero longitudinal resistance, as we have argued in the paragraphs above. Upon further increase of the voltage, one now probes localised states again in the form of closed equipotential lines situated in the upper tail of the broadened Landau level.

### 5.4.2 The role of the quantum Hall effect in metrology

In the preceding section, we have learnt about the essential role played by disorder in the quantum Hall effect, which should astonish us a lot. On the basis of our knowledge about quantum phenomena in general and quantum conductance in particular, we would expect ballistic transport in the absence of disorder that inevitably induces scattering. However, in the context of the quantum Hall effect, we have seen that disorder is necessary to trap the electrons (or holes) on the equipotential lines of the disorder potential if the filling factor is swept away from its integer value $v = 2n = hn_{el}/eB$, which allowed for the formation of $M = 2n$ edge channels. This point needs to be stressed: without disorder, one would not be able to pin the resistances and thus to obtain de values $R_H = h/e^2M$, e.g. for the Hall resistance, in terms of the universal constants $e, h$ and the integer $n$. Notice that the precise form of the potential landscape does not enter in this argument as long as the equipotential lines around the valleys and the summits of the potential
are closed. Indeed, one observes that in samples with an increased electron mobility, i.e. in samples with lower disorder, the size of the plateaus shrinks so that the latter are less well defined. In conclusion, contrary to a “normal” situation in quantum physics, the quantum Hall effect requires a disordered system in order to measure with high precision a quantum effect! This is probably a unique situation in physics, and it has an extremely important implication: it allows us to define with high precision the unit of resistance.

Soon after the discovery of the quantum Hall effect, scientists have realised this technological potential of the quantum Hall effect. Consequently, they decided at the 1990 International metrological conference “Conférence Générale des Poids et des Mesures” to use the quantum Hall effect (more precisely its occurrence at $\nu = 2$) as the standard of the resistance unit, dismissing the original Ohm to retirement. Indeed, its value is fixed in 1990 by the Klitzing

$$R_{K-90} = \frac{h}{e^2} = 25\,812,807\,\Omega. \qquad (5.97)$$

Later, in 2018, the Conférence des Poids et des Mesures decided to use only fundamental constants as the basic units, and the Klitzing is now a derived unit, defined by the fundamental units e and h.

### 5.4.3 The role of the spin in the quantum Hall effect and appearance of plateaus at $\nu = 2n + 1$

Until now we have considered that all Landau levels (in GaAs heterostructures) are twofold spin-degenerate as a consequence of the relatively weak Zeeman effect (see discussion in section 5.2.4). While this assumption is often justified, it ceases to be valid when the magnetic field is sufficiently strong to separate in energy the two spin branches of the last occupied Landau level. In this case, one observes quantum Hall plateaus also at odd-integer filling factors $\nu = 2n + 1$, as it is shown in figure 5.12(a). This is most naturally explained by the Zeeman effect and its spin splitting by an energy $\Delta_Z \propto B$ that increases linearly with the magnetic field [see also equation (5.66)], even if this energy scale is much smaller than the Landau-level separation. In order to observe a quantum Hall effect associated with the different spin branches of the Landau levels, both the thermal energy and the energy associated with the disorder potential must be smaller than the Zeeman splitting, $k_B T$, $\delta E_{dis} \ll \Delta_Z$. Notice that the Zeeman effect is on the order of $\Delta_Z \sim 3...4$ K at 10 T (or 0.25...0.3 meV), i.e. on the same order of magnitude as the
<figure><img src="image_56.png" /></figure>

Figure 5.12: (a) Measurement of the longitudinal and Hall resistances showing that the twofold spin degeneracy is lifted at high magnetic fields. Plateaus from the $\nu = 2n$ series are indicated by the blue numbers while the red numbers denote plateaus at $\nu = 2n + 1$, due to a lifted spin degeneracy. The measurements are performed at $T = 1.9$ K. Adapted from Suddards et al., New J. Phys. **14**, 083015 (2012). (b) Sketch of the Landau-level spectrum at an even-integer filling factor corresponding to a complete filling of all spin branches until the level $n-1$. (c) in the case of an odd-integer filling, only the lowest spin branch of the last Landau level (here $n$) is filled (in addition to the lower-lying Landau levels), corresponding to an odd number of edge channels $M = 2n + 1 = \nu$. The Zeeman effect (possibly enhanced by interaction effects) needs to be sufficiently strong to separate sufficiently in energy the spin branches and to place the Fermi level between them.
temperature at which the measurements shown in figure 5.12(a) have been
carried out.

If we take into account the Zeeman effect, the above picture of the quan-
tum Hall effect remains unaltered at even-integer filling factors $v = 2n$, as
it is shown in figure 5.12(b): the Fermi level is placed between two adja-
cent Landau levels (between $n - 1$ and $n$), and the occupied Landau levels
contribute two channels each at the sample edges, associated with the two
spin orientations. The only consequence of the Zeeman effect on the edge
channels is a slight local separation of the channels at the edge. In contrast
to even-integer filling factors, the Fermi level may also be placed in between
the two spin branches of the last Landau level $n$ if the Zeeman effect is suf-
ficiently strong. This situation, which corresponds to an odd-integer filling
factor $v = 2n + 1$, is sketched in figure 5.12(c) where only the lowest spin
branch of the Landau level $n$ is occupied. This Landau level gives there-
fore rise to only one edge channel while all other $n$ Landau levels contribute
two edge channels, for the two spin orientations. This is the origin of the
quantum Hall plateaus at odd-integer filling factors $v = 2n + 1$, which are
nevertheless not as well developped as the plateaus at even-integer fillings as
a consequence of the smaller Zeeman energy as compared to the gap between
adjacent Landau levels, as one may notice in figure 5.12(a).

5.4.4 Relativistic quantum Hall effect in graphene

The structure of the Landau levels that we have obtained from the quanti-
sation of Dirac fermions [see equation (5.48)] allows us to understand finally
the particularities of the quantum Hall effect in graphene, first measured in
2005 by the Manchester group around Geim and Novoselov [Novoselov et al.,
Nature **438**, 197 (2005)], who received the 2010 Nobel Prize in Physics for
their various milestone-experiments in graphene research, and by the group
at Columbia University around Kim and Störmer<fnref n="28" /> [Zhang et al., Nature
**438**, 201 (2005)]. The measurements of the latter group are reproduced in
figure 5.13. In panel (a), one notices the usual signature of the quantum Hall
effect – the appearance of plateaus in the Hall resistance accompanied by a
vanishing longitudinal resistance – when the magnetic field is varied while the
electronic density is kept constant. In comparison with 2D electron systems

<footnote n="28"><sup>28</sup>Störmer had already received the 1998 Nobel Prize in Physics for the discovery of the fractional quantum Hall effect, together with his colleague Tsui and the theoretician Laughlin.</footnote>
<figure><img src="image_57.png" /><figcaption>Figure 5.13: Measurement of the quantum Hall effect in graphene (in the Hall-bar geometry). (a) Measurement of the longitudinal and Hall resistances at an electronic density that is fixed by the back gate, as a function of the magnetic field. (b) Measurement of the same resistances at a fixed magnetic field (at 9 T), now as a function of the back-gate voltage that allows one to vary the electronic density in the graphene sample. (c) Sketch of the Landau levels and their associated density of states. From Zhang *et al.*, Nature **438**, 201 (2005).</figcaption></figure>
in GaAs heterostructures, it is much easier to sweep also the electronic density by the capacitive effect, with the help of the so-called gate voltage that is applied between the graphene sample and a metallic back gate separated from the graphene sheet by an insulating layer. The measurement of the Hall and longitudinal resistances at a fixed magnetic field (of 9 T), as a function of the gate voltage is shown in panel (b). Also here, one observes plateaus in the Hall resistance accompanied by zeros in the longitudinal resistance. Most saliently the plateaus are found at filling factors

$$ \nu = \pm(4n + 2), \qquad (5.98) $$

in contrast to the quantum Hall sequence $\nu = 2n$ (in the absence of a Zeeman effect) in GaAs heterostructures.

How can we understand this difference? One should first notice that the filling factors at which the quantum Hall effect occurs change in steps by four. This can be easily understood by the degeneracy of the graphene Landau levels, which show a twofold valley degeneracy in addition to the usual spin degeneracy, as we have discussed in section 5.2.4. In the absence of the Zeeman effect, which is again vanishingly small in the shown sample as compared to disorder broadening of the Landau levels,<fnref n="29" /> the graphene Landau levels are therefore fourfold degenerate. However, there is an additional offset of two in the filling-factor sequence. In order to understand this offset, let us remember that the Fermi level must be situated in the gap between two adjacent Landau levels in order to observe a quantum Hall effect. This is not the case at charge neutrality, where $n_{el} = 0$ is identical to $\nu = n_{el}/n_B = 0$. In the absence of a magnetic field, charge neutrality means that the Fermi level is situated exactly at the band contact points $K$ and $K'$, i.e. between the upper cone of the dispersion relation corresponding to the conduction band and the lower cone (valence band). In the presence of a quantising magnetic field, we can therefore already say that all Landau levels at positive energy ($\lambda = +$) are empty, while all levels at negative energy ($\lambda = -$) are fully occupied. However, as we have seen in section 5.2.3, the special $n = 0$ Landau level is formed precisely at zero energy even if there are no quantum states at zero magnetic field due to the vanishing density of states. This level must therefore be half-filled with electrons (or, equivalently, half-filled with holes) at charge neutrality, and the Fermi level is thus situated at the

<footnote n="29">²⁹Notice that additional quantum Hall effects at other fillings have later been measured in graphene samples of higher mobility, indicating a lifted spin (and also valley) degeneracy.</footnote>
centre of this Landau level $n = 0$. If we recall the above discussion of the
general features of the quantum Hall effect, this situation is at odds with
the appearance of a quantum Hall effect, which requires indeed a Fermi level
between adjacent Landau levels. Since the $n = 0$ Landau level (for massless
Dirac fermions, with $\Delta = 0$) is itself fourfold degenerate, the filling factor
must be $\nu = 2$ for the first quantum Hall effect to occur because the $n = 0$
Landau level is then completely filled and the Fermi level lies in between
the levels $n = 0$ and $n = 1$. Similarly, we may argue in terms of holes, in
which case the first quantum Hall effect occurs at $\nu = -2$, for the same
reasons. In conclusion, the $n = 0$ Landau level, which remains at zero energy
regardless of the strength of the magnetic field and that is half-filled at charge
neutrality ($n_{el} = 0$), is responsible for the filling-factor offset of $\Delta\nu = \pm 2$ in
the quantum Hall sequence (5.98) in graphene, while the steps in four reflect
the fourfold spin-valley degeneracy of the Landau levels.

Notions and concepts to retain

• Peierls substitution

• Landau quantisation, cyclotron variable and guiding centre

• Landau gauge

• Landau levels of electrons in parabolic bands and in (gapped) graphene

* degeneracy of the Landau levels, orbital *vs*. internal (spin, valley) degeneracy

• chiral edge channels, localised bulk states

• chiral transport

* longitudinal and Hall resistances as four-terminal resistances in the Landauer-Büttiker formalism

• percolation picture of the quantum Hall effect

• quantum Hall effect and resistance unit, metrology
# Chapter 6

## Introduction to topological matter

In chapter 2, we recalled the elements of the band structure formed by electrons in a periodic lattice given by the underlying atomic crystal. Bloch's theorem allows us to obtain their spectrum which consists of energy bands. These latter can be seen as energy levels of the Bloch Hamiltonian that vary continuously as a function of the wave vector, which is a good quantum number because of the discrete translation symmetry on the lattice. The calculation of energy bands has been, and still is, a major occupation of (numerical) studies in material sciences, be it in the framework of tight-binding models or density-functional calculations. This analysis of the band structure provides us indeed with a lot of information about the materials under scrutiny, and above all whether we are confronted with metals or insulators (or semiconductors if the gap is relatively small). Furthermore, the density of states, which is obtained directly from the band structure (and the dimensionality of the system, see section 3.1 in chapter 3), bears important information about the thermodynamical properties or yet possible instabilities of the electronic system, e.g. due to electronic interactions or coupling to other degrees of freedom. As an example, one might cite Stoner's criterion for the occurrence of ferromagnetism due to electron-electron interactions – the criterion appeals to the product of the coupling constant and the density of states, which must be larger than a critical value – or else electron-phonon coupling that is at the origin of superconductivity in a large number of materials.

Beyond these valuable pieces of information obtained from the spectrum,
wave functions also inform us about a number of properties in electronic sys-
tems. These properties are more subtle to grasp, from a theoretical as well
as from an experimental point of view, and have for quite a long time not
been treated in the reference text books of condensed-matter physics. It is
only since about 2005 that this community of physicists has become more
and more interested in the properties encoded in the Bloch states. The lat-
ter unveil indeed relevant geometric and topological properties of electrons
restricted to a single electronic band. Even if electrons are dynamically
restricted to a single electronic band, e.g. if one considers only adiabatic
processes excluding quantum jumps to other bands, those other bands are
nevertheless present and “communicate” with the electrons in the band of
interest. We show below that this communication involves *virtual transitions*
(as opposed to quantum transitions) of the electron to these other bands.
This type of banc coupling is taken into account by the so-called *Berry cur-
vature*, which is a geometric phase in the quantum mechanical wave function
that the electrons acquire in adiabatic processes within a single band. The
Berry curvature provides indeed information about the geometry of the elec-
tronic bands and their states and yields, once integrated over the entire first
Brillouin zone, a topological invariant: the *Chern number*.

After a historical introduction to topological concepts in condensed mat-
ter, especially in the context of the classification of defects (section 6.1), we
introduce the Berry connection, Berry curvature and Berry phase – often also
called Berry-ology in a somewhat nonchalant manner – as well as the Chern
number, in the description of electronic bands (section 6.3). In section 6.4,
we apply these concepts to a massive Dirac fermion. We have already seen
that Dirac fermions are responsible for the low-energy properties of certain
materials, such as graphene, and they emerge indeed naturally in the wider
class of topological materials. They allow us to describe the electronic bands
in the vicinity of topological phase transitions, from a local point of view
in reciprocal space, without resorting to a complete calculation of the Berry
curvature over the entire first Brillouin zone. Dirac fermions also allow us
to obtain a simple (qualitative) description of topological insulators and the
*bulk-edge correspondence* in 2D materials (or the *bulk-surface correspondence*
in 3D materials),<fnref n="1" /> which is the hallmark of topological materials. We fin-
ish the present chapter with a discussion of a correction to the electronic

<footnote n="1">¹Here, we use the term *bulk-edge correspondence* generically regardless of the dimensionality of the system.</footnote>
<figure><img src="image_58.png" /><figcaption>Figure 6.1: Continuous deformation of a coffee mug into a torus... or the sculpture in the “Grand Hall”.</figcaption></figure>

group velocity in Bloch bands – one can indeed still find in a large number of textbooks that the group velocity is given by

$$v_n = \frac{1}{\hbar} \nabla_k E_n(k).$$

However, Karplus and Luttinger understood, already in the 1950’s long before Berry’s seminal work on geometric phases, that this formula is incomplete and that an additional term must be added. In modern terms, this additional term is proportional to the Berry curvature, which is coupled to the time derivative of the wave vector. The understanding of the Karplus-Luttinger anomalous velocity is at the heart of section 6.5, where we approach this correction in the context of adiabatic pumping in a 1D crystal.

## 6.1 Topology in condensed-matter physics – defect classification

As much as the topological description of electronic bands is a rather modern field of condensed-matter research, topological concepts have been used since the 1970’s, especially in the classification of defects. From a general point of view, mathematical topology groups geometric objects into classes – called homotopy classes – where two objects, which we will consider for illustration made of a malleable material, can be deformed into each other by continuous deformations if they belong to the same class. For example, a cup (with its handle) can be continuously deformed into a donut (or torus) or into the black marble sculpture in the “Grand Hall” of our beloved Alma Mater (see figure 6.1). In contrast, this cup, or any of the objects in the class in question, cannot be deformed into a balloon (or a sphere) – to do so, one
would need to cut the handle of the cup in order to get rid of the hole, which is the common property between the objects of the homotopy class to which the cup and the torus belong. In the same way, a pretzel (with two holes) cannot be obtained by a continuous deformation from an object like the torus. Without formalizing this point for the moment, we notice that the different homotopy classes of objects are characterised by the number of holes that their elements have. This number plays the role of a *topological invariant* and remains a (topologically) protected quantity under the effect of continuous deformations. In other words, this invarant is necessarily an integer in the same way that there is no half-hole.

### 6.1.1 Characterisation of topological defects in order parameters

Naturally, in Physics, we are less interested in objects such as those described above than in quantities that describe for example a phase of matter. A theory that describes with great success the phases of matter and transitions between them is precisely given by Landau's theory. According to this theory, a phase transition is accompanied by the disappearance of an *order parameter* when we vary a control parameter (such as temperature) that drives the transition.<fnref n="2" /> In the context of magnetism, for example, the order parameter is the magnetisstion $\vec{M}(\mathbf{r})$ which takes a constant value everywhere in space in the ferromagnetic phase at low temperature (see the left-hand side of figure 6.2). This ferromagnetic order cedes its place at high temperature (above the critical temperature $T_c$) to a disordered paramagnetic phase, where the order parameter vanishes, $\vec{M}(\mathbf{r}) = 0$.

Let us examine now the symmetries of the encountered phases. Indeed, the phase transition is accompanied by a broken symmetry at low temperature: the disordered phase is symmetric in that the spins that form the paramagnetic phase point in all directions in space, whereas the ferromagnetic phase is ordered. This latter phase consists of spins that all point in the same direction in order to give a non-zero magnetisation, $\vec{M}(\mathbf{r}) = \vec{M}_0$. However, since no direction is a *priori* preselected due to the isotropy of space, this phase breaks the SO(2) symmetry of the original rotations. The order

<footnote n="2">²Here, we do not provide a complete review of Landau’s theory, because it is not the object of the course, but we limit ourselves only to a few aspects relevant for the discussion of topological properties.</footnote>
<figure><img src="image_59.png" /><figcaption>Figure 6.2: Phase transition between a ferromagnetic and a paramagnetic phase (with disordered spins), as a function of temperature. The magnetisation plays the role of an order parameter – it is nonzero in the ferromagnetic phase at low temperature and disappears at the *critical temperature* $T_c$ to give way to a high-temperature paramagnetic phase with $\vec{M} = 0$.</figcaption></figure>

parameter, i.e. the macroscopic magnetisation of the set of spins, disappears at the critical temperature which describes the transition (figure 6.2). Let us now try to characterise the low-energy excitations of the ordered system and start with the collective excitations. These are excursions of the order parameter [here the magnetisation $\vec{M}(\mathbf{r})$] around its value $\vec{M}_0$ when the $\mathbf{r}$ position changes in space. In the present case of a ferromagnet, these collective excitations are the long-wavelength *spin waves* (see figure 6.3). In fact, the *Goldstone's theorem* tells us that the energy of these collective excitations tends to zero in the limit of very long wavelengths. Without proving this theorem, the vanishing energy at long wavelengths can be understood relatively easily. In this limit, the spin wave becomes a global rotation of the magnetisation of $\vec{M}_0 \rightarrow \vec{M}'_0$. However, such a state has the same energy as the starting state because of the $SO(2)$ rotation symmetry of the model.

In addition to these collective spin-wave excitations, the order parameter can be disturbed in a more local and violent manner. Let us imagine a local disappearance (at a position $\mathbf{r}_0$) of the order parameter with a kind of monopole in its neighbourhood, where all the spins point for example in the direction of the vector associated with their position with respect to $\mathbf{r}_0$. This yields a hedgehog pattern [see figure 6.4(a)] if we consider all spins on a sphere with $\mathbf{r}_0$ at its center. This “magnetic monopole” is precisely a topological defect, which can be characterised by the *lasso rule* that we owe to the French physicist Kléman. This rule stipulates that, to catch a $d'$-dimensional defect in a $d$-dimensional space, one needs a lasso of dimension
<figure><img src="image_60.png" /><figcaption>Figure 6.3: Collective excitation (spin wave) of a ferromagnet. The magnetisation $\vec{M}(\mathbf{r})$ changes spatially and explores a certain part of the Bloch sphere around its equilibrium value $\vec{M}_0$.</figcaption></figure>

or

$$d = d' + r + 1. \qquad (6.1)$$

In order to catch a magnetic monopole (of dimension $d' = 0$) in a 3D ferromagnet, one then needs a lasso of dimension $r = 2$, i.e. a sphere that can encompass the defect. We may already anticipate that it is sufficient to analyze the order parameter on the surface of the sphere (our lasso) to know whether the lasso hunter has caught a topological defect or not. If for example all the spins on this sphere point in a perpendicular direction away from its centre, the hunter has indeed caught a magnetic monopole [see figure 6.4(a)].

Mathematically, this is formalized by maps (functions) from one compact space to another one. Let us consider again our ferromagnetic system, where we are interested in the evolution of the magnetisation when the whole sphere that envelopes the defect is explored. To each point on this sphere $S^2$ (the positions $\mathbf{r}$), we associate another vector – the magnetisation $\vec{M}(\mathbf{r})$ – which itself lives on a sphere $S^2$. Indeed, a point on this last sphere represents a possible orientation of the magnetisation, and the latter can thus be seen as a map from one sphere $S^2$ (that of the positions in real space) to another
<figure><img src="image_61.png" /><figcaption>Figure 6.4: Topological defects. (a) Magnetic monopole (topological defect of a 3D ferromagnet). All spins point in a direction away from the centre of the sphere, which encompasses the defect. (b) Vortex of negative charge in the *XY* model. (c) Vortex of positive charge in the *XY* model. (d) Image of the blue path of figure (b) in the space $S_M^1$ of the order parameter (*XY* phase or angle between the spin and the *x*-axis). (e) Image of the green path of figure (b) in the space $S_M^1$. (f) Image of the red path of figure (c) in the space $S_M^1$.</figcaption></figure>
sphere $S_M^2$ (that of spin orientations),<fnref n="3" />

$$ \vec{M}: S^2 \to S_M^2. \qquad (6.2) $$

It turns out that these maps are characterised by an integer number<fnref n="4" /> that describes the number of times that the magnetisation wraps up its sphere $S_M^2$ when all the sphere $S^2$ of the positions (i.e. the lasso) is explored. This integer is actually a topological invariant that cannot be changed via continuous deformations.

To illustrate this point, let us consider an even simpler case: a vortex. A vortex is precisely a defect of an order parameter such as that of a 2D superconductor. The superconductor is characterised by a complex $N$-particle wave function with a well-defined phase $\theta$, which takes values between 0 and $2\pi$. By identifying 0 and $2\pi$, due to the periodicity of the phase, the latter can be represented by a position on the unit circle, i.e. by an angle. According to the rule (6.1) of the lasso hunter, we need, in order to catch this $d' = 0$ dimensional defect in a $d = 2$ dimensional space, a $r = 1$ dimensional lasso, that is to say a real loop, and the term “lasso” now takes on its full meaning. The order parameter of the superconductor at a point $\mathbf{r}$ in the 2D plane is therefore the phase which can be represented by an arrow indicating the phase on the unit circle. The superconductor is thus analogous to a 2D ferromagnet with a magnetisation that is also constrained in the equatorial plane of the Bloch sphere.<fnref n="5" /> From a mathematical point of view, the topological defects can again be classified by maps, but this time we deal with maps from the $S^1$ space (the lasso surrounding the defect) to the $S_M^1$ space that is the unit circle, which represents the superconducting phase,

$$ \theta: S^1 \to S_M^1. \qquad (6.3) $$

These maps are also classified by an integer, $\pi_1(S^1) = \mathbb{Z}$. To unveil the meaning of this integer, we may consider closed paths (the lassos) in the 2D space and take a look at the phase (the $XY$ magnetisation) at each of the positions, in two patterns sketched in the figures 6.4(b) and (c). It is clear from the naked eye that the two patterns represent two different vortices,

<footnote n="3">³We have introduced the index $M$ to show that this second sphere is associated with the orientations of the magnetisation even if it has the same structure as the sphere $S^2$ that encloses the defect.</footnote>

<footnote n="4">⁴For mathematicians, this is the fundamental group $\pi_2(S^2) = \mathbb{Z}$.</footnote>

<footnote n="5">⁵This model is also called the 2D $XY$ model.</footnote>
which we try to understand by closed paths in the lasso-hunter image. Let us first consider the blue path in figure 6.4(b) as well as its image [figure 6.4(d)] on the circle $S_M^1$. Figure 6.4(b) indicates that the image of this path does not explore the full circle. It can thus be contracted to a single point, which corresponds to a zero winding number (or topological invariant). This reflects the simple fact that the blue path in figure 6.4(b) does not surround the (topological) defect. The situation changes when we take a look at the green path in the same figure that now surrounds the vortex. Indeed its image on the circle $S_M^1$ [figure 6.4(e)] now explores the whole circle in a counter-clockwise direction (and the reverse direction of the oriented path in position space). We thus assign a *vorticity* of $-1$ (this is the invariant or *topological charge*) to this defect, which is an *anti-vortex*. Figure 6.4(c) shows a vortex with a vorticity $+1$, and its identification is done in the same way as for the anti-vortex. To do so, we analyse the image of the path on the circle $S_M^1$ [figure 6.4(f)]. This image covers the circle in a clockwise direction and thus in the same direction as the path $S^1$ in position space. Notice finally that there are vortices with vorticity larger than $1$ – in this case, the image of the path around the topological defect covers the circle $S_M^1$ several times. This number of windings is precisely the meaning of the topological invariant $\pi_1(S^1) = \mathbb{Z}$, which represents here the vorticity.

## 6.2 Back to the quantum Hall effect

In the previous chapter, we saw that the quantum Hall effect shows a very high robustness to the disorder potential: the charged impurities in the bulk of the sample trap the electrons by forcing them to follow the equipotential lines of the disorder-potential landscape, and the electric transport involves only the extended states which are located at the edges of the sample. These states being chiral, the transport is directional in that the direction is imposed by the orientation of the magnetic field. It should be noted that these edge states and thus the electric transport in the quantum Hall effect, i.e. for filling factors close to $\nu = n$, are not affected by the concrete form of the disorder potential in the bulk – indeed if one potential landscape is replaced by another one, this only changes the positions where the bulk electrons are trapped, but it does not affect the transport at the edges.

We have already understood that the number of completely filled Landau levels conditions the number of edge channels. In 1982, Thouless, Kohmoto,
Nightingale and Nijs (TKNN) showed that the number of Landau levels can be seen as a topological invariant, in a similar way to the topological invariants introduced earlier in this chapter, in the characterisation of topological defects. Here, we do not reproduce the calculations giving rise to this topological number, but we discuss in the following sections a similar invariant in the framework of the Berry curvature. Most saliently, the quantum Hall effect indicates us a very general phenomenon of topological matter: the topological bulk invariant (here the number of completely filled Landau levels) conditions the edge properties (here the number of chiral channels). This is a manifestation of the bulk-edge correspondence that we will examine in more detail and in a constructive way in section 7.4 of the next chapter. The quantum Hall effect can thus be seen as the prototype of a topological insulator: it is a bulk insulator (in 2D) because of the position of the Fermi level between two adjacent Landau levels, accompanied by a (1D) edge conductor. Let us recall that the chiral 1D character of the edge states is at the origin of the precise conductance quantisation and consequently that of the Hall resistance. The great robustness of these edge states is the manifestation of what is called *topological protection* due to the topological invariant which, being an integer, cannot be changed continuously.

## 6.3 Berry-ology – Berry phase, connection and curvature

In this section, we discuss the electronic bulk properties of a topological material, in terms of the geometric properties of its band structure. The goal is to understand how the Chern number emerges as the natural topological invariant which describes the different energy bands obtained in band-structure calculations, namely in tight-binding calculations that have been recalled in the chapter 3. The content of this section is based on the work by TKNN, which we have already mentioned above, as well as that by Berry in 1984. In particular, we follow Berry's work, which is conceptually easier to grasp and that allows for a comparison with known quantities of electromagnetism. Indeed, we will find here quantities analogous to the vector potential, magnetic field and magnetic flux, but in reciprocal space. The discussion is again
based on the Bloch states<fnref n="6" /> $|u_n(\mathbf{k})\rangle$, which are solutions of the Schrödinger equation (2.31),

$$H(\mathbf{k})|u_n(\mathbf{k})\rangle = E_n(\mathbf{k})|u_n(\mathbf{k})\rangle, \qquad (6.4)$$

in terms of the Bloch Hamiltonian $H_k$. We shall see in this section that, on a closed path in the reciprocal space of wave vectors $\mathbf{k}$, the states $|u_n(\mathbf{k})\rangle$ evolve in a remarkable manner when they are forced to remain in the same band: in addition to the usual dynamical phase, the states acquire a geometric phase that encodes the topological properties of the band. To introduce the idea schematically, we may consider the $|u_n(\mathbf{k})\rangle$ states as maps of the first Brillouin zone $\mathcal{T}^d$, which can be seen as a $d$-dimensional torus and thus as a compact space (because of the periodicity of the reciprocal lattice), to the space of the $N_{\text{motif}}$ spinor states (2.32) associated with the Hilbert subspace $\mathcal{E}_n$ of the $n$-th band,

$$|u_n(\mathbf{k})\rangle : \mathcal{T}^d \to \mathcal{E}_n. \qquad (6.5)$$

Note that the space $\mathcal{E}_n$ is also compact because of the periodicity of the wave functions.<fnref n="7" /> We are then interested in the images (in the space $\mathcal{E}_n$) of the different closed paths in $\mathcal{T}^d$. More precisely, we consider a continuous evolution of the states $|u_n(\mathbf{k})\rangle$ along these paths. Remember from the discussion in section 2.2.3 that this would not be a meaningful task in the space of Bloch wave functions that are all orthogonal at different wave vectors in reciprocal space, $\langle\Psi_{n,\mathbf{k}}|\psi_{n',\mathbf{k}'}\rangle = \delta_{n,n'}\delta_{bk,\mathbf{k}'}$, because of the plane-wave factors in their wave functions. This is no longer the case of the Bloch states $|u_n(\mathbf{k})\rangle$, which may thus be expanded as

$$|u_n(\mathbf{k} + d\mathbf{k})\rangle = |u_n(\mathbf{k})\rangle + \nabla_{\mathbf{k}}|u_n(\mathbf{k})\rangle \cdot d\mathbf{k} = |u_n(\mathbf{k})\rangle + |\nabla_{\mathbf{k}}u_n\rangle \cdot d\mathbf{k}, \qquad (6.6)$$

The last equation illustrates only a notation to avoid possible confusions in the sequel. Indeed, the state $|\nabla_{\mathbf{k}}u_n\rangle$ is the Hilbert-space state obtained by

<footnote n="6">⁶Remember that we use the term Bloch states for the eigenstates in the Bloch basis, which represent the periodical part of the Bloch wave functions, see discussion in sections 2.2.2 and 2.2.3.</footnote>

<footnote n="7">⁷Formally, it is the projective space $\mathbb{CP}^{N_{\text{motif}}-1}$ of complex numbers – the $N_{\text{motif}}$ components of the states $|u_n(\mathbf{k})\rangle$ are indeed complex numbers. On the other hand, these $N_{\text{motif}}$ complex components are not all independent: they are constrained by the normalisation as well as the irrelevance of the global phase of the wave functions. Mathematically, this is the essence of the projective space $\mathbb{CP}^{N_{\text{motif}}-1}$, where we identify all the complex vectors $\vec{v}$ (with $N_{\text{motif}}$ components) which differ only by a complex multiplicative number $z$, i.e. all $\vec{v}'$ if $\vec{v}' = z\vec{v}$.</footnote>
taking the gradient of the state $|u_n(\mathbf{k})\rangle$. As the gradient is a linear operator, the state thus obtained is also part of the Hilbert space $\mathcal{E}_n$. Notice, however, that it is not a Hermitian operator that would satisfy $A = A^\dagger$: contrary to the Hermitian operators, which can be applied to the right ket as well as to the left bra, this is not the case for the gradient, for which such a change comes along with a sign change,

$$ \langle u_n(\mathbf{k}) | \nabla_{\mathbf{k}} u_n \rangle = - \langle \nabla_{\mathbf{k}} u_n | u_n(\mathbf{k}) \rangle. \qquad (6.7) $$

This convention of including the gradient in the ket thus allows us to avoid a confusion, which does not occur for Hermitian operators. Recall that the gradient is an anti-Hermitian operator – equation (6.7 is precisely a consequence of this anti-Hermiticity –, and in order to make it Hermitian, we have to multiply it by a factor $i$.

In the following, we will focus on this continuous evolution of states on a path in wave-vector space.

### 6.3.1 Adiabatic transport in reciprocal space

In order to spell out the program invoked above, we may consider a path in reciprocal space along which an electron evolves adiabatically. This means that the path $\mathbf{k}(t)$ can be parametrised by a time $t$ and that the variation of the wave vector is sufficiently slow. Quantitatively, we consider time scales larger than that given by the (inverse) gaps between the band $n$ which hosts the electron and its adjacent bands, $T \gg \hbar/\text{min}|E_n(\mathbf{k}) - E_m(\mathbf{k})|$. We can then use adiabatic theory, i.e. we exclude quantum jumps from one band to another, and the electron remains throughout this process in its initial band $n$. If the quantum state at time $t = 0$ is $|\psi(t = 0)\rangle = |u_n(\mathbf{k}_0)\rangle$, it is then given by

$$ |\psi(t)\rangle = e^{i\gamma(t)} |u_n(\mathbf{k}(t))\rangle, \qquad (6.8) $$

at a later time $t$. This means that it is necessarily an eigenstate of the Bloch Hamiltonian parameterised by the wave vector $\mathbf{k}(t)$,

$$ H(\mathbf{k}(t)) |u_n(\mathbf{k}(t))\rangle = E_n(\mathbf{k}(t)) |u_n(\mathbf{k}(t))\rangle, \qquad (6.9) $$

while its phase $\gamma(t)$ remains to be evaluated, using the Schrödinger equation

$$ i\hbar \frac{d}{dt} |\psi(t)\rangle = H(\mathbf{k}(t)) |\psi(t)\rangle. \qquad (6.10) $$
The left-hand side gives

$$i\hbar \frac{d}{dt} |\psi(t)\rangle = i\hbar \left[ i\dot{\gamma} |u_n((\mathbf{k}(t)))\rangle + \dot{\mathbf{k}} \cdot |\nabla_{\mathbf{k}} u_n\rangle \right] e^{i\gamma(t)}, \quad (6.11)$$

and the right-hand side may be written

$$H(\mathbf{k}(t))|\psi(t)\rangle = e^{i\gamma(t)} E_n(\mathbf{k}(t))|u_n(\mathbf{k}(t))\rangle, \quad (6.12)$$

which, by multiplying the last two equations from the left with the bra $\langle u_n(\mathbf{k}(t))|$ gives

$$\dot{\gamma} = i\dot{\mathbf{k}} \cdot \langle u_n(\mathbf{k}(t)) | \nabla_{\mathbf{k}} u_n(\mathbf{k}(t)) \rangle - \frac{1}{\hbar} E_n(\mathbf{k}(t)). \quad (6.13)$$

The integration of this last equation from $t=0$ to $t=T$ then gives

$$\begin{aligned} \gamma(T) - \gamma(t=0) &= \int_0^T dt \frac{d\mathbf{k}}{dt} i \langle u_n(\mathbf{k}(t)) | \nabla_{\mathbf{k}} u_n(\mathbf{k}(t)) \rangle - \frac{1}{\hbar} \int_0^T dt E_n(\mathbf{k}(t)) \\ &= \int_C d\mathbf{k} \cdot \vec{\mathcal{A}}_n(\mathbf{k}) - \frac{1}{\hbar} \int_0^T dt E_n(\mathbf{k}(t)). \end{aligned} \quad (6.14)$$

The last term is the simple dynamic phase that the quantum system acquires during its time evolution. The first term is of a different nature and much less common. It is a geometric phase, given by the line integral of the *Berry connection*, defined as

$$\vec{\mathcal{A}}_n(\mathbf{k}) = i \langle u_n(\mathbf{k}) | \nabla_{\mathbf{k}} u_n(\mathbf{k}) \rangle, \quad (6.15)$$

over the path $C$ in reciprocal space. Let us finally note that we are interested in closed paths rather than in open ones in view of a topological description of the band $E_n(\mathbf{k})$, so that $\mathbf{k}(T) = \mathbf{k}(t=0) = \mathbf{k}_0$. The geometric phase, which is called *Berry phase*, on the closed path $C$ is then written<fnref n="8" />

$$\Gamma(C) = \oint_C d\mathbf{k} \cdot \vec{\mathcal{A}}_n(\mathbf{k}). \quad (6.16)$$

This expression strongly resembles the Aharonov-Bohm phase, which we have briefly mentioned in the introduction,

$$\Gamma_{AB} = \frac{e}{\hbar} \oint d\mathbf{r} \cdot \mathbf{A}(\mathbf{r}), \quad (6.17)$$

<footnote n="8">⁸The dynamic phase is of no interest here.</footnote>
accumulated by an electron on a closed path in direct space (of positions). The Berry connection thus plays the role of a vector potential vector in reciprocal space (of wave vectors) and the Berry phase that of an Aharonov-Bohm phase. It seems then natural to ask oneself whether there is also the analog of a magnetic field. This is indeed the case: it is the Berry curvature which is obtained by taking the rotational (with respect to the wave vector) of the Berry connection

$$ \vec{\mathcal{B}}_n(\mathbf{k}) = \nabla_{\mathbf{k}} \times \vec{\mathcal{A}}_n(\mathbf{k}) = i \langle \nabla_{\mathbf{k}} u_n | \times | \nabla_{\mathbf{k}} u_n \rangle. \qquad (6.18) $$

This somewhat barbaric notation indicates that we must take the vector product with respect to the gradients of the wave functions. It can also be written component by component (with $\mu, \nu, \sigma = x, y, z$)

$$ \vec{\mathcal{B}}_n(\mathbf{k})_{\sigma} = i \epsilon_{\mu\nu\sigma} \langle \partial_{k_{\mu}} u_n | \partial_{k_{\nu}} u_n \rangle, \qquad (6.19) $$

where $\epsilon_{\mu\nu\sigma}$ is the antisymmetric Levi-Civita tensor<fnref n="9" /> and where we adopt the Einstein convention that the sum over all repeated (Greek) indices is implicit. As, in 2D materials, the Berry curvature is a vector oriented in the z-direction, this expression can be written as

$$ \mathcal{B}_n^z(\mathbf{k}) = i \left( \langle \partial_{k_x} u_n | \partial_{k_y} u_n \rangle - \langle \partial_{k_y} u_n | \partial_{k_x} u_n \rangle \right). \qquad (6.20) $$

### 6.3.2 Berry curvature

We have seen, in the previous paragraphs, that the Berry curvature emerges from the restriction of the electron dynamics to a single band, in the context of adiabatic transport. This may seem a bit abstract, and we try to give here a more physical illustration of this quantity, which can be interpreted in the context of inter-band coupling. Indeed, even if the electron always stays in the same band, the other neighbouring bands influence its motion. It is precisely this influence that is taken into account by the Berry curvature. In perturbation theory, this would be a second-order process where the electron can explore the other bands via virtual processes, that means virtual

<footnote n="9">⁹Recall that the antisymmetric tensor is $\epsilon_{xyz} = 1$ as well as for any cyclic permutation of indices and $-1$ for any anti-cyclic permutation, while it is 0 if the same index occurs twice.</footnote>
transitions to and from the other bands without involving a true quantum jump.

We know from electromagnetism that the magnetic field is gauge-invariant while the vector potential is not. This is also the case for the Berry curvature which is gauge-invariant while the Berry connection depends on the chosen gauge. This last dependence is manifested during a phase change in the wave function, $|u_n(\mathbf{k})\rangle = \exp(i\phi_k)|u'_n(\mathbf{k})\rangle$ – this is a gauge transformation if the phase $\phi_k$ does not introduce an additional singularity in the wave function. The Berry connection is then written

$$ \vec{\mathcal{A}}_n(\mathbf{k}) = -\nabla_{\mathbf{k}}\phi_{\mathbf{k}} + \vec{\mathcal{A}}'_n(\mathbf{k}), \qquad (6.21) $$

where $\mathcal{A}'_n(\mathbf{k}) = i\langle u'_n(\mathbf{k})|\nabla_{\mathbf{k}}u'_n(\mathbf{k})\rangle$ is the connection associated with the transformed state. On the other hand, we have

$$ \vec{\mathcal{B}}_n(\mathbf{k}) = \vec{\mathcal{B}}'_n(\mathbf{k}), \qquad (6.22) $$

because $\nabla_{\mathbf{k}} \times \nabla_{\mathbf{k}}\phi_{\mathbf{k}} = 0$, due to the absence of singularities in the phase $\phi_{\mathbf{k}}$, i.e. the phase is a differentiable function.

In order to make manifest the gauge invariance of the Berry curvature, it is useful to get rid of a formulation of it in terms of wave-function gradients. Such a gauge-invariant formulation of the Berry curvature can be found by inserting the closure relation

$$ \sum_m |u_m(\mathbf{k})\rangle\langle u_m(\mathbf{k})| = 1 $$

in the equation (6.19), where we sum over all bands,

$$ \begin{aligned} \mathcal{B}_n(\mathbf{k})_\sigma &= i\epsilon_{\mu\nu\sigma} \sum_m \langle \partial_{k_\mu} u_n | u_m(\mathbf{k}) \rangle \langle u_m(\mathbf{k}) | \partial_{k_\nu} u_n \rangle \\ &= i\epsilon_{\mu\nu\sigma} \left[ \sum_{m \neq n} \langle \partial_{k_\mu} u_n | u_m(\mathbf{k}) \rangle \langle u_m(\mathbf{k}) | \partial_{k_\nu} u_n \rangle \right. \\ & \qquad \left. + \langle \partial_{k_\mu} u_n | u_n(\mathbf{k}) \rangle \langle u_n(\mathbf{k}) | \partial_{k_\nu} u_n \rangle \right]. \end{aligned} \qquad (6.23) $$

In the last line, we have put apart the term for $m = n$ and restrict the sum to run over the other bands $m \neq n$. As a consequence of equation (6.7), the last term of equation (6.23), which we have isolated from the sum, is therefore symmetric under the exchange of the components $k_\mu$ and $k_\nu$,

$$ \langle \partial_{k_\mu} u_n | u_n(\mathbf{k}) \rangle \langle u_n(\mathbf{k}) | \partial_{k_\nu} u_n \rangle = \langle \partial_{k_\nu} u_n | u_n(\mathbf{k}) \rangle \langle u_n(\mathbf{k}) | \partial_{k_\mu} u_n \rangle. \qquad (6.24) $$
Now, since the tensor $\epsilon_{\mu\nu\sigma}$ is itself anti-symmetric under this exchange, it follows that the sum over the last term of equation (6.23) gives zero.

We still need to evaluate the matrix element $\langle u_m | \partial_{k_\nu} u_n \rangle$, where we have omitted the argument $\mathbf{k}$ in order to lighten the notations. This matrix element can be calculated using the Bloch Hamiltonian $H = H(\mathbf{k})$ and its eigenvalue equation

$$ 
\begin{aligned}
E_m \langle u_m | \partial_{k_\nu} u_n \rangle &= \langle u_m | H | \partial_{k_\nu} u_n \rangle \qquad (6.25) \\
&= \langle u_m | \partial_{k_\nu} (H | u_n \rangle) - \left\langle u_m \left| \frac{\partial H}{\partial k_\nu} \right| u_n \right\rangle \\
&= \langle u_m | \partial_{k_\nu} (E_n | u_n \rangle) - \left\langle u_m \left| \frac{\partial H}{\partial k_\nu} \right| u_n \right\rangle \\
&= \frac{\partial E_n}{\partial_{k_\nu}} \langle u_m | u_n \rangle + E_n \langle u_m | \partial_{k_\nu} u_n \rangle - \left\langle u_m \left| \frac{\partial H}{\partial k_\nu} \right| u_n \right\rangle,
\end{aligned}
$$

where we have used the Schrödinger equation in the first and third lines.

Let us first consider the situation with $m \neq n$, in which case the first term of the last line is zero because of the orthogonality of the wave functions, $\langle u_m | u_n \rangle = 0$. Consequently, we find

$$ (E_m - E_n) \langle u_m | \partial_{k_\nu} u_n \rangle = - \left\langle u_m \left| \frac{\partial H}{\partial k_\nu} \right| u_n \right\rangle \qquad (6.26) $$

or equivalently

$$ \langle u_m | \partial_{k_\nu} u_n \rangle = \frac{\left\langle u_m \left| \frac{\partial H}{\partial k_\nu} \right| u_n \right\rangle}{E_n - E_m} = \hbar \frac{\langle u_m | v_\nu | u_n \rangle}{E_n - E_m}, \qquad (6.27) $$

where we have used the expression (2.45),

$$ v_\nu(\mathbf{k}) = \frac{1}{\hbar} \frac{\partial H}{\partial k_\nu}, \qquad (6.28) $$

for the velocity operator in the Bloch basis. We finally notice that, for $m = n$, the auxiliary calculation (6.25) reproduces simply the group velocity of the $n$-th band,

$$ \left\langle u_n \left| \frac{\partial H}{\partial k_\nu} \right| u_n \right\rangle = \hbar \langle u_n | v_\nu(\mathbf{k}) | u_n \rangle = \frac{\partial E_n}{\partial k_\nu}. \qquad (6.29) $$
<figure><img src="image_62.png" /><figcaption>Figure 6.5: Left: closed path that an electron follows adiabatically in the first Brillouin zone. Right: projection of the path onto the n-th band. During the course of the path, the system changes adiabatically its energy $E_n(k(t))$, which is $E_n(k_0) = E_n(k(T))$, at the beginning of the path ($t = 0$) as well as at the end ($t = T$). The blue curve indicates the image $E(k(t))$ of the path $k(t)$. In the framework of perturbation theory, the system undergoes, all along the path, virtual transitions (red arrows) towards the other bands, here $E_m(k)$.</figcaption></figure>

Using this formulation of the matrix elements, we can rewrite the Berry curvature in terms of the velocity operators in the Bloch basis,

$$ \mathcal{B}_n(\mathbf{k})_\sigma = i\epsilon_{\mu\nu\sigma} \sum_{m \neq n} \frac{\langle u_n \left| \frac{\partial H}{\partial k_\mu} \right| u_m \rangle \langle u_m \left| \frac{\partial H}{\partial k_\nu} \right| u_n \rangle}{[E_n(\mathbf{k}) - E_m(\mathbf{k})]^2} \quad (6.30) \\ = i\hbar^2\epsilon_{\mu\nu\sigma} \sum_{m \neq n} \frac{\langle u_n | v_\mu | u_m \rangle \langle u_m | v_\nu | u_n \rangle}{[E_n(\mathbf{k}) - E_m(\mathbf{k})]^2}, $$

i.e. without using wave-function gradients.

In order to understand in more detail the particular expression (6.30) of the Berry curvature, we can interpret it in the light of perturbation theory. Here, we are dealing with infinitesimal perturbations on the path in reciprocal space. Indeed, by expanding the Bloch Hamiltonian

$$ H(\mathbf{k} + d\mathbf{k}) = H(\mathbf{k}) + d\mathbf{k} \cdot \nabla_{\mathbf{k}} H, \quad (6.31) $$

the first term can be seen as the non-perturbed Hamiltonian $H_0$ and $H' = d\mathbf{k} \cdot \nabla_{\mathbf{k}} H$ as the perturbation. Quantum-mechanical perturbation theory
teaches us that the wave function $|u_n(\mathbf{k} + d\mathbf{k})\rangle$, which is the eigenstate of the Hamiltonian $H(\mathbf{k} + d\mathbf{k})$, is written to first order as

$$|u_n(\mathbf{k} + d\mathbf{k})\rangle = |u_n(\mathbf{k})\rangle + \sum_{m \neq n} |u_m(\mathbf{k})\rangle \frac{\langle u_m(\mathbf{k}) | d\mathbf{k} \cdot \nabla_{\mathbf{k}} H | u_n(\mathbf{k}) \rangle}{E_n(\mathbf{k}) - E_m(\mathbf{k})}. \quad (6.32)$$

Now the direct expansion [see the equation (6.6)] of the wave function gives us

$$|u_n(\mathbf{k} + d\mathbf{k})\rangle = |u_n(\mathbf{k})\rangle + |\nabla_{\mathbf{k}} u_n\rangle \cdot d\mathbf{k}, \quad (6.33)$$

from which we get the same result,

$$\langle u_m | \nabla_{\mathbf{k}} u_n \rangle = \frac{\langle u_m | \nabla_{\mathbf{k}} H | u_n \rangle}{E_n(\mathbf{k}) - E_m(\mathbf{k})}, \quad (6.34)$$

as that obtained above [equation (6.27)].

The above argument does not provide us with new results but allows us to interpret the Berry curvature in a complementary picture that is sketched in figure 6.5. By following the C path in the first Brillouin zone, the electron adiabatically changes its energy and always remains in the same band $E_n(\mathbf{k})$. On the other hand, it undergoes virtual transitions to neighbouring bands $E_m(\mathbf{k})$, as it is stipulated by perturbation theory. Even if we consider adiabatic processes in which an electron cannot jump between the energetically separated bands, the distant bands nevertheless affect the dynamics of the particles restricted to a single band via virtual transitions that are taken into account by the Berry curvature. It follows that the Berry curvature is necessarily zero in (tight-binding) models with a single band. Indeed, we can convince ourselves of this fact directly by looking at the formula (6.30). It is a special case of a fundamental property of the Berry curvature: when it is summed over all the bands, the total curvature must vanish. This is indeed the case, as for each value of the wave vector $\mathbf{k}$, we have the sum rule<fnref n="10" />

$$\sum_n \vec{B}_n(\mathbf{k}) = 0. \quad (6.35)$$

$$\sum_n \sum_{m \neq n} \frac{\langle u_n | \frac{\partial H}{\partial \mathbf{k}_\mu} | u_m \rangle \langle u_m | \frac{\partial H}{\partial \mathbf{k}_\nu} | u_n \rangle}{[E_n(\mathbf{k}) - E_m(\mathbf{k})]^2}$$

is symmetric with respect to the exchange $\nu \leftrightarrow \mu$ while $\epsilon_{\nu\sigma}$ is antisymmetric, hence the cancellation of the sum.

<footnote n="10">¹⁰This can be seen directly in the formula (6.30) : by summing over the index $n$, the term</footnote>
### 6.3.3 Chern number

The Berry curvature allows us to define a topological invariant. Indeed, it inherits a periodicity in reciprocal space from the periodicity of the Bloch wave functions, $|\psi_{n,k}\rangle$, if the Bloch states are defined via the unitary transformation (2.39), $|u_n(k)\rangle = \exp(-i\mathbf{k} \cdot \hat{\mathbf{r}})|\psi_{n,k}\rangle$. Due to its periodicity, the Berry curvature can be viewed as a compact manifold $M$ so that

$$ \vec{\mathcal{B}}_n(\mathbf{k}) : \mathcal{T}^d \to \mathcal{M}, \qquad (6.36) $$

where $\mathcal{T}^d$ represents the compactified first Brillouin zone, which is isomorphic to a $d$-dimensional torus in a $(d+1)$-dimensional space, as we have mentioned above. From now on, we discuss $d=2$, in which case the Berry curvature is a periodic one-component function because it only possesses a $z$-component. The maps can then again be characterised by an integer that plays the role of our topological invariant. In the present case, this invariant is called first Chern number (or TKNN invariant), and it is obtained by integrating the Berry curvature over the entire first Brillouin zone,

$$ C_n = \int_{\text{1BZ}} \frac{d^2k}{2\pi} \mathcal{B}_n^z(\mathbf{k}) \in \mathbb{Z} \qquad (6.37) $$

To appreciate this point, it is useful to pass again by the analogy with electromagnetism. We have already mentioned that the Berry curvature can be seen as a magnetic field in reciprocal space. Recall that the magnetic flux through a surface $S$ is obtained by integrating the vector potential on the border $\partial S$ of this surface

$$ \phi = \oint_{\partial S} d\mathbf{r} \cdot \mathbf{A}(\mathbf{r}), \qquad (6.38) $$

which gives the Aharonov-Bohm phase $\Gamma_{AB} = 2\pi\phi/\phi_0$, in terms of flux quantum $\phi_0 = h/e$. In the same way, we can define the Berry flux

$$ \Phi(C) = \oint_C d\mathbf{k} \cdot \vec{\mathcal{A}}_n(\mathbf{k}), \qquad (6.39) $$

which is identical to the Berry phase $\Gamma(C) = 2\pi\Phi(C)$ modulo a factor $2\pi$. By Stokes' theorem, these integrals are equivalent to surface integrals of the
magnetic field or of the Berry curvature, and the Berry flux can be written as

$$ \Phi(C) = \int_{\Sigma} d^2k \mathcal{B}_n^z(k), \qquad (6.40) $$

where $\Sigma$ is the reciprocal-space surface enclosed by the path $C$. We will henceforth refer to this second formulation, which is equivalent to the first one if one can define a global gauge for the vector potential (the Berry connection). Without going into the mathematical details, this is not always possible, especially in the case we are interested in here – we speak of an *obstruction*. Indeed, when such a global gauge cannot be found, there are singularities in the Berry connection which can be seen as *magnetic monopoles*. As we know, these magnetic monopoles do not exist in classical electromagnetism, but Dirac has noticed that they are quite possible in quantum mechanics. They manifest themselves by a quantized flux through a closed surface around the monopole,

$$ \Phi(C) = 2\pi C_n, \qquad (6.41) $$

where $C_n$ is precisely the first Chern number (6.37).

This Chern number, introduced as a topological invariant (TKNN) associated with Bloch bands, may seem rather formal at first sight. However, it is, in the same way as the Berry curvature, a highly relevant quantity from a physical point of view. On the one hand, it classifies the topologically different phases and consequently the transitions between these phases. As it is an integer, it is a truism to say that it cannot change continuously. This means that the band structure itself cannot change continuously between two topologically different phases: in order to change the Chern number associated with a Bloch band, it is necessary that this last band crosses another one. This gives rise to *crossing points* between bands during a topological transition which we have already encountered in the context of massless Dirac fermions, as in graphene. This link between topological phase transitions and Dirac fermions is the subject of the next section.

More concretely, the Berry curvature gives rise to a corrective term to the group velocity of the electrons in the band $E_n(k)$. The velocity acquires a transverse component, and so does the conductivity of the electrons in the band. This is very similar to the Hall effect, and, indeed, there is even a quantized version of this effect in the form of the *quantum anomalous Hall effect* and the *quantum spin Hall effect*. While the appearance of the Berry term in the anomalous Karplus-Luttinger velocity is discussed at the
end of this chapter, the Landauer-Büttiker-type transport in the quantized versions of these effects is presented in the next chapter, where we review some spectacular experiments carried out in materials of a completely new generation.

## 6.4 Berry curvature of a massive Dirac fermion and “half Chern number”

As we mentioned at the end of the previous section, a topological phase transition manifests itself by a crossing between (at least two) bands so as to allow for a discrete change of the Chern number. The simplest case consists then of a 2D two-band model,<fnref n="11" /> which may be treated, in the vicinity of the topological phase transition, in terms of a Dirac fermion in the continuum limit. In this case, the continuum Dirac fermion is obtained by expanding the two-band model around the wave vector at which the band crossing occurs. In order to appreciate this point, let us first consider a semiconductor with a direct gap at $q = 0$, where the two “bands are inverted”. Directly at this point, the Hamiltonian is simply written as

$$H(\mathbf{q} = 0) = \begin{pmatrix} \Delta & 0 \\ 0 & -\Delta \end{pmatrix}, \qquad (6.42)$$

and the energy of the two states (bands) is $\pm\Delta$. This very simple model allows us to define what we mean by band inversion: it occurs when the gap parameter $\Delta$ changes its sign, while it necessarily goes through the value $\Delta = 0$ where the two states are degenerate. This particular value $\Delta = 0$ happens to determine the topological phase transition, which is described in more detail in the next section. In order to take into account the neighbourhood of this point in reciprocal space, it is sufficient to add a complex function $f(\mathbf{q})$ in the off-diagonal elements of the Hamiltonian. This complex function couples the two states. As we are interested only in the neighbourhood of the contact point, we can expand this function to lowest order in $\mathbf{q}$, and this yields

$$f(\mathbf{q}) \approx \hbar(\mathbf{v}_1 \cdot \mathbf{q} + i\mathbf{v}_2 \cdot \mathbf{q}), \qquad (6.43)$$

<footnote n="11">¹¹Often a system with $N$ bands can be decomposed into several systems with two bands. Even if this is not always the case, we adopt this viewpoint here, for the sake of a first simplified approach.</footnote>
where $\mathbf{v}_1$ and $\mathbf{v}_2$ are real parameters (of the physical dimension of a velocity). It can be shown (but we will not do it here) that there always exists a frame of reference so that $\mathbf{v}_1 = (v_1, 0)$ and $\mathbf{v}_2 = (0, v_2)$, and we still simplify the discussion by choosing $|v_1| = |v_2| = v$.<fnref n="12" /> Moreover, we omit for the moment a possible dispersion in the diagonal terms which is not of interest here.<fnref n="13" /> We thus realize that the Hamiltonian in the vicinity of the (avoided) crossing point at $\mathbf{q} = 0$ is nothing other than that of a Dirac fermion [see the equation (2.71) of the section 2.3],

$$H_{\xi}(\mathbf{q}) = \begin{pmatrix} \Delta & \xi\hbar v(q_x - i\xi q_y) \\ \xi\hbar v(q_x + i\xi q_y) & -\Delta \end{pmatrix}, \qquad (6.44)$$

where we recall that $\xi = \pm$ keeps track of the valley.<fnref n="14" /> We notice here that the bands are well separated when $\Delta \neq 0$, and indeed $\Delta = 0$ denotes the transition between two (topologically) distinct phases depending on whether $\Delta$ is positive or negative. The Dirac Hamiltonian can also be cast into the form

$$H_{\xi}(\mathbf{q}) = \varepsilon(q) \begin{pmatrix} \cos \theta & \xi \sin \theta e^{-i\xi\phi} \\ \xi \sin \theta e^{i\xi\phi} & -\cos \theta \end{pmatrix}, \qquad (6.45)$$

where $\varepsilon(q) = \sqrt{\Delta^2 + \hbar^2 v^2 q^2}$ and

$$\cos \theta = \frac{\Delta}{\varepsilon(q)}, \qquad \sin \theta = \frac{\hbar v q}{\varepsilon(q)}, \qquad \tan \phi = \frac{q_y}{q_x}, \qquad (6.46)$$

in terms of the modulus $q = |\mathbf{q}|$ of the wave vector. We have already seen that the spectrum of the Hamiltonian consists of two branches ($\lambda = \pm$), given by $E_{\lambda}(\mathbf{q}) = \lambda\varepsilon(q)$, independent of the valley $\xi$, but we are more particularly interested here in the eigenstates, which can be written as

$$|u_{\lambda=+}^{\xi}\rangle = \begin{pmatrix} \cos \frac{\theta}{2} \\ \xi \sin \frac{\theta}{2} e^{i\xi\phi} \end{pmatrix} \quad \text{and} \quad |u_{\lambda=-}^{\xi}\rangle = \begin{pmatrix} -\xi \sin \frac{\theta}{2} e^{-i\xi\phi} \\ \cos \frac{\theta}{2} \end{pmatrix}. \qquad (6.47)$$

<footnote n="12">¹²This can always be obtained by doing a scale transformation of the components of the wave vector.</footnote>

<footnote n="13">¹³One can also show that the dispersive terms on the diagonal scale like $q^2$. We will get back to this kind of terms in the model proposed by Bernevig, Hughes and Zhang in section 7.3.2.</footnote>

<footnote n="14">¹⁴This is natural because we will see in the following, based on very general arguments, that Dirac fermions emerge in pairs in electronic band structures. Moreover remember that we have seen, in the concrete case of graphene and boron nitride, that there are two low-energy Dirac fermions, located at $K$ and $K'$ at the corners of the first Brillouin zone.</footnote>
These expressions for the eigenstates help us to easily calculate the Berry connection

$$ \vec{\mathcal{A}}_{\lambda, \xi}(\mathbf{q}) = i \langle u_{\lambda}^{\xi} | \nabla_{\mathbf{q}} u_{\lambda}^{\xi} \rangle. \qquad (6.48) $$

Let us first calculate the gradient for the branch (band) of positive energy,

$$ |\nabla_{\mathbf{q}} u_{+}^{\xi}\rangle = \left( \xi \left[ \frac{1}{2} \cos\left(\frac{\theta}{2}\right) \nabla_{\mathbf{q}}\theta + \sin\left(\frac{\theta}{2}\right) (i\xi \nabla_{\mathbf{q}}\phi) \right] e^{i\xi\phi} \right), \qquad (6.49) $$

which gives<fnref n="15" />

$$ \langle u_{+}^{\xi} | \nabla_{\mathbf{q}} u_{+}^{\xi} \rangle = i\xi \sin^2\left(\frac{\theta}{2}\right) \nabla_{\mathbf{q}}\phi. \qquad (6.50) $$

In the same way, we find for the negative-energy branch

$$ \langle u_{-}^{\xi} | \nabla_{\mathbf{q}} u_{-}^{\xi} \rangle = -i\xi \sin^2\left(\frac{\theta}{2}\right) \nabla_{\mathbf{q}}\phi, \qquad (6.51) $$

and thus for the Berry connection

$$ \vec{\mathcal{A}}_{\lambda, \xi}(\mathbf{q}) = -\xi\lambda \sin^2\left(\frac{\theta}{2}\right) \nabla_{\mathbf{q}}\phi = -\frac{\xi\lambda}{2}(1 - \cos\theta)\nabla_{\mathbf{q}}\phi. \qquad (6.52) $$

Equation (6.52) allows us to obtain the Berry curvature associated to a massive Dirac fermion.<fnref n="16" />

$$ \begin{aligned} \vec{\mathcal{B}}_{\lambda, \xi}(\mathbf{q}) &= \nabla_{\mathbf{q}} \times \vec{\mathcal{A}}_{\lambda, \xi}(\mathbf{q}) = \frac{\lambda \xi}{2} [\nabla_{\mathbf{q}} \cos \theta \times \nabla_{\mathbf{q}} \phi - (1 - \cos \theta) \nabla_{\mathbf{q}} \times \nabla_{\mathbf{q}} \phi] \\ &= \frac{\lambda \xi}{2} \nabla_{\mathbf{q}} \cos \theta \times \nabla_{\mathbf{q}} \phi \end{aligned} \qquad (6.53) $$

As $\cos \theta = \Delta/\sqrt{\Delta^2 + \hbar^2 v^2 q^2}$, we find directly

$$ \nabla_{\mathbf{q}} \cos \theta = - \frac{\hbar^2 v^2 \Delta}{\varepsilon(q)^3} \mathbf{q}. \qquad (6.54) $$

<footnote n="15">¹⁵The terms proportional to $\nabla_{\mathbf{q}}\theta$ cancel out in the scalar product.</footnote>

<footnote n="16">¹⁶The second term of the first line is indeed zero. One can show, e.g. with the help of the Cauchy-Riemann differential equations in spherical coordinates, that $\nabla_{\mathbf{q}}\phi = 2\pi\delta^2(\mathbf{q})$ in terms of the delta function, i.e. this term only contributes when $\mathbf{q} = 0$. However, we have $\cos \theta = 1$ at $\mathbf{q} = 0$.</footnote>
Furthermore, we have<fnref n="17" />

$$ \nabla_q \phi = \frac{1}{q^2} \begin{pmatrix} -q_y \\ q_x \\ 0 \end{pmatrix}, \qquad (6.55) $$

and consequently $\vec{B}_{\lambda,\xi}(\mathbf{q}) = \mathcal{B}_{\lambda,\xi}^z(\mathbf{q})\mathbf{u}_z$, with

$$ \mathcal{B}_{\lambda, \xi}^{z}(\mathbf{q}) = -\frac{\lambda \xi \hbar^2 v^2 \Delta}{2 \varepsilon(q)^3} = -\frac{\lambda \xi \hbar^2 v^2 \Delta}{2 (\Delta^2 + \hbar^2 v^2 q^2)^{3/2}} = -\frac{\lambda \xi \operatorname{sgn}(\Delta)}{2} \frac{\lambda_C^2}{(1 + \lambda_C^2 q^2)^{3/2}}, \qquad (6.56) $$

where $\operatorname{sgn}(\Delta)$ denotes the sign of the gap parameter $\Delta$ and where we introduced a characteristic length

$$ \lambda_C = \frac{\hbar v}{|\Delta|} = \frac{\hbar}{m_D v}, \qquad (6.57) $$

in terms of the Dirac mass $m_D = |\Delta|/v^2$. It is indeed clear from its definition that the physical dimension of the Berry curvature is the square of a length. In the case of a Dirac fermion, this characteristic length has a very particular interpretation: it is the *Compton length* of relativistic quantum mechanics (modulo a factor 2). Recall that the Compton length (for an electron of mass $m_0$) is given by $\lambda_C = h/m_0c$, in terms of the speed of light $c$. We have already mentioned, in the context of graphene, that this velocity is replaced by the Fermi velocity, as it can be seen in equation (6.57). In high-energy physics, the Compton length, which is $\lambda_C = 2.4 \times 10^{-12}$ m, is a lower limit of subatomic resolution in optical spectroscopy (X-rays or $\gamma$-rays). Physically, a photon with a wavelength given by the Compton length has an energy $E = hc/\lambda_C = m_0c^2$ which equals the mass energy of an electron. This means that a photon of this wavelength (or more precisely of its half)

$$ \nabla_q \tan \phi = \frac{1}{\cos^2 \phi} \nabla_q \phi = \begin{pmatrix} -q_y/q_x^2 \\ 1/q_x \\ 0 \end{pmatrix} $$

because $\tan \phi = q_y/q_x$. Also, we have $\cos \phi = q_x/q$, hence the result.

<footnote n="17">¹⁷This can be calculated indirectly, by first calculating</footnote>
possesses a sufficient energy to create an electron-positron pair.<fnref n="18" /> In our condensed-matter-physics situation, a photon with a wavelength of $h/2m_Dv$ would have sufficient energy to promote an electron from the valence band to the conduction band, separated by the gap $2\Delta$. Such transitions are in contradiction with our basic hypothesis that the electron remains in its band during adiabatic processes. Note that this conceptual link between the Compton length and the Berry curvature is not unexpected: in both cases, it is a measure of coupling between bands if one interprets the the positron branch as the valence band and thus of the limit of validity for adiabatic processes. If we are confronted with processes that require length scales below the Compton length, the adiabatic theorem breaks down.

Equation (6.56) indicates a reciprocity relation between the value of the Berry curvature at $q = 0$ and its width in wave-vector space: when the value of the Berry curvature $B_{\lambda,\xi}(q = 0) = -\lambda\xi\lambda_C^2/2$ is increased, by increasing the Compton length (for example by decreasing the gap between the bands), the width of the curvature is reduced in reciprocal space; $q_C = 1/\lambda_C$ is indeed the characteristic wave vector at which the Berry curvature has half of its maximal value. This reciprocity is manifested in the calculation of the Berry phase, which is

$$ 
\begin{aligned}
\Gamma(q) &= \int_{|q'| \le q} d^2q' B_{\lambda,\xi}(q) = -\frac{\lambda\xi \operatorname{sgn}(\Delta)}{2} 2\pi\lambda_C^2 \int_0^q dq' \frac{1}{(1 + \lambda_C^2 q^2)^{3/2}} \\
&= -\lambda\xi \operatorname{sgn}(\Delta) \pi \left( 1 - \frac{1}{\sqrt{1 + \lambda_C^2 q^2}} \right) \qquad (6.58)
\end{aligned}
$$

on a circular path of radius $q$. The Berry flux associated with this Dirac fermion can be easily calculated in the limit $q \to \infty$, and one finds

$$ \tilde{C}_{\lambda,\xi} = \frac{\Gamma(q \to \infty)}{2\pi} = -\frac{1}{2}\lambda\xi\operatorname{sgn}(\Delta), \qquad (6.59) $$

which is independent of the characteristic length $\lambda_C$.

<footnote n="18">¹⁸This process is called *Compton scattering*. Strictly speaking, this diffusion takes place only in the vicinity of a massive spectator particle, such as a nucleus, in order to preserve the momentum in this scattering process. Indeed, the electron-positron pair could be described in its rest frame, i.e. in the frame of reference in which its centre of mass is at rest, and the total pair momentum is zero. This is in contradiction with the photon that gave birth to this pair and whose momentum $p = E/c \ge 2m_0c$ is non-zero in any frame of reference, hence the need for a heavier object to absorb this momentum.</footnote>
<figure><img src="image_63.png" /><figcaption>Figure 6.6: (a) Berry curvature, calculated over the entire first Brillouin zone for the complete gapped-graphene model ($\Delta \neq 0$, or boron nitride), according to equation (2.64), for $\Delta/t = 0, 1$. The wave vectors are measured in units of $1/a$ and the Berry curvature in units of $a^2$. The Berry curvature is concentrated around the points $K$ and $K'$ at the corners of the first Brillouin zone, with opposite contributions, while it vanishes in the rest of the first Brillouin zone. These contributions can then be summed up individually, because $1/\lambda_C \ll 1/a$, as we have assumed in the continuum limit. (b) Berry curvature of a single massive Dirac fermion calculated in the low-energy model, using equation (6.56), for $\lambda_C = 10a$. Adapted from J.-N. Fuchs et al., Eur. Phys. J. B **77**, 351 (2010).</figcaption></figure>
Notice that this total flux is reminiscent of the Chern number, which we introduced in the previous section [see equation (6.37)] and that is obtained by integrating the Berry curvature over the entire first Brillouin zone. In the latter situation, we have argued that this flux must be an *integer* and not a *half-integer* number as that obtained in the present case. However, this is no contradiction: now the flux does not need to be an integer because we have calculated the flux of a Dirac fermion in the continuum limit where the first Brillouin zone has been replaced by a non-compact space. Indeed, the integral in equation (6.58) is performed over the whole space $\mathbb{R}^2$ (in the limit $q \to \infty$). Nevertheless this calculation is valid when the Berry curvature of a Dirac fermion is concentrated in a region of size $1/\lambda_C \ll 1/a$, where $1/a$ is the characteristic separation between different Dirac points in the first Brillouin zone.<fnref n="19" /> The Berry curvature for boron nitride (gapped graphene) with a constant gap $\Delta/t = 0.1$, calculated for the model (2.64), is shown in figure 6.6(a). There we see that the Berry curvature is highly concentrated around the points at $K$ and $K'$ while it vanishes in the other locations of the first Brillouin zone. This validates our hypothesis that we can add directly the contributions from the two valleys to obtain the full Chern number. For comparison, figure 6.6(b) shows the characteristic shape of the Berry curvature in the vicinity of a single valley, according to equation (6.56).

Let us analyze further the result (6.59). The fact that a Dirac fermion contributes $\pm 1/2$ to the total Chern number, which must be an integer if we consider the whole first Brillouin zone, indicates that Dirac fermions necessarily come in *pairs* in periodic band structures. This phenomenon has been studied for the first time by Nielsen and Ninomiya in 1983, in the framework of quantum-field theory, and is called the *fermion doubling theorem*. While it came as a surprise in high-energy physics, where physicists tried to analyse the Dirac equation numerically on a lattice, it arises naturally in band-structure calculations, such as in graphene, where we have seen that the Dirac fermions are located at the inequivalent $K$ and $K'$ points at the corners of the first Brillouin zone.

From now on, we are going to call the quantity $\tilde{C}$ *half Chern number*. While it is a rather suspicious quantity from a pure mathematical point of view, it turns out to be quite a practical concept in the discussion of topological phase transitions. The Chern number of a band is then obtained

<footnote n="19">¹⁹Typically $a$ is on the order of the lattice spacing, but this is not a necessary condition.</footnote>
by summing the half Chern numbers in all valleys $\xi$,

$$C_{\lambda} = \sum_{\xi} \tilde{C}_{\lambda, \xi} = -\frac{\lambda}{2} \sum_{\xi} \xi \operatorname{sgn}(\Delta_{\xi}). \qquad (6.60)$$

This expression directly indicates that it is necessary to have gap parameters $\Delta_{\xi}$ which themselves change sign from one valley $\xi$ to another one in order to obtain a non-zero Chern number, $C_{\lambda} \neq 0$. Note that this is not the case for boron nitride, where the sign of $\Delta$ is the same in both valleys $K$ and $K'$, a situation represented in figure 6.6. In the following, we speak of an inverted gap when the sign of $\Delta$ is opposite in the two different valleys. We emphasise that this gap inversion is not visible if we consider only the band spectrum which depends only on the modulus square of the gap, $|\Delta|$. This link between gap inversion and Chern insulator (or topological insulator) happens to be extremely fruitful in the description of topological phases and topological phase transitions, which we discuss in the next section, as well as in the understanding of the bulk-edge correspondence (see section 7.4 of the following chapter).

### 6.4.1 Description a topological phase transition in terms of a half Chern number

We have seen, in the previous section, that the half Chern number associated with a massive Dirac fermion provides us with relevant information about the band structure. In particular the sum of the different half Chern numbers of a band $\lambda$ yields its Chern number. The latter is the topological invariant which distinguishes the different topological phases: one speaks of two different topological phases when their Chern numbers, or more generally their topological invariants, are different. Of course, talking about phases only makes sense if the band which interests us is specified. We therefore place the Fermi level in the gap between the two bands so as to have an insulator, and it is the Chern number of the completely filled band that determines the topological phase.<fnref n="20" /> Since the Chern number is necessarily an integer, it cannot be changed continuously in a transition between two different topological phases. Now we have seen in the previous section that the Chern number can be changed by changing the half Chern number of a Dirac fermion, that

<footnote n="20">²⁰This choice is arbitrary for the moment, but it is natural when we consider the associated transport properties which is the subject of the next section.</footnote>
<figure><img src="image_64.png" /><figcaption>Figure 6.7: Topological phase transition. We assume that we have at our disposal a (theoretical) knob that allows us to continuously change the gap parameter $\Delta_{\xi=+}$ in the $\xi = +$ valley while the other valley ($\xi = -$) is not affected by this change. It is purely a spectator. (a) There is a massive Dirac fermion (with $\Delta_{\xi=+} < 0$) in the $\xi = +$ valley which contributes $\tilde{C}_{\lambda=+,\xi=+} = 1/2$ to the Chern number (of the conduction band). Here, the spectator valley contributes herself $\tilde{C}_{\lambda=+,\xi=-} = -1/2$ to the Chern number, which thus vanishes. (b) The gap closes ($\Delta_{\xi=+} = 0$) at the transition, where the contribution to the Chern number is undefined. (c) The gap opens again but with $\Delta_{\xi=+} > 0$ ("inverted gap"). Consequently, the Dirac fermion contributes $\tilde{C}_{\lambda=+,\xi=+} = -1/2$ to the Chern number, which is then $C_{\lambda=+} = -1$ for the conduction band, since the contribution from the spectator valley has not changed, $\tilde{C}_{\lambda=+,\xi=-} = -1/2$. Let us recall that, in the valence band, the contributions are opposite to those of the conduction band, because of the sum rule (6.35).</figcaption></figure>
is to say by reversing the sign of its gap parameter. This requires to be able to act, by some theoretical button (to be specified eventually), on the gap parameter of only one of the Dirac fermions, let us say the one in the $\xi$ valley, while that in the other valley $-\xi$ is not affected (this Dirac fermion is thus a pure spectator of the transition). We have sketched this situation in figure 6.7. The transition takes place when $\Delta_{\xi} = 0$, which separates a region of parameters with $\Delta_{\xi} < 0$, with a half Chern number $\tilde{C}_{\lambda,\xi} = \lambda/2$ in the band $\lambda$, from a region with $\Delta_{\xi} > 0$, or $\tilde{C}_{\lambda,\xi} = -\lambda/2$. As it should be, the Chern number changes then by one, i.e. $\Delta C_{\lambda,\xi} = 1$. We insist on the fact that the gap vanishes directly at the transition, and the bands now touch each other in an isolated point in the first Brillouin zone. We recall that $\Delta_{\xi} = 0$ means the presence of massless Dirac fermions: two types of insulator (with $\Delta \neq 0$) are then separated by a semimetal (with $\Delta = 0$) that then resembles the situation we have encountered in graphene. In the following, we call the insulator with a Chern number $C_{\lambda=-} = 0$ a trivial insulator while the one with $C_{\lambda=-} \neq 0$ is called Chern insulator. This is our first example of a topological insulator. Gap closing is the quintessence of a topological phase transition in band structures: a topological phase transition is associated with the emergence of a massless Dirac fermion in the band structure. We will study in the next chapter explicit models illustrating such a situation, in the framework of the Haldane and the Kane-Mele models.

## 6.5 Adiabatic pumping and Karplus-Luttinger anomalous velocity

Before discussing explicit models that show topological phase transitions, we will end this chapter with a curiosity associated with the Berry curvature: the latter alters the (average) velocity of a wave packet in the n-th band, which is written

$$ \dot{\mathbf{r}}_n = \frac{1}{\hbar} \nabla_{\mathbf{k}} E_n(\mathbf{k}) - \dot{\mathbf{k}} \times \vec{B}_n(\mathbf{k}), \qquad (6.61) $$

where $\mathbf{r}_n$ is here an average position of the wave packet in the n-th band. The first term is nothing else than the usual group velocity (3.26) while the second one is the so-called anomalous Karplus-Luttinger velocity, which we will study in the following. Before going into the calculations, let us notice the symmetry with respect to the other semi-classical equation which determines
the electron dynamics and that is simply the more familiar Newton equation,

$$ \hbar \dot{\mathbf{k}} = -e \vec{\mathcal{E}} - e \dot{\mathbf{r}}_n \times \mathbf{B}, \qquad (6.62) $$

where $\vec{\mathcal{E}} = -\nabla_{\mathbf{r}}\Phi(\mathbf{r})$ is an electric field emanating from a scalar potential $\Phi(\mathbf{r})$ and $\mathbf{B}$ the magnetic field. The equation (6.61) then corroborates our interpretation of the Berry curvature in terms of a magnetoelectric field in reciprocal space, in comparison with Newton's equation (6.62).

In order to establish the anomalous contribution to the average velocity of an electron, we proceed similarly to the gauge-invariant formulation of the Berry curvature presented in section 6.3.2. We consider first a 1D Hamiltonian $H(q, t)$ with parameters that can vary in time. At each instant, the states $|u_n(q, t)\rangle$ form a complete basis satisfying the stationary Schrödinger equation

$$ H(q, t)|u_n(q, t)\rangle = E_n(q, t)|u_n(q, t)\rangle, \qquad (6.63) $$

where $q$ is the 1D wave vector which will be identified with $k_x$ in the following. We adopt a first-order perturbative treatment for the state

$$ |\psi_n(q, t)\rangle = e^{i\gamma(t)} \left[ |u_n(q, t)\rangle - i\hbar \sum_{m \neq n} |u_m(q, t)\rangle \frac{\langle u_m(q, t)|\dot{u}_n(q, t)\rangle}{E_n(q, t) - E_m(q, t)} \right]. \qquad (6.64) $$

Contrary to the initial discussion of the Berry phase in section 6.3 [see equation (6.8)], we now take into account possible quantum jumps between the bands during the evolution of the wave packet.<fnref n="21" /> The last term in equation (6.64) indeed indicates that the $|\psi_n\rangle$ state has corrective components in the $m \neq n$ bands. We now evaluate the average velocity in the state $|\psi_n(q, t)\rangle$ using the relation (6.28),

$$ v(q, t) = \frac{1}{\hbar} \frac{\partial H(q, t)}{\partial q}, \qquad (6.65) $$

<footnote n="21">²¹Formally, this perturbative approach finds its origin in Floquet theory in the framework of a periodic pumping. Without going into the details of this approach, it considers the Hamiltonian $H' = H(t) - i\partial/\partial t$, where the second term is seen as a perturbation of $H(t)$. Its treatment in first-order perturbation theory gives the expression (6.64) for the state $|\psi_n(q, t)\rangle$.</footnote>
between the velocity operator and the Bloch Hamiltonian $H(q, t)$,

$$v_n(q,t) = \langle \psi_n(q,t) | v(q,t) | \psi_n(q,t) \rangle \approx \frac{1}{\hbar} \left\langle u_n(q,t) \left| \frac{\partial H(q,t)}{\partial q} \right| u_n(q,t) \right\rangle \\ +i \sum_{m \neq n} \left\langle u_m(q,t) \left| \frac{\partial H(q,t)}{\partial q} \right| u_m(q,t) \right\rangle \frac{\langle \dot{u}_n(q,t) | u_m(q,t) \rangle}{E_n(q,t) - E_m(q,t)} + \text{c.c.}, \qquad (6.66)$$

where we have omitted terms beyond linear order in $1/(E_n - E_m)$. Note that the first term can be written as

$$\frac{1}{\hbar} \left\langle u_n(q,t) \left| \frac{\partial H(q,t)}{\partial q} \right| u_n(q,t) \right\rangle = \frac{1}{\hbar} \frac{\partial E_n(q,t)}{\partial q}, \qquad (6.67)$$

which is nothing else than the usual group velocity obtained in equation (6.29). The terms in the second line of equation (6.66) then give the anomalous velocity, which we can write down in a simpler way using

$$\langle \dot{u}_n(q,t) | u_m(q,t) \rangle = \frac{\left\langle u_n(q,t) \left| \frac{\partial H(q,t)}{\partial t} \right| u_m(q,t) \right\rangle}{E_n(q,t) - E_m(q,t)}, \qquad (6.68)$$

which is obtained in the same way as equation (6.27) by replacing the derivative with respect to $k_ν$ with $\partial/\partial t$. We thus find for the average velocity

$$v_n(q,t) = \frac{1}{\hbar} \frac{\partial E_n(q,t)}{\partial q} \qquad (6.69) \\ +i \sum_{m \neq n} \frac{\langle u_n(q,t) \left| \frac{\partial H}{\partial t} \right| u_m(q,t) \rangle \langle u_m(q,t) \left| \frac{\partial H}{\partial q} \right| u_n(q,t) \rangle}{[E_n(q,t) - E_m(q,t)]^2} + \text{c.c.}$$

In the expression of the velocity (6.69) we recognise in the second term an expression that is very similar to that of the Berry curvature in its gauge-invariant formulation (6.30). One just has to replace the wave vector $q \to k_x$ and the time dependence by $k_y$. In order to see this link in more detail, let us get back to the 2D case and consider that the time dependence comes from an electric field applied in the y-direction. We can then use the Peierls substitution with $\mathbf{k}(t) = \mathbf{q} + (e/\hbar)\mathbf{A}(t)$, where the homogeneous vector potential gives rise to the electric field $\partial\mathbf{A}/\partial t = -\vec{\mathcal{E}} = -\mathcal{E}\mathbf{u}_y$. Because both quantities are spatially homogeneous, the presence of the vector potential does not
affect the symmetry properties of the system given by the discrete translations, and thus Bloch's theorem remains valid. This is to be contrasted with the situation of a homogeneous magnetic field the vector potential of which depends (linearly) on the position and thus breaks the discrete translation symmetry, as we discussed in the previous chapter where we revisited Landau quantisation. The band structure and the Berry curvature are therefore not affected by this potential, and the components of the wave vector may finally be written $k_x(t) = q_x = q$ and $k_y(t) = q_y - (e/\hbar)\mathcal{E}t$. This implies for the derivatives

$$ \frac{\partial}{\partial q} = \frac{\partial}{\partial k_x}, \quad \frac{\partial}{\partial t} = \dot{k}_y \frac{\partial}{\partial k_y} = -\frac{e}{\hbar} \mathcal{E} \frac{\partial}{\partial k_y}, \qquad (6.70) $$

so that the velocity (6.69) in the x-direction becomes

$$ \begin{aligned} v_n^x(\mathbf{k}) &= \frac{1}{\hbar} \frac{\partial E_n(\mathbf{k})}{\partial k_x} \\ &\quad + i \dot{k}_y \sum_{m \neq n} \frac{\left\langle u_n(\mathbf{k}) \left| \frac{\partial H}{\partial k_y} \right| u_m(\mathbf{k}) \right\rangle \left\langle u_m(\mathbf{k}) \left| \frac{\partial H}{\partial k_x} \right| u_n(\mathbf{k}) \right\rangle}{[E_n(\mathbf{k}) - E_m(\mathbf{k})]^2} + \text{c.c.} \\ &= \frac{1}{\hbar} \frac{\partial E_n(\mathbf{k})}{\partial k_x} - \dot{k}_y \mathcal{B}_n^z(\mathbf{k}), \end{aligned} \qquad (6.71) $$

as we saw at the beginning of the section in the framework of the semi-classical equations of motion (6.61).

### 6.5.1 Quantum anomalous Hall effect

Note that the last equation (6.71) can be written in terms of the electric field as

$$ v_n^x(\mathbf{k}) = \frac{1}{\hbar} \frac{\partial E_n(\mathbf{k})}{\partial k_x} + \frac{e}{\hbar} \mathcal{E} \mathcal{B}_n^z(\mathbf{k}), \qquad (6.72) $$

or, in a vector notation, as

$$ \mathbf{v}_n(\mathbf{k}) = \frac{1}{\hbar} \nabla_{\mathbf{k}} E_n(\mathbf{k}) + \frac{e}{\hbar} \vec{\mathcal{E}} \times \vec{\mathcal{B}}_n(\mathbf{k}). \qquad (6.73) $$

The last term gives rise to a drift insofar as the anomalous velocity in the x-direction is affected, through the Berry curvature, by the electric field in the y-direction. This is strikingly reminiscent of the Hall effect. Indeed,
we are confronted, here, with an anomalous Hall effect. In order to make this connection explicit, let us recall that the current density is obtained by summing over all the occupied states, and the contribution of the n-th band reads

$$ \mathbf{j}_n = -\frac{e}{S} \sum_{\text{occ.}} \mathbf{v}_n(\mathbf{k}) = -\frac{e}{\hbar} \int_{\text{occ.}} \frac{d^2k}{(2\pi)^2} \nabla_{\mathbf{k}} E_n(\mathbf{k}) - \frac{e^2}{\hbar} \vec{\mathcal{E}} \times \int_{\text{occ.}} \frac{d^2k}{(2\pi)^2} \vec{\mathcal{B}}_n(\mathbf{k}), \qquad (6.74) $$

where $\int_{\text{occ}}$ denotes the integral over all occupied states and S is the total area. The second term gives a transverse conductivity in the form

$$ \sigma_H = \frac{e^2}{\hbar} \int_{\text{occ.}} \frac{d^2k}{2\pi} \mathcal{B}_n^z(\mathbf{k}), \qquad (6.75) $$

which is called anomalous Hall conductivity. Written in this form, it is not yet quantized – it is only a fraction of the Chern number – because the integral is taken only over the occupied states of the band. On the other hand, when the n-th band is completely filled, the integral includes all the wave vectors of the first Brillouin zone, and consequently the Hall conductivity of a fully occupied band is quantized,

$$ \sigma_H = \frac{e^2}{\hbar} \int_{1BZ} \frac{d^2k}{2\pi} \mathcal{B}_n^z(\mathbf{k}) = \frac{e^2}{\hbar} C_n, \qquad (6.76) $$

in terms of the Chern number $C_n$. Let us finally emphasise that in the case of a completely filled band, the first term of the equation (6.74) does not contribute to the conductivity because of the periodicity of the energy in reciprocal space. This is an unexpected result, to say the least: a completely filled band, with a non-zero Chern number, can conduct a current (in the direction transverse to the applied electric field) even if it is separated by a gap from the other bands. We are thus confronted with some kind of chimera: a conducting insulator which probably represents the most remarkable phenomenon associated with a topological insulator. More precisely, it is called a Chern insulator, because the conductivity is determined by the Chern number. On the other hand, the calculation which provided this strange result does not tell us where the associated current flows. A look at the quantum Hall effect suggests that the current flows again at the sample edges. We will show in detail in the next chapter, where we discuss different types of
2D topological insulators, that this well-educated guess turns out to be the right one.

We thus notice at the end of this chapter that the Berry curvature and the Chern number – two concepts introduced in a relatively abstract way within adiabatic transport – give rise to physically relevant and measurable phenomena. On the one hand, the topological classification of the bands via the Chern number (each number represents a different homotopy class) allows us to describe topological phase transitions: because the Chern number varies discontinuously, the gap between two adjacent bands must close during a topological phase transition. Exactly at the transition a massless Dirac fermion emerges. On the other hand, the Berry curvature plays an important role in the electronic transport as it intervenes in the anomalous (Karplus-Luttinger) velocity. It bends the trajectories of the electrons in a way similar to a real magnetic field, and one obtains an anomalous (quantum) Hall effect. In the same way as for the quantum Hall effect, in the presence of a magnetic field, the appearance of the anomalous Hall effect requires a broken time-reversal symmetry – otherwise there would not be a privileged direction for the drift that is at the origin of the transverse conductivity. The discussion of the role time-reversal symmetry plays in topological materials as well as of concrete models that realise a quantum anomalous Hall effect are the focus of the next chapter, where we discuss 2D topological insulators.

## Notions and concepts to retain

* topological invariants, homotopy classes

* Berry-ology (Berry connection, curvature, phase and flux), Chern number

* half Chern number, fermion doubling

* topological phase transitions and gap closing

* adiabatic pumping, Karplus-Luttinger anomalous velocity, (quantum) anomalous Hall effect
# Chapter 7

## Two-dimensional topological insulators

In the preceding chapter, we introduced – in a relatively abstract manner – the topological classification of materials with a band structure based on Bloch's theorem. This approach is also called *topological band theory*. The present chapter aims at a more concrete illustration of these rather novel concepts with the help of some paradigmatic 2D band models. The first one is the Haldane model, which consists of a tight-binding model on the honeycomb lattice reminiscent of that used to discuss the band structure of graphene. In addition to our discussion of graphene bands in section 2.3, where we concentrated on nearest-neighbour hopping between sites of different sublattices, now also next-nearest neighbour hopping is taken into account. Moreover, the hopping matrix elements acquire a phase to take into account a possible inhomogeneous flux distribution inside the hexagons of the lattice, while the total flux is set to zero. Indeed, this somewhat artificial assumption was meant to be a theoretical illustration of an unexpected phenomenon, an answer to a question raised by Haldane in 1988 (Nobel Prize in Physics, 2016): is it possible to have a quantum Hall effect in the *absence* of a magnetic field? With the help of his variant of the graphene model, he could answer this question positively. Even if this question might seem a bit far-fetched at first sight, it turned out to be quite a visionary one. Without calling it in this manner, he proposed the first topological (Chern) insulator, which displays the quantum anomalous Hall effect discussed in section 6.5.1 of the last chapter. As we saw there, one needs to break time-reversal symmetry in order to obtain this effect, and this is achieved by the inhomogeneous flux,
even if it is zero on the average.

While Haldane's work had remained a curiosity of theoretical condensed-matter physics during quite a long time,<fnref n="1" /> it regained interest in 2005 when the theoreticians Kane and Mele proposed a generalisation of the Haldane model that restores time-reversal symmetry, taking advantage of the electrons' spin. In short, the Kane-Mele model consists of two separate copies of the Haldane model, one for each spin orientation, so that spin-↑ electrons form a Chern insulator and spin-↓-electrons another one, with opposite Chern numbers. As the Haldane model, the Kane-Mele model is a variant of the tight-binding model for electrons in graphene. Kane and Mele understood that, in order to encounter separate the dynamics of spin-↑ and spin-↓ electrons, one needs take into account *spin-orbit coupling*. Since this coupling is weak in graphene, their model is first of all of conceptual interest rather than a realistic representation of a physically relevant material. Indeed, one needs to look for materials with a strong *spin-orbit coupling* that is usually induced by heavy atoms. In these materials, the lattice dynamics of the electrons (such as the hopping parameters in tight-binding models) depend on their spin orientation.

Shortly after their seminal proposal, three other theoreticians, Bernevig, Hughes and Zhang, showed that such a situation, described by a variant of the Kane-Mele model, may arise in CdTe/HgTe heterostructures. Since one is still confronted with two spin copies of the Haldane model, the topological insulator described by these models should display yet another type of quantum (anomalous) Hall effect: while spin-↑ electrons experience a Hall drift in one direction due to the Berry curvature, spin-↓ electrons are expected to be deviated in the opposite direction. This link between spin and direction of propagation is the origin of the *quantum spin Hall effect*.

A detailed discussion of these two variants of the Hall effect, the quantum anomalous and the quantum spin Hall effects, based on the Haldane and the Kane-Mele models, respectively, is the central part of this chapter (see sections 7.2 and 7.3.1). The variant proposed by Bernevig, Hughes and Zhang is discussed in section 7.3.2. Furthermore, similarly to the quantum Hall effect, the electronic transport in these two effects takes place at the sample edges, as a consequence of the so-called *bulk-edge correspondence* (see section

<footnote n="1">¹This is mainly due to the fact that at that time there were no truly 2D materials available and even less with broken time-reversal symmetry. Indeed, one would need to consider magnetic 2D systems because not only the magnetic field but also magnetic moments break this symmetry.</footnote>
7.4). Since time-reversal symmetry plays a central role in the formation of topological insulators, and topological materials in general, we start this chapter with a brief review of this symmetry in section 7.1.

## 7.1 Time-reversal symmetry

Since quite a long time, physicists asked themselves the question about the time arrow: what would happen if we invert the sense of the time flow? Evidently, there is a privileged direction since people get older and not younger, or a coffee cup breaks into pieces when it falls on the ground but its pieces never reform the cup. In our macroscopic world, there is a clear sense in which time flows, and the question might seem stupid. However, most fundamental laws of physics are invariant under time reversal, i.e. they are time-reversal symmetric. One does not even need to search in the realm of quantum physics since even the classical (Newton) equations of motion are time-reversal-symmetric because they are given in terms of second derivatives in time.

In view of this observation, let us analyse how several basic quantities transform if we change the time arrow. Naturally, we have $t \to -t$ for time itself while the position of an object remains unchanged, $\mathbf{r} \to \mathbf{r}$. Remember that, in quantum physics, these quantities are observables, i.e. (Hermitian) operators that act on quantum states in a Hilbert space, and transformations are therefore represented by operators themselves, acting on the observables. Time-reversal symmetry is therefore represented by an operator $T$, which needs to be constructed in a second step and that transforms the Hilbert-space states as

$$|\psi'\rangle = T|\psi\rangle \qquad (7.1)$$

and the observables $\mathcal{O}$ as

$$\mathcal{O}' = TOT^{-1}. \qquad (7.2)$$

We encounter such transformations quite generally, e.g. in the context of the description of symmetries (recall the discrete translation symmetries used in section 2.1 used in the construction of Bloch wave functions). In most situations that you have encountered so far in quantum physics, these symmetries are represented by *unitary* operators, $U^\dagger = U^{-1}$, as it is the case for translation or rotation symmetries. However, in 1931, the Hungarian physicist Wigner showed that not only unitary, but also *anti-unitary* operators are
eligible in the representation of a symmetry.<fnref n="2" /> An anti-unitary operator $U_a$ is therefore an anti-linear operator that is obtained by combining a unitary operator $U$ with complex conjugation $K$,

$$U_a = UK. \qquad (7.3)$$

Wigner's observation now leaves us with the task to check whether the symmetries we are interested in are represented by unitary or anti-unitary operators. Notice that translations and rotations are obtained from infinitesimal operations that lead to continuous Lie groups. For this reason, they are given by unitary operators, in agreement with what you have learned from your past quantum-physics classes. Time-reversal symmetry is, however, a discrete symmetry, and we already know, based on the above discussion, that the position and momentum operators, $\mathbf{r}$ and $\mathbf{p}$, respectively, must satisfy

$$\mathbf{r}' = \mathbf{T}\mathbf{r}\mathbf{T}^{-1} = \mathbf{r} \quad \text{et} \quad \mathbf{p}' = \mathbf{T}\mathbf{p}\mathbf{T}^{-1} = -\mathbf{p}. \qquad (7.4)$$

These rules simply reflect the fact that the position remains unaffected by time reversal, while the momentum of particle changes to its opposite value. What about some other physical quantities, such as the angular momentum of a particle or its spin? Since the angular momentum is the vector product of position and momentum, it must also change its sign

$$\mathbf{T}\mathbf{L}\mathbf{T}^{-1} = -\mathbf{L}, \qquad (7.5)$$

and so must the spin $\mathbf{S}$, which is an intrinsic angular momentum of the particles,

$$\mathbf{T}\mathbf{S}\mathbf{T}^{-1} = -\mathbf{S}. \qquad (7.6)$$

This is indeed in line with our intuitions because the sense of rotation should change when we invert the flow of time.

If we require that the quantum-mechanical equations of motion be invariant under time reversal, in order to match the classical Newtonian ones,

<footnote n="2">²The usual argument for the choice of unitary operators is based on the requirement that the matrix elements of the observable $\langle\psi|\mathcal{O}|\psi'\rangle$ must remain invariant and thus also the scalar products $\langle\psi|\psi'\rangle$ themselves. In quantum field theory, however, transition probabilities are the physically relevant quantities, and they are given in terms of the modulus square of such matrix elements. Wigner then argued that the requirement of invariant matrix elements is too restrictive. Indeed, the requirement of keeping the transition probabilities invariant allows one to choose anti-unitary operators on equal footing with unitary operators.</footnote>
we must first recall that they are based on the canonical quantisation rules induced by the commutation relations for position and momentum. The commutators must therefore remain invariant themselves,

$$[x_l, p_k] = i\hbar\delta_{k,l} \quad \text{et} \quad T[x_l, p_k]T^{-1} = T(i\hbar\delta_{k,l})T^{-1}. \qquad (7.7)$$

However, we have $T[x_l, p_k]T^{-1} = -[x_l, p_k]$ because only the components of the momentum operator change sign. In order to obtain time-reversal invariant commutation relations, we therefore need to make sure that

$$TiT^{-1} = -i, \qquad (7.8)$$

which is achieved by the complex conjugation in the operator $T$, i.e. by its anti-linearity.<fnref n="3" />

A remarkable property of an anti-unitary operator, such as time reversal, is that its square has the eigenvalues $\pm 1$,

$$T^2|\psi_{\pm}\rangle = \pm|\psi_{\pm}\rangle, \qquad (7.9)$$

$$\dot{x}_l = \frac{1}{i\hbar}[x_l, H],$$

whose average yields the Ehrenfest theorem and which yields

$$T\dot{x}_l T^{-1} = T\left(\frac{1}{i\hbar}[x_l, H]\right) T^{-1}.$$

On the left-hand side, we have $T\dot{x}_l T^{-1} = -\dot{x}_l$, and on the right-hand side we find

$$T\left(\frac{1}{i\hbar}[x_l, H]\right) T^{-1} = T\left(\frac{1}{i\hbar}\right) T^{-1} T[x_l, H] T^{-1} = T\left(\frac{1}{i\hbar}\right) T^{-1}[x_l, H] = T\left(\frac{1}{i\hbar}\right) T^{-1} i\hbar \dot{x}_l.$$

If time reversal were represented by a unitary operator with $TiT^{-1} = i$, this would yield the absurd result that $\dot{\mathbf{r}} = -\dot{\mathbf{r}}$. It is therefore still the anti-linearity (7.8) that ensures the correct sign.

<footnote n="3">³The anti-linearity or anti-unitarity can also be seen directly from the quantum-mechanical equations of motion induced by the commutator. Consider the usual non-relativistic particle described by the Hamiltonian $H = |\mathbf{p}|^2/2m$, which is clearly time-reversal invariant, $H' = H$ since it appeals to the modulus square of the momentum operator, which itself changes sign. However, we must make sure that $\dot{\mathbf{r}}' = \mathbf{p}'/m$ in the frame of reference with inverted time arrow, as much as we had $\dot{\mathbf{r}} = \mathbf{p}/m$ in the original frame. Otherwise we would not retrieve the correct equations of motion via Ehrenfest's theorem. Let us recall the Heisenberg equations of motion,</footnote>
Instead of providing a mathematical proof of this property, let us check whether the operator $T$ satisfies this condition upon an explicit construction of the time-reversal operator. This operator must naturally satisfy the relations (7.5) for the angular-momentum operator and (7.6) for the spin. Recall that the components of the angular-momentum and spin operators satisfy the commutation relations

$$[L_x, L_y] = i\hbar L_z \quad \text{and} \quad [S_x, S_y] = i\hbar S_z, \qquad (7.10)$$

respectively, and consequently one of the components must be purely imaginary. By convention, we choose the $y$-components $L_y$ and $S_y$ to be imaginary so that the conditions

$$TL_yT^{-1} = -L_y \quad \text{and} \quad TS_yT^{-1} = -S_y \qquad (7.11)$$

are satisfied by the anti-linearity of the operator $T$, as long as the unitary part $U$ leaves them invariant

$$UL_yU^{-1} = L_y \quad \text{and} \quad US_yU^{-1} = S_y, \qquad (7.12)$$

while changing the sign of the other (real) components,

$$UL_{x/z}U^{-1} = -L_{x/z} \quad \text{and} \quad US_{x/z}U^{-1} = -S_{x/z}. \qquad (7.13)$$

Indeed, we already know a unitary operator that satisfies these equations: it is simply a rotation by $\pi$ around the $y$-axis,

$$U = e^{-i\pi J_y/\hbar}, \qquad (7.14)$$

where $J_y = L_y + S_y$ is the $y$-component of the total angular momentum operator, which is precisely the generator of rotations around the $y$-axis. The anti-unitary operator that we are looking for therefore reads

$$T = e^{-i\pi J_y/\hbar} K. \qquad (7.15)$$

While the angular momentum operator has integer numbers as eigenvalues, in units of $\hbar$, the spin part can take half-integer values. Consequently, if we take in a somewhat unusual manner the $y$-axis as our quantisation axis for the spin, we have $S_y|s, m_y\rangle = m_y\hbar|s, m_y\rangle$, where the quantum number $m_y$ is an integer if $s$ is an integer itself and a half integer for half-integer spins $s$.
Therefore, the application of the square of the time-reversal operator (7.15) on the eigenstates $|s, m_s\rangle$ of the spin operator yields

$$ T^2|s, m_s\rangle = e^{-2i\pi J_y/\hbar}|s, m_y\rangle = e^{-2i\pi m_y}|s, m_y\rangle = \pm|s, m_y\rangle, \quad (7.16) $$

with a sign + for integer spins (bosons) and – for half-integer spins (fermions).

What we have shown here in a rather abstract manner, can be illustrated explicitly and more directly in the case of a spin-1/2 particle, with $S_y = (\hbar/2)\sigma_y$, in terms of the Pauli matrix $\sigma_y$. In this case a series expansion of the unitary operator$^4$

$$ U = e^{-i\pi\sigma_y/2} = \sum_{n=0}^{\infty} \frac{1}{n!} \left(\frac{-i\pi}{2}\right)^n \sigma_y^n = \cos\left(-\frac{\pi}{2}\right) 1 + i \sin\left(-\frac{\pi}{2}\right) \sigma_y \quad (7.17) $$

directly gives

$$ U = -i\sigma_y \quad \text{and} \quad T = -i\sigma_y K. \quad (7.18) $$

We thus have

$$ T^2 = -i\sigma_y K(-i\sigma_y)K = -i\sigma_y(-i\sigma_y)K^2 = -\sigma_y^2 = -1 \quad (7.19) $$

since $i\sigma_y$ is real and $K^2 = 1$.

We terminate this section with a note on the behaviour of electromagnetic fields under time-reversal transformations. We have already mentioned above that Newton's law $m\ddot{r} = F$ is invariant under time-reversal symmetry. Consequently, the forces $F$ must themselves be time-reversal-invariant,

$$ TFT^{-1} = F. \quad (7.20) $$

The force generated by the electric and magnetic fields on a particle of mass $m$ and charge $e$ is given by

$$ F = e(\mathbf{E} + \mathbf{v} \times \mathbf{B}). \quad (7.21) $$

$$ \cos \theta = \sum_{m=0}^{\infty} \frac{(-1)^m}{(2m)!} \theta^{2m} \quad \text{and} \quad \sin \theta = \sum_{m=0}^{\infty} \frac{(-1)^m}{(2m+1)!} \theta^{2m+1}. $$

<footnote>$^4$In this expansion, we use the fact that $\sigma_y^{2m} = 1$ and thus $\sigma_y^{2m+1} = \sigma_y$ for any integer value of $m$. Furthermore, we make use of the series expansion of the trigonometric functions</footnote>
In order to have time-reversal-invariant forces, the fields and their associated potentials must therefore satisfy the following transformation laws

$$ \begin{gathered} TET^{-1} = \mathbf{E}, \quad TV(\mathbf{r})T^{-1} = V(\mathbf{r}) \\ \text{and} \\ TBT^{-1} = -\mathbf{B}, \quad T\mathbf{A}(\mathbf{r})T^{-1} = -\mathbf{A}(\mathbf{r}). \end{gathered} \quad (7.22) $$

This ensures indeed that also Maxwell's laws are invariant under time-reversal symmetry. Notice that we have already made use of the antisymmetry of the magnetic field under time-reversal symmetry in the discussion of the Onsager relations relating the transmission and reflection coefficients in the electric conductance for quantum mesoscopic conductors in section 4.1 of chapter 4.

### 7.1.1 Kramers' theorem

We are now well prepared to understand *Kramers' theorem* which states that electrons (or more generally fermions with half-integer spin) always occur in pairs in time-reversal invariant systems.<fnref n="5" /> Before discussing this theorem in the context of Bloch bands, let us consider first a general Hamiltonian *H* that is time-reversal invariant, i.e. that respects

$$ THT^{-1} = H. \quad (7.23) $$

Furthermore, let $|\psi\rangle$ be an eigenstate of the Hamiltonian. In this case $T|\psi\rangle$ is an equally valid eigenstate of the Hamiltonian at the same energy – this is precisely the sense of a symmetry. In order to prove Kramers' theorem, we thus need to show that $T|\psi\rangle$ is an eigenstate different from $|\psi\rangle$. This means that $|\psi\rangle$ and $T|\psi\rangle$ are not collinear, $T|\psi\rangle \neq c|\psi\rangle$, where c is a complex number. Indeed, if the contrary $T|\psi\rangle \neq c|\psi\rangle$ were correct, this would lead to an apparent contradiction, as it can be shown by applying once more the operator T to the (wrong) equation. On the one hand side, we would then have

$$ T^2|\psi\rangle = -|\psi\rangle \quad (7.24) $$

since we are dealing with a fermionic state. On the other side, the equation would give

$$ T(T|\psi\rangle) = Tc|\psi\rangle = c^*T|\psi\rangle = c^*c|\psi\rangle = |c|^2|\psi\rangle, \quad (7.25) $$

<footnote n="5">⁵We owe this theorem to the Dutch physicist Kramers, who formulated it in 1930.</footnote>
where we have made use of the fact that $T$ acts on the complex number $c$ by complex conjugation. One would therefore draw the absurd conclusion that $|c|^2 = -1$, so that $|\psi\rangle$ and $T|\psi\rangle$ must be linearly independent, and the energy level $E$ must be at least twofold degenerate.<fnref n="6" /> For free fermions (in the absence of a magnetic field) this is reflected by the twofold spin degeneracy.

Let us now adapt these arguments to spin-1/2 electrons in Bloch bands described by the Bloch Hamiltonian $H_s(\mathbf{k})$ with the two spin orientations $\uparrow$ ($s = +$) and $\downarrow$ ($s = -$). The condition (7.23) must now be rewritten as

$$TH_s(\mathbf{k})T^{-1} = H_{-s}^*(-\mathbf{k}) = H_s(\mathbf{k}) \qquad (7.26)$$

for a time-reversal-invariant Bloch Hamiltonian and

$$T|u_{n,s}(\mathbf{k})\rangle = |u_{n,-s}(-\mathbf{k})\rangle \qquad (7.27)$$

for its eigenstates since $T$ changes $\mathbf{k}$ to $-\mathbf{k}$ and $s$ to $-s$. This yields a symmetry for the spectrum where the energy of the state $|u_{n,-s}(-\mathbf{k})\rangle$ is necessarily the same as that of the state $|u_{n,s}(\mathbf{k})\rangle$, i.e.

$$E_{n,s}(\mathbf{k}) = E_{n,-s}(-\mathbf{k}). \qquad (7.28)$$

In view of this result, two comments need to be made.

1. The bands are not necessarily twofold spin degenerate at every wave vector $\mathbf{k}$: at each wave vector, there are indeed two states per band for the spin $s = +$ and $s = -$, but without having necessarily $E_{n,+}(\mathbf{k}) = E_{n,-}(\mathbf{k})$. This latter global spin degeneracy of the bands is not enforced by Kramer's theorem even if we have considered until now bands obtained from tight-binding models in which the hopping parameters do not depend on the spin orientation. However, this is no longer the case in systems where the direction of propagation depends explicitly on the spin orientation. While it may seem unusual at first sight, this situation is quite commonly encountered in a large class of materials with strong spin-orbit coupling. This phenomenon will be briefly discussed a little bit later in this section.

<footnote n="6">⁶Naturally, the argument does not apply to bosonic states $|\phi\rangle$, where $T^2|\phi\rangle = +|\phi\rangle$. The argument would then yield $|c|^2 = 1$, which is not a contradiction.</footnote>
<figure><img src="image_65.png" /><figcaption>Figure 7.1: Illustration of Kramers’ theorem for Bloch bands in time-reversal-symmetric systems. While the twofold spin degeneracy of the bands can generally be lifted, Kramer’s theorem constrains the two spin branches (blue for spin ↑ and red for spin ↓) of each band to cross at special points in the first Brillouin zone that are themselves invariant under time-reversal symmetry, called time-reversal-invariant momenta (TRIM). These TRIM are the Γ point at the centre of the first Brillouin zone and the points G/2 at the zone borders. Furthermore, the spin branches respect the symmetry $E_{n,↑}(\mathbf{k}) = E_{n,↓}(-\mathbf{k})$.</figcaption></figure>

2. The index $s$ normally denotes the spin, $s = ↑, ↓$, along an (arbitrary) quantisation axis. However, Kramer’s theorem is not necessarily re-stricted to the physical spin. More generally, it denotes an internal degree of freedom that can take two values, also called a $\mathbb{Z}_2$ index. In the case of materials with spin-orbit coupling, this $\mathbb{Z}_2$ index can gen-erally denote a more complex quantity than the spin since the latter (or more precisely its quantisation axis) may vary as a function of the wave vector. Having this issue in mind, we nevertheless consider this degree of freedom to be the $z$-component of the physical spin in order to avoid further complications in the physical discussion.<fnref n="7" />

However, there are some special points in the first Brillouin zone, where the bands are necessarily twofold degenerate if time-reversal symmetry is respected, i.e. if the spectrum satisfies the condition (7.28). The first one of these points is readily identified: since at these special points we have $\mathbf{k} = -\mathbf{k}$, it is the $\Gamma$ point at the centre ($\mathbf{k} = 0$) of the first Brillouin zone, where $E_{n,s}(0) = E_{n,-s}(0)$. In other words, even if the spin degeneracy of the electronic bands is generally lifted, the two spin branches necessarily cross at this point. Is it the only crossing point of the two spin branches? Recall

<footnote n="7">⁷Notice that this is indeed the correct $\mathbb{Z}_2$ index in the case of an *intrinsic* spin-orbit coupling in the 2D systems under discussion in this section.</footnote>
that the spectrum is periodic in reciprocal space and two points are therefore equivalent if they are related by a reciprocal lattice vector,

$$k \sim k + G, \quad E_{n,s}(k) = E_{n,s}(k + G). \quad (7.29)$$

The bands therefore have crossing points at the wave vectors

$$k_{\text{TRIM}} = -k_{\text{TRIM}} + G \quad \Leftrightarrow \quad k_{\text{TRIM}} = \frac{G}{2} \quad (7.30)$$

at the borders of the first Brillouin zone. We may thus identify a total of four (in 2D) or eight (in 3D) time-reversal invariant momenta (TRIM) in the first Brillouin zone. Kramers' theorem imposes a double degeneracy or, equivalently, a band crossing at these TRIM, as it is represented schematically in figure 7.1. To recapitulate, Kramers' theorem for electronic bands states that the spin degeneracy in each band can in principle be lifted, even in the absence of a magnetic field, but the two spin branches of each band necessarily cross at the TRIM unless time-reversal symmetry is broken.

## Time-reversal symmetry in systems without spin-orbit coupling

The reader may now legitimitely raise the question whether the situation described above, where the spin degeneracy of the bands is globally lifted apart from the TRIM, actually exists in real materials. Remember that this is not the case in the systems we have studied so far, be it the tight-binding description of the square and triangular lattices or that of graphene (see chapter 2): there, the hopping parameters do not depend on the electronic spin. Therefore, we have

$$H_{\uparrow}(\mathbf{k}) = H_{\downarrow}(\mathbf{k}), \quad (7.31)$$

so that all bands are globally twofold spin-degenerate, $E_{\uparrow}(\mathbf{k}) = E_{\downarrow}(\mathbf{k})$, for any lattice vector $\mathbf{k}$. In this case, time-reversal symmetry (7.26) imposes the identity

$$T H_s(\mathbf{k}) T^{-1} = H^*_{-s}(-\mathbf{k}) = H^*_s(-\mathbf{k}) = H_{-s}(\mathbf{k}) \quad (7.32)$$

and thus

$$E_{n\uparrow}(\mathbf{k}) = E_{n,\downarrow}(\mathbf{k}) = E_{n,\uparrow}(-\mathbf{k}) = E_{n,\downarrow}(-\mathbf{k}) \quad (7.33)$$

for the spectrum. Each of the bands $n$ is therefore symmetric under $\mathbf{k} \to -\mathbf{k}$ and twofold spin-degenerate at any wave vector.
A hint of spin-orbit coupling

How to escape this rather boring situation where the electronic bands are globally spin-degenerate? Indeed, such a global degeneracy prevents us from obtaining topologically non-trivial band structures in time-reversal-symmetric systems, as we discuss in detail below. From the above argumentation, one sees that we would require spin-up electrons to move differently in the crystal than spin-down electrons. Within a tight-binding description, this amounts to having *spin-dependent* hopping parameters that are provided precisely by the *spin-orbit coupling*. Without going into the details – a full description of the spin-orbit coupling and its manifestation in tight-binding models is far beyond the scope of the present lecture notes – the effect finds its origin in relativistic quantum mechanics, namely in the Dirac equation. In relativistic quantum mechanics, the Dirac equation plays the same central role as the Schrödinger equation in non-relativistic quantum physics. Among its most amazing implications is the appearance of negative-energy solutions that have soon be interpreted as *anti-particles* and that can to some extent be viewed as the high-energy analogue of holes in a valence band. In the presence of an electromagnetic field, the original negative- and positive-energy solutions get mixed. However, the fields are usually associated with an energy scale that is much smaller than the gap between the particle and the anti-particle branches<fnref n="8" /> so that the electromagnetic field can be treated within perturbation theory. The magnetic field then yields the well-known Zeeman contribution in the low-energy Hamiltonian describing to lowest order the electron branch. Similarly the electric field gives rise to a term

$$H_{SO} = \lambda_{SO}(\mathbf{p} \times \mathbf{E}) \cdot \boldsymbol{\sigma}, \qquad (7.34)$$

where $\boldsymbol{\sigma}$ represents the physical spin, and the constant $\lambda_{SO}$ is the coupling strength. The expression (7.34) is precisely the spin-orbit Hamiltonian and can be interpreted roughly as a Zeeman term generated by a *magnetic* field $\mathbf{B}' \sim \mathbf{p} \times \mathbf{E}$.<fnref n="9" /> Notice finally that the spin-orbit term (7.34) respects time-reversal symmetry, $TH_{SO}T^{-1} = H_{SO}$, since it is proportional to the product

<footnote n="8">⁸Remember that this gap is on the order of a MeV= $10^6$ eV for electrons and positrons.</footnote>

<footnote n="9">⁹The above-mentioned $Z_2$ quantum number arises naturally from the diagonalisation of the spin-orbit term. It is simply the z-component of the spin, $s = \uparrow, \downarrow$, if both the motion is restricted to the 2D $xy$-plane as well as the variation of the electrostatic potential, which is at the origin of the electric field. In this case, the vector product $\mathbf{p} \times \mathbf{E}$ is oriented in the z-direction, which is then the natural quantisation axis for the spin. This remains valid in the case of an atomic spin-orbit coupling, generated by a scalar product $\boldsymbol{\sigma} \cdot \mathbf{L}$ between the</footnote>
of the electron's spin and its momentum, which are both odd under $T$ while the electric field $E$ is even, and $\lambda_{SO}$ is a real constant to ensure the spin-orbit term to be Hermitian.

### 7.1.2 Berry curvature in time-reversal-symmetric systems

We finish this section about time-reversal symmetry with an analysis of its impact on the Berry curvature, which we have identified as a basic ingredient in the description of 2D topological materials. Let us first investigate a system with no spin-orbit coupling, e.g. a tight-binding model in which the hopping parameters do not depend on the electronic spin. In this case, we may simply omit the spin index, as we have intuitively done in the discussion of the tight-binding models in the introductory chapter 2, and account for it as a global degeneracy factor of 2 when it comes to investigating its conduction or other statistical properties. The eigenstates associated with a band $E_n(\mathbf{k})$ are then transformed under the action of $T$ as

$$|u_n(-\mathbf{k})\rangle = T|u_n(\mathbf{k})\rangle = |u_n(\mathbf{k})\rangle^*, \qquad (7.35)$$

since the time-reversal-symmetry operator is reduced to a simple complex conjugation. Consequently, the Berry curvature (6.20) respects

$$ \begin{aligned} \mathcal{B}_n^z(-\mathbf{k}) &= i \langle \partial_{k_x} u_n(\mathbf{k}) | \partial_{k_y} u_n(\mathbf{k}) \rangle^* - k_x \leftrightarrow k_y \qquad (7.36) \\ &= i \langle \partial_{k_y} u_n(\mathbf{k}) | \partial_{k_x} u_n(\mathbf{k}) \rangle - k_x \leftrightarrow k_y = -\mathcal{B}_n^z(\mathbf{k}), \end{aligned} $$

where we have used the property $\langle\phi|\psi\rangle^* = \langle\psi|\phi\rangle$ of the scalar product. Our straight-forward calculation shows that, in the absence of spin-orbit coupling, the Berry curvature of any single isolated band is an antisymmetric function of the wave vector in time-reversal symmetric systems. As a consequence,

<footnote>spin $\sigma$ and the atomic orbital momentum $\mathbf{L}$, in which case the $\mathbb{Z}_2$ quantum number may refer to the coupled angular momenta $\sigma + \mathbf{L}$. However, there may be situations, also in 2D electronic systems, where the electrostatic potential varies in the direction perpendicular to the $xy$-plane, e.g. if the 2D system is put on a substrate or encapsulated by different materials below and above. In this case, one encounters the so-called *Rashba* spin-orbit coupling, in which case the $\mathbb{Z}_2$ quantum number is a more complex object than the simple $z$-component of the spin. In the remainder of these notes, we will however stick to the simpler situation where the $\mathbb{Z}_2$ quantum number is identified with the $z$-component of the electronic spin.</footnote>
the Chern number of each of these bands is zero since it is the integral of the Berry curvature over the entire Brillouin zone. The Hall conductivity [see equation (6.76)] is equally zero, and a time-reversal-symmetric system thus cannot display a quantum anomalous Hall effect.

Let us know investigate the Berry curvature in a time-reversal-symmetric system where the spin degeneracy is lifted by a non-vanishing spin-orbit coupling. At first sight, it might seem senseless to calculate a Berry curvature for each of the spin branches of an electronic band $n$ since they necessarily cross at the TRIM, as we have discussed above (see figure 7.1), in contrast to our assumption of adiabatic transport. Notice however that because of the $\mathbb{Z}_2$ quantum number, i.e. because the $z$-component of the spin is a good quantum number here, one may diagonalise the Hamiltonian in each spin sector. Transitions with a spin change are therefore discarded, and we may sensefully associate a Berry curvature with each of the spin branches of a band $n$. In this case, equation (7.27) implies

$$ \mathcal{B}_{n,-s}^{z}(-\mathbf{k}) = i \left( \langle \partial_{k_x} T u_{n,s}(\mathbf{k}) | \partial_{k_y} T u_{n,s}(\mathbf{k}) \rangle - k_x \leftrightarrow k_y \right), \quad (7.37) $$

and we may use the property

$$ \langle T\phi|T\psi\rangle = \langle\psi|\phi\rangle \quad (7.38) $$

for an anti-unitary operator.<fnref n="10" /> Since the derivative with respect to the wave vector $k_{x/y}$ commutes with $T$, one finds

$$ \begin{aligned} \mathcal{B}_{n,-s}^{z}(-\mathbf{k}) &= i \left( \langle T \partial_{k_x} u_{n,s}(\mathbf{k}) | T \partial_{k_y} u_{n,s}(\mathbf{k}) \rangle - k_x \leftrightarrow k_y \right) && (7.39) \\ &= i \left( \langle \partial_{k_y} u_{n,s}(\mathbf{k}) | \partial_{k_x} u_{n,s}(\mathbf{k}) \rangle - k_x \leftrightarrow k_y \right) = -\mathcal{B}_{n,s}^{z}(\mathbf{k}). \end{aligned} $$

$$ \langle T\phi|T\psi\rangle = \sum_j (T\phi)_j^*(T\psi)_j, $$

with $(T\psi)_j = (UK\psi)_j = \sum_k U_{j,k} \psi_k^*$ and $(T\phi)_j^* = (UK\phi)_j = \sum_l U_{j,l}^* \phi_l$, where we recall that the operator $T$ consists of a unitary operator $U$ times complex conjugation $K$. The scalar product may therefore be rewritten as

$$ \langle T\phi|T\psi\rangle = \sum_{j,k,l} \psi_k^* U_{j,l}^* U_{j,k} \phi_l = \sum_{j,k,l} \psi_k^* (U^\dagger)_{l,j} U_{j,k} \phi_l = \sum_{k,l} \psi_k^* \delta_{k,l} \phi_l = \langle\psi|\phi\rangle, $$

where we have made use of the unitarity $U^\dagger = U^{-1}$ and $\sum_j (U^{-1})_{l,j} U_{j,k} = \delta_{k,l}$.

<footnote n="10">¹⁰We have already used implicitly this property in the spinless case, where the operator $T$ is reduced to complex conjugation, but it is valid more generally. In order to see this, let us consider the scalar product in an orthonormal basis,</footnote>
Time-reversal symmetry therefore does not impose the anti-symmetry of the Berry curvature in each spin branch of the band $n$, but it relates the curvatures of the two branches of opposite spin. Most saliently, the integral of the Berry curvature $B_{n,s}(\mathbf{k})$ over a symmetric area in reciprocal space does no longer necessarily vanish, and the associated Chern number

$$C_{n,s} = \int_{1BZ} \frac{d^2k}{2\pi} \mathcal{B}_{n,s}^z(\mathbf{k}) \qquad (7.40)$$

can now be non-zero, in contrast to the case without spin-orbit coupling. Notice, however, that the property (7.39) of the Berry curvature implies

$$C_{n,\uparrow} = -C_{n,\downarrow}, \qquad (7.41)$$

i.e. the Chern numbers for the two spin branches of the band $n$ are opposite. Therefore, if the two spin branches of the band $n$ are completely filled, as it is the case for an insulator, the two spin contributions to the Hall conductivity sum up to zero,

$$\sigma_H = \frac{e^2}{h} (C_{n,\uparrow} + C_{n,\downarrow}) = 0, \qquad (7.42)$$

as it was the case in the absence of spin-orbit coupling. In conclusion, time-reversal symmetry prohibits a (quantum) anomalous Hall effect. Notice that this is also the case if the band $n$ is not completely filled. While it is not the entire first Brillouin zone that is then occupied by electrons in the band $n$, the contributions from states $\mathbf{k}$ and $-\mathbf{k}$ cancel each other. Indeed, if a state $\mathbf{k}$ with energy $E_{n,\uparrow}(\mathbf{k})$ is occupied by a spin-$\uparrow$ electron, so is the state $-\mathbf{k}$ with a spin-$\downarrow$ electron since they have the same energy $E_{n,\downarrow}(\mathbf{k}) = E_{n,\uparrow}(-\mathbf{k})$, but opposite Berry curvature.

In contrast to the transverse electronic current, which must be zero, it is possible to have a non-zero transverse spin current, in which case the current densities of the two spin branches are not added but subtracted. Following the lines of section 6.5.1 of the preceding chapter, one may have the following transverse spin current density

$$\mathbf{j}_{\text{spin}}^{H} = \frac{\hbar}{2S} \sum_{\text{occ.}} [\mathbf{v}_{n,\uparrow}(\mathbf{k}) - \mathbf{v}_{n,\downarrow}(\mathbf{k})] = \frac{1}{4\pi} \vec{\mathcal{E}} \times \int_{\text{occ.}} \frac{d^2k}{2\pi} [\vec{\mathcal{B}}_{n,\uparrow}(\mathbf{k}) - \vec{\mathcal{B}}_{n,\downarrow}(\mathbf{k})], \qquad (7.43)$$

where we have made use of equation (6.74),<fnref n="11" /> taking into account only the transverse part of the densities that depends on the Berry curvature. For a

<footnote n="11">¹¹The prefactor is different and obtained by substituting $-e \to \hbar/2$, simply because we are dealing with spin and not with charge transport here.</footnote>
time-reversal-symmetric system, the symmetry property (7.39) of the Berry curvature yields the spin-Hall conductivity

$$ \sigma_{\text{spin}}^{H} = \frac{1}{2\pi} \int_{\text{occ.,}\uparrow} \frac{d^2k}{2\pi} \mathcal{B}_{n,\uparrow}(\mathbf{k}) \qquad (7.44) $$

and, in the case of an insulator with $n$ occupied bands, the quantised value

$$ \sigma_{\text{spin}}^{H} = \frac{1}{2\pi} \sum_{n \text{ occ.}} C_{n,\uparrow}, \qquad (7.45) $$

which is at the origin of the *quantum spin Hall effect*, which we discuss in detail in section 7.4.3.

## 7.2 Haldane model – a topological insulator without time-reversal symmetry

Based on the preceding discussion, we have understood that topologically interesting bands require (i) a broken time-reversal symmetry or (ii) materials with a sufficient spin-orbit coupling to lift the global spin degeneracy of the bands. Even if the second situation requires a more complex theoretical description, it is more often encountered in real materials than the first one, which necessitates generically some form of magnetism to break time-reversal symmetry. However, due to its relative conceptual simplicity, we first discuss the so-called *Haldane model*,<fnref n="12" /> which belongs to the first class of systems with an explicitly broken time-reversal symmetry. The model is nothing other than our tight-binding model for (gapped) graphene, discussed in section 2.3 in which additional second-nearest neighbour hopping is taken into account, in the form of the complex hopping amplitude $t' \exp(\pm i\phi)$. It is precisely the complex character of the hopping amplitude that is at the origin of a broken time-reversal symmetry, for $\phi \neq 0, \pi$.

Before presenting the model step by step, let us first give a brief motivation for it. Indeed, the Haldane model may be viewed as a model system which allows one to describe in a simple way a topological phase transition triggered by a control parameter. We have already seen, in section 6.4.1,

<footnote n="12">¹²The model was first proposed by the theoretician Haldane in 1988.</footnote>
<figure><img src="image_66.png" /><figcaption>Figure 7.2: Topological phase transition (reproduction of figure 6.7). We consider to possess a theoretical knob (a control parameter) that allows us to change continuously the gap parameter $\Delta_{\xi=+}$ in the valley $\xi = +$ while the “spectator” valley ($\xi = -$) is not affected by this change.</figcaption></figure>
that such a transition can be described in the framework of low-energy massive Dirac fermions, as it is sketched out in figure 7.2 (reproducing figure 6.7 of a topological phase transition). Remember that a broken time-reversal symmetry allows us to modify the (mass) gap of the Dirac fermion in the valley $\xi$ independently of that in the valley $-\xi$, because we do not have $E_n(-\mathbf{k}) = E_n(\mathbf{k})$ then. This is possible due to the complex second-nearest neighbour hopping, and we shall see that the phase $\phi$ plays precisely the role of the control parameter that allows us to drive the topological phase transition.

### 7.2.1 Tight-binding model

As mentioned above, the Haldane model is a variant of the tight-binding model for gapped graphene or boron nitride. Again the honeycomb lattice imposes a 2 × 2-matrix structure of the Bloch Hamiltonian

$$H(\mathbf{k}) = \begin{pmatrix} h_{AA}(\mathbf{k}) & h_{AB}(\mathbf{k}) \\ h_{BA}(\mathbf{k}) & h_{BB}(\mathbf{k}) \end{pmatrix}, \qquad (7.46)$$

where the off-diagonal matrix elements

$$h_{BA}(\mathbf{k}) = h_{AB}(\mathbf{k})^* = -t(1 + e^{i\mathbf{k}\cdot\mathbf{a}_1} + e^{i\mathbf{k}\cdot\mathbf{a}_2})$$

are identical to those of the graphene Bloch Hamiltonian since they represent nearest-neighbour hopping between sites on the different sublattices. However, the diagonal elements differ from the graphene Hamiltonian: in addition to the gap parameter $\Delta$, which takes into account the difference in the on-site energy on the two different sublattices (see section 2.3), Haldane considered also directional hoppings between second-nearest neighbours on the same sublattice. The directional character of these hoppings is represented by the green arrows in figure 7.3 and respects the following rule: for a hopping event from a site $k$ to $l$, the electron acquires a phase $\phi$ if the arrow points from $k$ to $l$ while it takes a phase $-\phi$ for a hopping in the direction opposite to that of the arrow (see left panel of figure 7.3).

Before we discuss the physical origin of these complex hoppings, in terms of Aharonov-Bohm phases, let us just use the above rules as the rules of a game in which the arrows are arranged in a pattern represented in the topmost right hexagon of the right panel in figure 7.3. The arrows are oriented in a clockwise manner around the internal hexagon delimited by the
<figure><img src="image_67.png" /><figcaption>Figure 7.3: Haldane model. Left: when hopping from a site $k$ to $l$ the electron acquires a phase $\phi$ and $-\phi$ for the inverse hopping from $l$ to $k$. Right: representation of the second-nearest-neighbour hoppings on the honeycomb lattice, where we have added a third (redundant) Bravais-lattice vector $\mathbf{a}_3 = \mathbf{a}_1 - \mathbf{a}_2$.</figcaption></figure>

second-nearest-neighbour hoppings. Let us now study its implications for the contribution to the tight-binding model, where we concentrate on hoppings between sites on the B sublattice. Consider in a first step hoppings along the vectors $\mathbf{a}_1$ and $-\mathbf{a}_1$. According to our discussion of tight-binding models in section 2.2.1 [see equation (2.36)], the hopping along $\mathbf{a}_1$ yields the term $t' \exp(i\mathbf{k} \cdot \mathbf{a}_1) \exp(-i\phi)$ – the first phase term is just the usual phase acquired by the wave function in a translation by $\mathbf{a}_1$ while the second one is the new one according to the rules exposed above. Since the arrow is pointing in the direction opposite to $\mathbf{a}_1$, the phase comes along with a minus sign. Similarly, the hopping along $-\mathbf{a}_1$ gives $t' \exp(-i\mathbf{k} \cdot \mathbf{a}_1) \exp(+i\phi)$, and both contributions add up to yield a term

$$2t' \cos(\mathbf{k} \cdot \mathbf{a}_1 - \phi) \qquad (7.47)$$

in the matrix element $h_{BB}(\mathbf{k})$ of the Bloch Hamiltonian. Let us continue with the contributions which arise due to hopping in the directions $\mathbf{a}_2$ and $\mathbf{a}_3 = \mathbf{a}_1 - \mathbf{a}_2$. The hoppings along these two directions are oriented in the same direction as the green arrows and thus come along with a positive phase contribution, and they add up to

$$2t' [\cos(\mathbf{k} \cdot \mathbf{a}_2 + \phi) + \cos(\mathbf{k} \cdot \mathbf{a}_3 + \phi)] \qquad (7.48)$$

once we also take into account hoppings along the opposite directions $-\mathbf{a}_2$ and $-\mathbf{a}_3$, respectively. In order to write the full matrix element $h_{BB}(\mathbf{k})$ in a
compact manner, we redefine the second-nearest-neighbour vectors as

$$ \mathbf{b}_1 = -\mathbf{a}_1, \quad \mathbf{b}_2 = \mathbf{a}_2, \quad \mathbf{b}_3 = \mathbf{a}_3, \qquad (7.49) $$

such that one obtains<fnref n="13" />

$$ \begin{aligned} h_{BB}(\mathbf{k}) &= -\Delta + 2t' \sum_{j=1}^{3} \cos(\mathbf{k} \cdot \mathbf{b}_j + \phi) && (7.50) \\ &= -\Delta + 2t' \sum_{j=1}^{3} [\cos(\mathbf{k} \cdot \mathbf{b}_j) \cos(\phi) - \sin(\mathbf{k} \cdot \mathbf{b}_j) \sin(\phi)], \end{aligned} $$

where, in addition to hopping, we have also taken into account the onsite energy $-\Delta$ on the sites of the B sublattice.

Now that we have obtained the expression for the matrix element $h_{BB}(\mathbf{k})$ associated with the B sublattice, it is simple to obtain that, $h_{AA}(\mathbf{k})$, on the A sublattice. Indeed, with respect to the B sublattice, the green arrows are inverted so that hoppings in the direction $\mathbf{a}_1$ are in the same direction as that of the green arrows, while they are in the opposite direction for $\mathbf{a}_2$ and $\mathbf{a}_3$. Consequently the relative signs in the arguments of the cosines are inverted, and one finds

$$ \begin{aligned} h_{AA}(\mathbf{k}) &= \Delta + 2t' \sum_{j=1}^{3} \cos(\mathbf{k} \cdot \mathbf{b}_j - \phi) && (7.51) \\ &= \Delta + 2t' \sum_{j=1}^{3} [\cos(\mathbf{k} \cdot \mathbf{b}_j) \cos(\phi) + \sin(\mathbf{k} \cdot \mathbf{b}_j) \sin(\phi)] \end{aligned} $$

for the corresponding matrix elements of the Bloch Hamiltonian, where we have taken into account again the onsite energy on A-sublattice sites, $-\Delta$. The Bloch Hamiltonian of the Haldane model is now readily decomposed in the basis of Pauli matrices,

$$ H(\mathbf{k}) = h_0(\mathbf{k}) \mathbb{I}_{2 \times 2} + \vec{h}(\mathbf{k}) \cdot \vec{\sigma}, \qquad (7.52) $$

where $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$, and $\mathbb{I}_{2 \times 2}$ is the $2 \times 2$ unit matrix here. Furthermore,

<footnote n="13">¹³The rewriting of the lattice vectors allows us to have the same relative sign between $\mathbf{k} \cdot \mathbf{b}_j$ and the phase $\phi$ in all three terms and thus a simpler notation.</footnote>
we have defined the functions

$$
\begin{align}
h_0(\mathbf{k}) &= 2t' \sum_{j=1}^{3} \cos(\mathbf{k} \cdot \mathbf{b}_j) \cos(\phi), \nonumber \\
h_x(\mathbf{k}) &= -t [1 + \cos(\mathbf{k} \cdot \mathbf{a}_2) + \cos(\mathbf{k} \cdot \mathbf{a}_3)], \nonumber \\
h_y(\mathbf{k}) &= -t [\sin(\mathbf{k} \cdot \mathbf{a}_2) + \sin(\mathbf{k} \cdot \mathbf{a}_3)], \nonumber \\
\text{and} \quad h_z(\mathbf{k}) &= \Delta + 2t' \sum_{j=1}^{3} \sin(\mathbf{k} \cdot \mathbf{b}_j) \sin(\phi). \tag{7.53}
\end{align}
$$

This decoposition has two advantages. First, one may directly read off the
spectrum (the two bands)

$$
E_{\lambda}(\mathbf{k}) = h_0(\mathbf{k}) + \lambda |\vec{h}(\mathbf{k})|, \qquad (7.54)
$$

where $\lambda = \pm$ is again the band index. Second, it allows us to see easily
that the Haldane model breaks time-reversal symmetry. Indeed, if the model
had this symmetry, equation (7.32), $H^*(-\mathbf{k}) = H(\mathbf{k})$, would impose that
the functions $h_0(\mathbf{k})$, $h_x(\mathbf{k})$, and $h_z(\mathbf{k})$ be even in the wave vector $\mathbf{k}$, while
$h_y(\mathbf{k}) = -h_y(-\mathbf{k})$ would need to be odd because of the imaginary character
of the Pauli matrix $\sigma_y$. One notices that all functions respect this symmetry
apart from $h_z(\mathbf{k})$, which is even only if $\phi = 0$ or $\phi = \pi$ so that the second-
nearest-neighbour hoppings would be real. However, for a general value of
$\phi$, this is not the case, and time-reversal symmetry is therefore broken.

The evolution of the energy bands upon variation of t', for fixed values<fnref n="14" />
of the phase φ = π/3 and the gap parameter Δ = t/2, is shown in figure 7.4.
Even if the chosen value of the phase generically breaks time-reversal sym-
metry, as mentioned in the preceding paragraph, the sectrum is nevertheless
symmetric at t' = 0 [panel (a)], for which we retrieve the bands of boron ni-
tride (or gapped graphene) discussed in section 2.3 (see figure 2.7). However,
for any non-zero value of t', this symmetry is clearly broken, E(-k) ≠ E(k),
as one may notice from figure 7.4(b), where the bands are shown for a value
t'/t = 0.07. Indeed, the gap is smaller in the K than in the K' valley and
is closed for a value t'/t = 1/9 [see figure 7.4(c)]. This value marks the
topological phase transition, which we generically discussed in section 6.4.1
and that is schematically shown in figures 6.7 or 7.2. The gap parameter in
the K' valley only changes marginally and does not invert its sign so that

<footnote n="14">¹⁴These values are arbitrary and just chosen to illustrate the topological phase transition.</footnote>
<figure><img src="image_68.png" /><figcaption>Figure 7.4: Evolution of the energy bands in the Haldane model upon variation of the second-nearest-neighbour hopping strength. The energies are plotted in units of $t$, and we have chosen a gap parameter $\Delta/t = 0.5$ and a phase $\phi = \pi/3$, for pure illustration. The principle figures show the bands for $k_y = 0$ while the insets provide a glimps at the full band structure, as a function of $k_x$ and $k_y$. (a) For $t' = 0$, we find the same (time-reversal-symmetric) spectrum as that of boron nitride discussed in section 2.3. (b) For $t'/t = 0.07$, the gap in the valley K is decreased with respect to that at K', the latter being the spectator valley. (c) The gap is closed at $t'/t = 1/9 \approx 0.11$. (d) At $t'/t = 0.18$ the gap is again opened, albeit with an inverted sign.</figcaption></figure>
$K'$ is our spectator valley and does not play a role in the topological phase transition. Panel (d) then shows the spectrum for $t'/t = 0.18$, where the gap parameter of the active valley $K$ has changed its sign and where we thus obtain a Chern insulator. Indeed, the Chern numbers of the two bands for $t'/t < 1/9$ are continuously connected to the time-reversal-symmetric model of boron-nitride ($t' = 0$ here), where the Chern numbers of each of the bands are zero. At the topological phase transition only the contribution of the $K$ valley to the total Chern number of each band changes the sign: the half-Chern number changes from $\tilde{C}_K = \pm 1/2$ to $\mp 1/2$ so that the total Chern number of the bands is now $C = \pm 1$. The full discussion and the quantitative determination of the Chern numbers, within a low-energy model, is the object of the following section 7.2.2.

## Physical origin of the phase in the hopping parameters

Before discussion in more detail the topological phase transition of the Haldane model and the transition line in parameter space, let us get back to the physical meaning of the phase $\phi$, which characterises the second-nearest-neighbour hoppings and that is at the origin of the broken time-reversal symmetry. Until now, it has only been a mathematical curiosity used in some “game rules”. However, from a physical point of view, we have already encountered a situation in which an electron acquires an additional phase in a displacement, in addition to the usual phase $k \cdot r$. It occurs in the Aharonov-Bohm effect, which we have briefly invoked in the introduction. In the presence of a magnetic field, or more precisely a vector potential $A$ that yields a non-zero flux trapped inside a closed path, the additional Aharonov-Bohm phase acquired on a path from a position $k$ to another one $l$ is given by $\phi_{k,l} = (e/\hbar) \int_k^l d\mathbf{r} \cdot \mathbf{A}$. Indeed, for a closed path $\partial\Sigma$ enclosing an area $\Sigma$, the total phase is given by the total flux $\Phi/(h/e) = (e/h) \int_{\partial\Sigma} d\mathbf{r} \cdot \mathbf{A}$ threading the area in units of the flux quantum $\phi_0 = h/e$. When Haldane proposed his model, he considered the rather unconventional situation where the total flux threading each hexagon of the lattice is vanishing, since there is no external magnetic field, while there is nevertheless an inhomogeneous flux inside. Physically, Haldane argued that such a situation could be encountered in the presence of an adatom at the centre of each hexagon that carries a magnetic moment.<fnref n="15" />

<footnote n="15">¹⁵Notice that at the moment, where these lines are written, several experimental groups try to produce such a situation in graphene with additional magnetic atoms.</footnote>
<figure><img src="image_69.png" /><figcaption>Figure 7.5: Different contributions to the total flux threading a hexagon of the honeycomb lattice in the Haldane model.</figcaption></figure>

In order to convince ourselves that the model described above with the
directional hopping and the acquired phases corresponds to such a situation
with a net zero flux per hexagon, we may cut the green arrows in figure 7.3
connecting second-nearest neighbours into three equal parts each contribut-
ing a phase $\phi' = \phi/3$ to the total phase.<fnref n="16" /> We can now calculate the flux
contributions of the different parts of a hexagon as shown in figure 7.5. Let
us start with the central hexagon delimited by the blue arrows (inside the
black hexagon, which itself represents the unit cell of the honeycomb lat-
tice with the lattice sites at its corners). All six arrows of the central blue
hexagon are oriented in the same (clockwise) direction and contribute each
$\phi' = \phi/3$ to the flux so that the flux through this hexagon is $6\phi' = 2\phi$. In
order to get a zero average flux it must be compensated by the other parts
that complete the black hexagon. There are two other types of parts, the six
equilateral triangles delimited by the red arrows (in addition to a blue arrow
for each triangle) and the six outer triangles delimited by the borders of the
black hexagon and the green arrows. The six equilateral triangles contribute
each a flux of $\phi'$ since the red arrows are oriented in a clockwise manner
and thus yield a phase of $2\phi'$ while the single blue arrow is oriented in an
anti-clockwise manner such as to contribute a phase of $-\phi'$. The second type
of triangles contribute a flux of $-2\phi'$ to the total flux since only the green

<footnote n="16">¹⁶This choice is not unique, and there exists other equivalent ones. It is a so-called *gauge choice*.</footnote>
arrows, which are oriented in an anti-clockwise manner around the triangles, carry a phase of $-\phi'$ each.<fnref n="17" /> Summing up all contributions to the flux, we obtain indeed a zero net flux through the unit cell represented by the black hexagon, $\Phi = 0$, and thus a zero flux threading the full 2D system as one would expect for a system in the absence of a magnetic field.

### 7.2.2 Low-energy model

As mentioned in the previous section, the Haldane model displays a topological phase transition that we have illustrated in figure 7.4 for a particular choice of parameters, where we have varied the strength of the second-nearest-neighbour hopping. In order to obtain the full transition line in parameter space separating a trivial from a topological (Chern) insulator or the phase diagram, we expand to lowest order the Bloch Hamiltonian of the Haldane model around the two valleys $K$ and $K'$. Since the off-diagonal terms are identical to those of graphene discussed in section 2.3 of chapter 2, we can simply copy them from equation (2.71),

$$h_{AB}(\xi \mathbf{K} + \mathbf{q}) = h_{BA}^*(\xi \mathbf{K} + \mathbf{q}) = \hbar v(\xi q_x - i q_y), \quad (7.55)$$

with $v = 3ta/2\hbar$. As for the diagonal terms, we may limit the discussion to the term $h_z(\mathbf{k})$ since the term $h_0(\mathbf{k})$ represents only an energy offset of the bands and does not bare topological information: since it comes along with the $I_{2 \times 2}$ matrix it is diagonal in any basis. The lowest-order term (at zero order) in the expansion then reads

$$h_z(\xi \mathbf{K}) = \Delta + 2t' \sum_{j=1}^{3} \sin(\xi \mathbf{K} \cdot \mathbf{b}_j) \sin(\phi) = m_{\xi} v^2, \quad (7.56)$$

with the *effective gap parameter*

$$m_{\xi} v^2 = \Delta - \xi 3 \sqrt{3} t' \sin(\phi), \quad (7.57)$$

where we have used

$$\mathbf{K} \cdot \mathbf{a}_1 = -\mathbf{K} \cdot \mathbf{a}_2 = \frac{2\pi}{3} \quad \text{et} \quad \mathbf{K} \cdot \mathbf{a}_3 = \frac{4\pi}{3} \quad (7.58)$$

<footnote n="17">¹⁷The black borders of the second type of triangles do not contribute since they represent nearest-neighbour hopping between sites on different sublattices that do not carry an Aharonov-Bohm phase for our gauge choice.</footnote>
[see the equations (2.67) of section 2.3], $K' = -K$ and

$$ \sin(\xi 2\pi/3) = -\sin(\xi 4\pi/3) = \xi \sqrt{3}/2. $$

We eventually find again a model of massive Dirac fermions in the vicinity of the points $K$ ($\xi = +$) and $K'$ ($\xi = -$) described in terms of the Hamiltonian

$$ H_{\xi}^{H}(\mathbf{q}) = \begin{pmatrix} m_{\xi}v^2 & \hbar v(\xi q_x - i q_y) \\ \hbar v(\xi q_x + i q_y) & -m_{\xi}v^2 \end{pmatrix}. \qquad (7.59) $$

It is clear from this expression and equation (7.57) that the gap parameter is explicitly *valley-dependent*. Recall that this dependence of the gap parameter on the valley index is a direct consequence of the broken time-reversal symmetry. One also notices from equation (7.57) that one may change the value and the sign of the (effective) gap parameter by varying the hopping amplitude $t'$ or the phase $\phi$. If we choose and arbitrary value of the phase $\phi$ in the interval $0 < \phi < \pi$, the gap in the $K$ valley changes sign at a value<fnref n="18" />

$$ t'_{c} = \frac{\Delta}{3\sqrt{3}\sin(\phi)} \qquad (7.60) $$

while the gap in the $K'$ valley the gap remains positive for all (positive) values of $t'$ as long as the gap parameter $\Delta$ is positive.<fnref n="19" /> In order to see that the values satisfying equation (7.60), which describe critical lines in the $\Delta - \phi$ diagram, are indeed associated with a topological phase transition, recall from equation (6.60) that the Chern number of the band $\lambda$ is given by the sum of the contributions from both valleys,

$$ C_{\lambda} = -\frac{\lambda}{2}[\operatorname{sgn}(m_{K}) - \operatorname{sgn}(m_{K'})]. \qquad (7.61) $$

As mentioned above for our choice of positive $t'$ and $\Delta$, $m_K$ changes sign at the critical value (7.60) while that of $m_{K'}$ remains the same (positive). The total Chern number therefore reads

$$ C_{\lambda} = 0 \quad \text{for } t' < t'_{c} \quad \text{and} \quad C_{\lambda} = \lambda \quad \text{for } t' < t'_{c}, \qquad (7.62) $$

<footnote n="18">¹⁸For the case of the parameters chosen in figure 7.4 [$\Delta/t = 0.5$ and $\sin(\phi = \pi/3) = \sqrt{3}/2$], the gap is then closed at $t'_{c} = 1/9 \simeq 0.11$, situation that corresponds to panel (c) of the figure.</footnote>

<footnote n="19">¹⁹For negative values of $\Delta$ or $t'$, the valleys $K$ and $K'$ switch their respective roles as active and spectator valley.</footnote>
<figure><img src="image_70.png" /><figcaption>Figure 7.6: Phase diagram of the Haldane model as a function of $\phi$ and $\Delta/t'$. The phases are characterised by the Chern number, and the insets depict schematically the corresponding energy bands.</figcaption></figure>

and we are indeed confronted with a topological phase transition described by a change of the Chern number when the gap is closed.

The complete phase diagram of the Haldane model is shown in figure 7.6 as a function of the parameters $\Delta/t'$ and $\phi$, and the insets recall the band structure associated with the different phases. The transition lines have been obtained from equation (7.60), which may be rewritten as

$$ (\Delta/t')_c = \pm 3\sqrt{3} \sin(\phi), \qquad (7.63) $$

and separate the topological phases with non-zero Chern number from the topologically trivial ones.

We saw in section 6.5.1 that a topological insulator with non-zero Chern number displays the quantum anomalous Hall effect if the Fermi level is in the gap between the energy bands. This is also the case here, and the phase characterised by a Chern number $C_\lambda = -\lambda$ therefore comes along with a transverse conductivity $\sigma_H = e^2/h$ (since the band with $\lambda = -$ is completely filled) while that with $C_\lambda = \lambda$ has a Hall conductivity of $\sigma_H = -e^2/h$ even if no external magnetic field is present. It is for that reason that Haldane originally called the quantum anomalous Hall effect also *quantum Hall effect*
*without a magnetic field.* However, as we have now extensively discussed, one requires nevertheless a broken time-reversal symmetry that is usually caused by a magnetic field, *e.g.* in the case of the quantum Hall effect. Remember that, here, we needed to rely on an inhomogeneous flux distribution in the unit cell that may be physically caused by additional magnetic atoms even if the flux vanishes on the average.

The above reasoning shows us that a topological insulator with broken time-reversal symmetry (Chern insulator) provides a non-zero Hall conductivity, but we have not yet understood where the current is transported since the bulk is insulating. This point will be discussed in detail in section 7.4 where we will see that the analogy between a Chern insulator and the quantum Hall effect can be pushed even further.

### 7.2.3 Experimental realisation of the Haldane model with cold atoms

Before continuing with transport properties, let us discuss an experimental implementation of the Haldane model with the help of cold atoms and optical lattices, from 2014 [Jotzu et al., Nature 515, 237 (2014)]. This experiment does not consist of a “natural” *condensed-matter* realisation of the Haldane model but of a quantum simulation where the lattice is created by an optical laser field and the electrons replaced by (neutral) atoms. The optical lattice consists of standing waves that are stabilised with the help of optical mirrors [see figure 7.7(a)], and the atoms are attracted to the minima (or maxima) of the laser field via a dipolar interaction.<fnref n="20" /> If the atoms are trapped in the light-field minima, the hopping parameter *t* is related to the tunnel barrier between adjacent minima. Contrary to the original Haldane model, the experimental setup used here does not provide a honeycomb but a square lattice, with inequivalent sites. As a consequence, the first Brillouin zone, depicted in figure 7.7(b), is itself a square, and the positions of the Dirac points are indicated by the orange and green crosses. Naturally, the tight-binding model associated with this particular lattice is not the same as that of the Haldane model, but it shares the main common features, and the calculation of the associated band structure is essentially identical to the one performed above. It does not provide novel insight and we therefore do not

<footnote n="20">²⁰The sign of the interaction determines whether the atoms prefer the minima or maxima of the laser field.</footnote>
<figure><img src="image_71.png" /><figcaption>Figure 7.7: Realisation of the Haldane model with cold atoms in an optical lattice. (a) Experimental setup of the lasers and mirrors. (b) Sketch of the first Brillouin zone with the position of the Dirac points. (c) Fraction of atoms in the upper band, which plays the role of the conduction band in the electronic case, after one Bloch oscillation, as a function of the gap parameter $\Delta_{AB}$ at $\phi = 0$. (d) Fraction of atoms in the upper band as a function of $\phi$ at $\Delta_{AB} = 0$. After Jotzu et al., Nature **515**, 237 (2014).</figcaption></figure>
repeat the calculations.

The mirrors, which are supported by piezo-electric elements, allow one to imprint a phase $\phi$ on the second-nearest-neighbour hoppings, and this breaks time-reversal symmetry in the present setup. Furthermore the amplitude of the laser field allows for a control of the gap parameter $\Delta = \Delta_{AB}$, which is again the difference in onsite energy on the two different sublattices.

In the experiment, the bosonic $^{40}$K atoms at first occupy the band minimum at the $\Gamma$-point in the lowest energy band, which corresponds to the valence band in electronic systems. However, we cannot characterise the situation as an insulating one since we are dealing with bosonic atoms: there is thus no Fermi level that one could place in between the energy bands. In order to explore the full bands, the atoms must therefore be drawn away from the $\Gamma$-point, and that can be achieved with the help of a force $F$ that is equivalent to an electric field in the above-mentioned electronic case and that changes the wave vector $\hbar k_x = \hbar k_{x,0} + Ft$ in reciprocal space. We retrieve here Bloch oscillations because of the periodicity in reciprocal space: once an atom reaches the border of the first Brillouin zone at a wave vector $k_b$, it reappears at the opposite edge at the equivalent wave vector separated from the first one by a reciprocal-lattice vector.

One now measures the fraction of atoms that are transmitted to the upper band (the analogue of the valence band) after a full Bloch oscillation, as a function of the parameters $\phi$ and $\Delta_{AB}$. The underlying mechanism, which allows the atoms to be transferred from the lower to the upper band, is *Landau-Zener tunneling* across the gap separating the bands. The fraction of transferred atoms is therefore maximal if the gap between the bands is closed on the path taken by the atoms in a cycle, and it may be lowered by the opening of a gap. For a phase $\phi = 0$, the measured fraction of transferred atoms is shown in figure 7.7(c), and we notice indeed that the fraction is maximal when the gap is zero, $\Delta_{AB} = 0$. However, also for $\Delta_{AB} = 0$, one may open a gap by varying the phase $\phi$, as we saw in section 7.2.2. This is shown in figure 7.7(d). Since the gap is given by the modulus of the gap parameter (7.57), we expect a symmetry $\phi \leftrightarrow -\phi$ in the transferred fraction as a function of $\phi$. This is clearly found experimentally in the figure, and we also see the doubling of the period since the fraction depends on $|\sin(\phi)|$.

Finally, it is possible to vary both parameters $\Delta_{AB}$ and $\phi$. Measuring the maxima of the fraction of atoms transferred from the lower to the upper band
<figure><img src="image_72.png" /><figcaption>Figure 7.8: Experimental determination of the phase diagram of the Haldane model. Plotting the maximum of the measured fraction of transferred atoms from the lower to the upper band corresponds to plotting the vanishing of the effective gap parameter (7.57), as a function of $\Delta_{AB}$ and $\phi$ [panel (b)]. The lines where the effective gap vanishes are precisely the transition lines in the phase diagram [panel (c)]. After Jotzu et al., Nature 515, 237 (2014).</figcaption></figure>
allows us to measure the lines in the phase diagram where the gap vanishes,

$$ \Delta_{\xi} = \Delta_{AB} - \xi 3\sqrt{3}t' \sin(\phi) = 0. \qquad (7.64) $$

This is nothing other than the transition line in the phase diagram, which we theoretically discussed in figure 7.6 and that corresponds to the condition (7.63). The result is depicted in figure 7.8, and one obtains a good qualitative agreement with the theoretically expected phase diagram (figure 7.6). The dots indicate the measured values while the continuous lines are the theoretically expected transition lines, within an experimental uncertainty represented by the dashed lines. This uncertainty is due to that in the experimental determination of the precise maximum of the distribution of transferred atoms.

## 7.3 Topological insulators with time-reversal symmetry

In the preceding section, we have encountered a particular topological insulator, the Chern insulator, that requires a broken time-reversal symmetry. As mentioned above, its formation requires – in the absence of a magnetic field – underlying ferromagnetic (or antiferromagnetic) properties of the material that breaks precisely this symmetry. One may naturally ask the question whether there are topologically interesting band structures and topological insulators that exist even in the presence of time-reversal symmetry. And if so, what is their specificity in comparison with Chern insulators? This is the object of the present section where we discuss two models, the Kane-Mele model and the Bernevig-Hughes-Zhang model, that describe such types of topological insulators.

### 7.3.1 Kane-Mele model – restoration of time-reversal symmetry

The first time-reversal-symmetric tight-binding model, which displays topological insulating phases, was proposed by Kane and Mele in 2005. They imagined a variant of graphene with spin-orbit coupling. The model consists indeed of two copies of the Haldane model, one for spin-↑ and one for spin-↓ electrons. As we have seen in the discussion of section 7.2, none of the two
copies satisfies $E_{n,s}(-\mathbf{k}) = E_{n,s}(\mathbf{k})$, where $s = \uparrow, \downarrow$ denotes the spin, but we have also seen that time-reversal symmetry is less restrictive and only requires the condition $E_{n,-s}(-\mathbf{k}) = E_{n,s}(\mathbf{k})$ to be satisfied, which is ensured if equation (7.26),

$$H_s(-\mathbf{k})^* = H_{-s}(\mathbf{k}), \qquad (7.65)$$

is satisfied. This can be achieved if spin-$\uparrow$ electrons accumulate the same phases as in the Haldane model discussed in the previous section (see figure 7.5), while spin-$\downarrow$ electrons accumulate opposite phases on the same second-nearest-neighbour hoppings. The Kane-Mele model is therefore described by the $4 \times 4$ Bloch Hamiltonian

$$H_{4 \times 4}(\mathbf{k}) = \begin{pmatrix} H_{\uparrow}(\mathbf{k}) & 0 \\ 0 & H_{\downarrow}(\mathbf{k}) \end{pmatrix}, \qquad (7.66)$$

with $H_{\downarrow}(\mathbf{k}) = H_{\uparrow}^*(-\mathbf{k})$. Since both $2 \times 2$ blocks are Haldane models, we may directly use the low-energy limit for them if we are interested only in the (massive) Dirac points in terms of which we may understand the topological properties and possible topological phase transitions. In total, we thus obtain four (massive) Dirac fermions: in each valley we have a Dirac fermion for each spin orientation.

Let us now take a look at the low-energy limit of the Kane-Mele model and discuss first the diagonal terms, which are given in terms of the effective masses. Since the spin-$\uparrow$ block of the Hamiltonian (7.66) is precisely the Bloch Hamiltonian of the Haldane model, the associated mass is directly given by equation (7.57),

$$m_{\xi, \uparrow} v^2 = \Delta - \xi 3 \sqrt{3} t' \sin(\phi). \qquad (7.67)$$

In order to obtain the associated masses for the spin-$\downarrow$ block, we do not need to perform any calculation, but we simply need to recall that the direction of the second-nearest-neighbour hoppings is inverted with respect to the spin-$\uparrow$ electrons so that the sign of the phase $\phi$ needs to be changed. Consequently, the associated mass term is given by

$$m_{\xi, \downarrow} v^2 = \Delta + \xi 3 \sqrt{3} t' \sin(\phi). \qquad (7.68)$$

These two expressions for the Dirac masses can be written in a compact form in terms of the spin index $s = \pm$ ($s = +$ for spin-$\uparrow$ and $s = -$ for spin-$\downarrow$ electrons) as

$$m_{\xi, s} v^2 = \Delta - \xi s 3 \sqrt{3} t' \sin(\phi). \qquad (7.69)$$
<figure><img src="image_73.png" /><figcaption>Figure 7.9: Schematic representation of the band structure of the Kane-Mele. The pair of blue bands represents those of spin-↑ and the pair of red bands corresponds to spin-↓ electrons. The contributions of the associated Dirac fermions (the half-Chern numbers) to the Chern number of the two spin branches of the valence band are indicated respecting the colour code.</figcaption></figure>

Now consider the off-diagonal terms. One notices that time-reversal symmetry, $H_{\downarrow}(\mathbf{k}) = H_{\uparrow}^{*}(-\mathbf{k})$, relates electrons with opposite spins in the two different valleys. If one reverses the spin, one needs to search an electronic state in the other valley. In the case of the x-component of the wave vector, which comes along with the real Pauli matrix $\sigma_x$, we must therefore glue a factor $s \times \xi$ to the associated off-diagonal element to enforce time-reversal symmetry, in contrast to the y-component where the sign change is directly compensated by the complex conjugation of the Pauli matrix $\sigma_y$. The spin blocks of the low-energy limit of the Kane-Mele model may therefore be written in the compact form

$$H_{\xi,s}^{KM}(\mathbf{q}) = \hbar v(\xi s q_x \sigma_x + q_y \sigma_y) + m_{\xi,s} v^2 \sigma_z. \quad (7.70)$$

The generic band structure of the Kane-Mele model is depicted in figure 7.9. Time-reversal symmetry enforces the band symmetry (7.28), $E_{\lambda,s}(\mathbf{k}) = E_{\lambda,-s}(-\mathbf{k})$, as it is clearly visible in the figure, and each pair of bands associated with a fixed spin orientation $s$ has the form of the bands in the Haldane model. The trick of the Kane-Mele model consists of allowing for a gap inversion for spin-↑ electrons in the K valley without changing the sign of the gap parameter in the K' valley, similarly to the case of the Haldane model, as long as this gap inversion is accompanied by one in the K' valley
for spin-↓ electrons. To see that this induces interesting topological properties, let us play again the bookkeeper's game of half-Chern numbers. As we have discussed above, each of the four Dirac fermions yields a contribution of

$$C_{\lambda;\xi,s} = -\frac{\lambda}{2} \xi s \operatorname{sgn}(m_{\xi,s}) \qquad (7.71)$$

to the Chern number of the band $\lambda$. If we add the contributions of both spin branches, the total Chern number for either the valence or the conduction band $C_{\lambda} = \sum_{\xi,s} C_{\lambda,\xi,s}$ vanishes. Remember that, for an insulator, necessarily both spin branches of the valence band are completely filled, and the vanishing Chern number therefore yields a boring zero Hall conductivity, as we have seen in section 7.1.2.

However, the fact that we have more branches in the Kane-Mele model than in a usual tight-binding model with only two spin copies allows us to make other combinations of the Chern numbers. Most saliently the Chern number associated with each of the spin branches does not necessarily vanish if we are confronted with inverted gaps, as sketched in figure 7.9, in which case the Chern number of the spin branch $s$ in the band $\lambda$ reads

$$C_{\lambda,s} = -\frac{\lambda s}{2} [\operatorname{sgn}(m_{K,s}) - \operatorname{sgn}(m_{K',s})] = \pm 1. \qquad (7.72)$$

In spite of the vanishing Hall conductivity (and $C_{\lambda} = 0$), we may nevertheless consider the change from $C_{\lambda,s} = 0$ to $\pm 1$ as a topological phase transition. It is also accompanied by a change in a physically measurable quantity, in the form of the spin Hall conductivity. As we have seen in section 7.1.2, this spin Hall conductivity is quantised in the insulating regime [see equation (7.45)] similarly to the charge conductivity for a Chern insulator. We will see later in this chapter, in section 7.4, that the quantum spin Hall effect manifests itself, very much as the quantum (anomalous) Hall effect, by particular conducting edge channels that provide a conductance that is well described within the Landauer-Büttiker formalism. However, before discussing this edge transport and the bulk-edge correspondence, we finish this section by a short discussion of another 2D model that displays the same topological properties as the Kane-Mele model but that is more directly inspired by existing materials.
<figure><img src="image_74.png" /><figcaption>Figure 7.10: Adapted from Bernevig et al., Science **314**, 1757 (2006). (A) Band structure of HgTe (left panel) and CdTe (right panel) in the vicinity of the Γ point. The Fermi level is situated at zero energy. While CdTe is a trivial insulator, the bands Γ₆ and Γ₈ are inverted in HgTe. Since there is no gap between the (inverted) bands at the Fermi level, the system is metallic and not a topological insulator. (B) CdTe/HgTe/CdTe heterostructure. Because of their like symmetry, the same bands in the different materials must be connected over the interface, with an inversion in the central HgTe part, where confinement quantisation leads to a quantised motion in the z-direction (perpendicular to the stack of layers). If the width *d* of the HgTe layer is below a critical value *d*<sub>c</sub>, the hole level H1 (dashed red line, associated with the red band Γ<sub>8</sub>) is lower in energy than the electronic level E1 (dashed blue line, associated with the blue band Γ<sub>6</sub>) so that the band inversion is not efficient. The situation changes for *d* > *d*<sub>c</sub>, where the electron level E1 is now below the hole level H1. In this case, the HgTe well becomes a 2D topological insulator due to confinement in the z-direction, with free (plane-wave) motion in the *xy*-plane.</figcaption></figure>
### 7.3.2 Bernevig-Hughes-Zhang model

Kane and Mele proposed their model in 2005 with the idea in mind to describe a realistic situation in graphene once spin-orbit coupling is taken into account. However, it turns out that the relevant spin-orbit coupling in graphene is extremely small due to the small atomic mass of the carbon atoms – the spin-orbit coupling is proportional to the fourth power of the atomic number Z – and later estimates indicated that the spin-orbit coupling in graphene would be in the μeV range, far below the relevant meV regime that governs most graphene samples.<fnref n="21" /> The Kane-Mele model is therefore mainly used as a paradigmatic example for a topological insulator that respects time-reversal symmetry.

A more realistic model, from a materials-sciences point of view, was proposed by Bernevig, Hughes, and Zhang in 2006 (BHZ model). The model was inspired by quantum wells based on mercury telluride (HgTe) and cadmium telluride (CdTe), which could be grown by MBE (see introduction, section 1.5). Cadmium telluride is an ordinary (non-topological) insulator, and its band structure in the vicinity of the Γ point is shown in the right panel of figure 7.10(A). It consists of a doubly degenerate Γ₆ conduction band separated by a substantial gap (~1.7 eV) from two doubly degenerate valence bands Γ₈. The degeneracy at the TRIM Γ is a consequence of time-reversal symmetry, but notice that the Kramers pairs in the valence bands (the Γ₈ bands) are in general not given by the true spin but some pseudospin ↑, ↓ that combines the physical spin and an atomic orbital momentum due to an atomic spin-orbit coupling. While the Γ₆ band arises from atomic orbitals with s-wave character, i.e. with vanishing orbital momentum (L = 0), and the pseudospin of the conduction bands $m_J = \pm 1/2$ is identical to the physical spin, this is not the case in the Γ₈ bands. The latter are associated with atomic orbitals with a p-wave character (L = 1), and the total angular momentum is given by $J = S + L = 3/2$. Consequently, the Γ₈ bands consist of two pairs of branches, $m_J = \pm 1/2$ (light hole band) and $m_J = \pm 3/2$ (heavy hole band), each of which is doubly degenerate [see figure 7.10(A)].

As compared to CdTe, the band structure of HgTe is more complex. One may first notice that the bands Γ₆ and Γ₈ are *inverted* as compared to those in CdTe. In view of the above discussions, this is a necessary ingredient to obtain a topological insulator. However, one also notices that the two

<footnote n="21">²¹One may nevertheless mention today’s efforts to find sufficiently high-mobility graphene samples and very sensitive probes to study the spin-orbit coupling in graphene.</footnote>