branches of the $\Gamma_8$ bands have opposite curvature, i.e. band masses with opposite sign, but still touch each other at the $\Gamma$ point. Due to the absence of a band gap, HgTe is therefore a 3D metal rather than a (topological) insulator.

In order to obtain a (2D) insulator, one can fabricate a heterostructure that consists of a HgTe layer (of thickness $d$) sandwiched between two (thick) CdTe layers [see figure 7.10(B)]. The confinement in the $z$-direction perpendicular to the layers leads to the formation of a 2D electron gas with free propagation in the $xy$-plane and to a quantised spectrum for the motion in the $z$-direction, similar to that of a quantum well. We are interested in the lowest confinement levels $E_1$ and $H_1$ associated with the two bands $\Gamma_6$ and $\Gamma_8$, respectively, which are depicted in figure 7.10(B).$^{22}$ If the layer thickness $d$ is smaller than a critical value, $d < d_c$, the confinement is strong and the $H_1$ level associated with the $\Gamma_8$ band remains below the $E_1$ level of the $\Gamma_6$ band because the band inversion in the HgTe well is not yet efficient as compared to the same bands in the CdTe layers to which the bands are connected. The levels are inverted above a value $d > d_c$, with $H_1$ above $E_1$, in which case we obtain indeed a 2D topological insulator.

Let us now try to understand the low-energy model associated with the band structure described above and convince ourselves that it describes indeed a topological material, as we would expect from the above arguments about band inversion. In contrast to the (tight-binding) models, which we have encountered so far, we try to inspect the structure of the model in the vicinity of the $\Gamma$ point within an approach that is reminiscent of k.p theory. Since we are confronted with two doubly degenerate bands, we already know that we need to eventually consider a $4 \times 4$ Hamiltonian. The $E_1$ states, which appear in a pair of $m_J = \pm 1/2$ spins, disperse as

$$E_{E1}(\mathbf{q}) = E_{E1}(d) + \hbar^2 \frac{q_x^2 + q_y^2}{2m_E} \qquad (7.73)$$

if we discard coupling to the other bands for a moment. We have already seen from the band structure in figure 7.10 that the $\Gamma_8$ bands, which give rise to the hole states, are associated with a quartet due to their spin $J = 3/2$. Notice, however, that due to the larger band mass of the $m_J = \pm 3/2$ states their confinement energy is smaller than that of the $m_J = \pm 1/2$ states so

<footnote>$^{22}$Recall that these levels are the $n = 1$ levels of the series $E_{\Gamma,\pm} = \pm \hbar^2 \pi^2 n^2 / m_{\pm} d^2$, with a plus sign for the band $\Gamma_6$ and a minus sign for the band $\Gamma_8$.</footnote>
that they are first shifted above the $E_1$ states in the inverted regime of the HgTe well, while the latter remain below the $E_1$ bands. We may therefore restrict the 2D model to the heavier $m_J = \pm 3/2$ bands responsible for the $H_1$ states that thus disperse as

$$E_{H1}(\mathbf{q}) = E_{H1}(d) - \hbar^2 \frac{q_x^2 + q_y^2}{2m_H}, \qquad (7.74)$$

the negative sign indicating that we are confronted with hole-type bands. The energies $E_{E1}(d)$ and $E_{H1}(d)$ determine the offset of the electron and hole bands, respectively, and they are governed by the confinement potential in the $z$-direction, as we have discussed above. In the case of a square-well confinement, these energies are given by

$$E_{E1}(d) = E_{E1}(d = \infty) + \frac{\pi^2 \hbar^2}{m_E d^2} \quad \text{et} \quad E_{H1}(d) = E_{H1}(d = \infty) - \frac{\pi^2 \hbar^2}{m_H d^2}, \qquad (7.75)$$

where we have for the HgTe part in the 3D bulk limit $E_{E1}(d = \infty) < E_{H1}(d = \infty)$ due to the band inversion. Since the $E_1$ bands move upwards in energy as a consequence of the confinement, while the $H_1$ bands shift downwards, there is a crossing of the bands at a critical thickness $d_c$. Confinement thus restores the “natural” order, i.e. it cancels the band inversion in the HgTe layer for a thickness below this critical value.

Now that we have established the structure of the diagonal elements of the low-energy band Hamiltonian, let us turn to the off-diagonal elements. We choose a basis of states $\{|H1, m_J = 3/2\rangle, |E1, m_J = 1/2\rangle, |H1, m_J = -3/2\rangle$, and $|E1, m_J = -1/2\rangle\}$ that turns out to be convenient in view of the couplings. Remember that the $E_1$ bands have an $s$-wave character, while the $H_1$ bands are of $p$ type so that their coupling requires an angular-momentum transfer of $\Delta L_z = \pm 1$ that is precisely provided by a coupling term $\hbar v(q_x \pm iq_y)$, and it couples the states $|H1, m_J = 3/2\rangle$ and $|E1, m_J = 1/2\rangle$ on the one hand side and $|H1, m_J = -3/2\rangle$ and $|E1, m_J = -1/2\rangle$ on the other hand.<fnref n="23" />

The low-energy Hamiltonian is therefore composed of two $2 \times 2$ blocks,

<footnote n="23">²³One may also take into account terms of higher order in $q$ to provide couplings between the other states. However such terms do not affect the topological properties of HgTe/CdTe quantum wells that are well described in terms of the present low-energy model obtained by a restriction to quadratic order in the diagonal elements and linear order in the off-diagonal ones.</footnote>
as in the case of the Kane-Mele model,

$$H_{\text{BHZ}}(\mathbf{q}) = \begin{pmatrix} H_{\uparrow}(\mathbf{q}) & 0 \\ 0 & H_{\downarrow}(\mathbf{q}) \end{pmatrix} \qquad (7.76)$$

with

$$H_{\uparrow}(\mathbf{q}) = \begin{pmatrix} E_{\text{H1}}(d) - \frac{\hbar^2 q_x^2 + q_y^2}{2m_H} & \hbar v(q_x - i q_y) \\ v(q_x + i q_y) & E_{\text{E1}}(d) + \frac{\hbar^2 q_x^2 + q_y^2}{2m_E} \end{pmatrix} \qquad (7.77)$$

and $H_{\downarrow}(\mathbf{q}) = H_{\uparrow}^{*}(-\mathbf{q})$, to respect time-reversal symmetry. It is important to emphasise that the “spin” $s=\uparrow,\downarrow$ does not represent the z-component of the physical spin. It should be viewed rather as a pseudospin denoting the two blocks represented by the sign of $m_J$ ($s=\uparrow$ for $m_J = +1/2$ and $+3/2$, and $s=\downarrow$ for $m_J = -1/2$ and $-3/2$) even if the two blocks are related by time-reversal symmetry as it would be the case for true spins! Notice that the above Hamiltonian may be cast into its canonical form

$$H_s(\mathbf{q}) = h_0(\mathbf{q})\mathbb{I}_{2\times2} + \begin{pmatrix} \Delta(d) - \frac{\hbar\mathbf{q}^2}{2m} & \hbar v(sq_x - iq_y) \\ \hbar v(sq_x + iq_y) & -\Delta(d) + \frac{\hbar\mathbf{q}^2}{2m} \end{pmatrix}, \qquad (7.78)$$

where

$$\Delta(d) = \frac{1}{2} [E_{\text{H1}}(d) - E_{\text{E1}}(d)] \qquad (7.79)$$

is the thickness-dependent gap parameter and

$$h_0(\mathbf{q}) = \frac{E_{\text{H1}}(d) + E_{\text{E1}}(d)}{2} - \frac{\hbar^2 \mathbf{q}^2}{2M} \qquad (7.80)$$

is an unimportant global energy shift that does not affect the topological properties of the system since it couples to the one matrix $\mathbb{I}_{2\times2}$. Furthermore, to cast the Hamiltonian into the above canonical form, in terms of Pauli matrices, we needed to redefine the effective masses as

$$\frac{1}{M} = \frac{1}{2} \left( \frac{1}{m_H} + \frac{1}{m_E} \right) \quad \text{and} \quad \frac{1}{m} = \frac{1}{2} \left( \frac{1}{m_H} - \frac{1}{m_E} \right). \qquad (7.81)$$

The expression (7.78) of the BHZ model allows us to test whether, or under which conditions, we obtain a topological insulator, via a calculation of the Berry flux. Similarly to the Haldane and Kane-Mele models, we use the Berry flux through the total $\mathbb{R}^2$ plane to obtain insight into the total Chern
number of the bands. Remember that we cannot a priori expect that this Berry flux is indeed the Chern number since the $\mathbb{R}^2$ plane used in the continuum models is not a compact manifold. With respect to the calculations of the Berry phase and flux of a massive Dirac fermion, presented in section 6.4, the gap parameter of the BHZ model depends now on the wave vector. For this reason it turns out to be simpler to calculate the Berry phase as the line integral of the Berry connection

$$ \Gamma_{\lambda,s}(q) = \oint_{|\mathbf{q}'|=q} d\mathbf{q}' \cdot \vec{\mathcal{A}}_{\lambda,s}, \qquad (7.82) $$

on a closed circular path of radius $q$. The Berry connection for the band $\lambda$ is calculated in the same manner as in section 6.4, where we obtained equation (6.52),

$$ \vec{\mathcal{A}}_{\lambda,s}(\mathbf{q}) = -\frac{s\lambda}{2}[1 - \cos\theta(q)]\nabla_{\mathbf{q}}\phi, \qquad (7.83) $$

if we substitute

$$ \cos\theta(q) = \frac{\Delta(d) - \hbar^2 q^2 / 2m}{\sqrt{[\Delta(d) - \hbar^2 q^2 / 2m]^2 + \hbar^2 v^2 q^2}}. \qquad (7.84) $$

Since $\oint_{|\mathbf{q}'|=q} d\mathbf{q}' \cdot \nabla_{\mathbf{q}'}\phi = 2\pi$, we thus find for the Berry phase on a circle of radius $q$

$$ \Gamma_{\lambda,s}(q) = -2\pi\lambda s \frac{1 - \cos\theta(q)}{2}, \qquad (7.85) $$

which converges, in the limit $q \to \infty$, to the contribution

$$ C_{\lambda,s} = \frac{\Gamma_{\lambda,s}(q \to \infty)}{2\pi} = -\lambda s \frac{1 - \cos\theta(q \to \infty)}{2} \qquad (7.86) $$

to the Chern number. We may already notice an essential difference with respect to the Berry flux of a massive Dirac point, which has been shown to be $\pm 1/2$ because $\cos\theta(q \to \infty) = 0$. Here, however, the limit reads $\cos\theta(q \to \infty) = -1$ as a consequence of the quadratic term in the diagonal elements of the Hamiltonian (7.78), which become dominant in the limit $q \to \infty$. For this reason, we now obtain an *integer* number,

$$ C_{\lambda,s} = -\lambda s. \qquad (7.87) $$

This means that, contrary to massive Dirac fermions that occur necessarily in pairs, there is no need for a second point in the first Brillouin zone that
provides a non-zero contribution to the Chern number in an underlying lattice model. Notice furthermore that the relative sign between the (thickness-dependent) gap parameter and the quadratically dispersing term $\hbar^2q^2/2m$ is essential for a non-zero Chern number. If we had the opposite sign – this is the case for a thickness below the critical value, $d < d_c$ – we would obtain a zero Chern number and thus a trivial insulator, as we intuitively expect indeed for non-inverted bands. Finally, due to time-reversal symmetry, the Chern numbers of the two different (pseudo)spin branches of each band have an opposite sign and thus compensate each other in the same manner as in the Kane-Mele model. With respect to the latter, the main difference is the number of relevant points in the first Brillouin zone: in the case of the Kane-Mele model, we have obtained a total number of four massive Dirac fermions, two in each valley $K$ and $K'$ for the two different spin orientations. In the BHZ model describing the HgTe/CdTe quantum well, we only have the two (pseudo)spin branches at the $\Gamma$ point. As mentioned at the beginning of the present section, the BHZ model provides the same conceptual insight into the properties of 2D time-reversal-symmetric topological insulators as the Kane-Mele model. However, in addition, it describes an existing CdTe/HgTe/CdTe heterostructure that has been used in the discovery of the quantum spin Hall effect – the probably most prominent manifestation of time-reversal-symmetric topological insulators – described in detail in section 7.4.3.

## 7.4 Bulk-edge correspondence

In the presentation of the quantum Hall effect, we argued that the electronic system can be viewed as a bulk insulator: the Fermi level is situated between two Landau levels or it is fixed in the tails of a disorder-broadened Landau level by localised states corresponding to closed equipotential lines around the minima (or maxima in the case of holes) of the potential landscape. At the same time, the system possesses as many conducting edge channels as completely filled Landau levels (or spin-valley subbranches of the levels) below the Fermi level in the bulk of the system. This is a manifestation of a very general property of topological materials that is known as the *bulk-edge correspondence* (or *bulk-boundary correspondence*)<fnref n="24" /> the detailed

<footnote n="24">²⁴In the present section, we discuss 2D topological insulators and mainly use the term bulk-edge correspondence to emphasise the 1D character of the edge channels. For topological materials in general dimensions (e.g. in the following chapter), it is more appropriate</footnote>
discussion of which is the object of the present section. In the general case of 2D topological materials, the number of channels is determined by a bulk topological invariant (in our case the Chern number). Furthermore, the sign of this invariant indicates the sense of propagation of the edge channels, which are again *chiral* as in the case of the quantum Hall effect, where the chirality was imposed by the orientation of the magnetic field.

In the present discussion of the bulk-edge correspondence, an original approach is chosen. It is based on two ingredients that should be familiar now. The first one is the picture of a topological phase transition discussed in section 6.4.1 and illustrated in figure 6.7 (and reproduced in figure 7.2), where the parameter that controls the transition (the gap parameter) changes its sign across the interface between a topological material and a trivial insulator such as the vacuum. The second ingredient is, perhaps somewhat unexpectedly, Landau-level quantisation even if there is no physical magnetic field present. The approach is based on the assumption that we have a smooth interface between the two types of materials even if we may eventually let the interface width tend to zero. In a hand-waving-type argument, we could say that if the interface is situated at the position $x=0$ and we have a topological insulator with an inverted gap parameter at $x < 0$, $\Delta_{\xi}(x < 0) < 0$, and a trivial one with $\Delta_{\xi}(x > 0) > 0$ at positive values of $x$, the Chern number “changes” by $\pm 1$ over the interface and the gap must close at $x=0$, where a metallic state is formed, in the form of a massless Dirac fermion. As simple as this picture may seem,<fnref n="25" /> it turns out to be globally valid and will guide us through this section.

### 7.4.1 Massive Dirac fermions and mass change at an interface

In order to keep the discussion of the interfaces conceptually as simple as possible, we illustrate the above ideas again with the help of the Haldane model in the low-energy limit, i.e. in terms of a massive Dirac fermion, and we omit complications related to the spin for the moment. Furthermore, we consider a single sample edge situated at $x=0$, keeping in mind that there is

<footnote>to talk about the bulk-boundary correspondence.</footnote>

<footnote n="25">²⁵The situation is naturally a bit more involved, since the variation of the gap parameter in the *x*-direction has the annoying consequence that the wave vector $q_x$ is no longer a conserved quantity due to its non-commutativity with the operator *x*.</footnote>
<figure><img src="image_75.png" /><figcaption>Figure 7.11: Sign change of the gap parameter $\Delta_{\xi}$ in the valley $\xi$ across an interface between an inverted gap ($\Delta_{\xi} < 0$, on the left-hand side) and a direct gap ($\Delta_{\xi} > 0$, on the right-hand side). We consider an interface of width $\ell$ over which the gap varies.</figcaption></figure>

a second one at $x = -L$ for a sample of width $L$. We get back to this second edge later on in the section. Such an interface is schematically represented in figure 7.11, where we consider a sign change of the gap parameter (in the active valley $\xi$) across an interface of width $\ell$. Furthermore, we suppose that this width is rather large with respect to a characteristic intrinsic length scale associated with massive Dirac fermions that is nothing other than the effective Compton length $\lambda_C = \hbar v/|\Delta|$ [see equation (6.57) of section 6.4], as we shall explicitly see below. For the variation of the gap parameter, we may choose a simple interpolation formula

$$ \Delta_{\xi}(x/\ell) = \Delta \tanh(x/\ell), \qquad (7.88) $$

where we have set the parameter $\Delta$ to be positive, $\Delta > 0$. As in the case of the topological phase transition discussed in section 6.4, the gap in the spectator valley $-\xi$ only varies marginally and does not change sign. It does therefore not affect the edge physics. In the vicinity of the edge at $x = 0$, the low-energy electronic properties are then described by the Hamiltonian

$$ H_{\text{bord}}^{\xi} = \begin{pmatrix} \Delta_{\xi}(x/\ell) & \hbar v(\xi q_x - i q_y) \\ \hbar v(\xi q_x + i q_y) & -\Delta_{\xi}(x/\ell) \end{pmatrix}. \qquad (7.89) $$

Before an explicit calculation of its spectrum, we may already inspect some general features. As a consequence of the gap-parameter variation in the $x$-direction and the non-commutativity $[x, q_x] = i$, the $x$-component of the wave
vector is no longer a good quantum number. In contrast to this component, the system remains translationally invariant in the y-direction. Consequently, we expect to obtain a family of 1D bands, $E_n(q_y)$, that are labeled by a discrete quantum number $n$ associated with the quantisation of the motion in the x-direction.

For a diagonalisation of the Hamiltonian (7.89), we first exchange the Pauli matrices $\sigma_y$ and $\sigma_z$. Technically, this may be achieved with the help of the unitary transformation $U = \exp(i\pi\sigma_x/4)$ that rotates the spin quantisation axes around the x-axis by an angle of $\pi/2$,

$$H_{\text{bord}}^{\xi'} = U H_{\text{bord}}^{\xi} U^{-1} = \begin{pmatrix} \hbar v q_y & \hbar v \xi q_x + i \Delta_{\xi}(x/\ell) \\ \hbar v \xi q_x - i \Delta_{\xi}(x/\ell) & -\hbar v q_y \end{pmatrix}. \quad (7.90)$$

Naturally, the spectrum is unaffected by this rotation as by any unitary transformation. This Hamiltonian, along with the interpolation formula (7.88) for the gap parameter can be solved exactly, albeit in a cumbersome manner.<fnref n="26" /> The solution turns out to be much simpler if we linearise the gap function (7.88) in the vicinity of the edge, $x = 0$, where we obtain $\Delta_{\xi}(x/\ell) \simeq \Delta x/\ell$ then, so that the approximate Hamiltonian reads

$$H_{\text{bord}}^{\xi'} = \begin{pmatrix} \hbar v q_y & \hbar v (\xi q_x + i x / \ell_S^2) \\ \hbar v (\xi q_x - i x / \ell_S^2) & -\hbar v q_y \end{pmatrix}. \quad (7.91)$$

Here we have defined a new length

$$\ell_S = \sqrt{\ell \hbar v / \Delta} = \sqrt{\ell \lambda_C} \quad (7.92)$$

that is the geometric average of the interface width $\ell$ and the effective Compton length $\lambda_C$.

If you take a closer look at the Hamiltonian (7.91) for the electronic properties of the edge, you will realise that we have already encountered (and diagonalised) it in a different context. It has indeed the same structure as the Hamiltonian (5.35), which describes massive Dirac fermions in a magnetic field. This may be seen clearly if we introduce the usual ladder operators

$$\hat{a} = \frac{\ell_S}{\sqrt{2}}(q_x - i x / \ell_S^2) \quad \text{et} \quad \hat{a}^\dagger = \frac{\ell_S}{\sqrt{2}}(q_x + i x / \ell_S^2), \quad (7.93)$$

<footnote n="26">²⁶The Hamiltonian has indeed a super-symmetric structure and may be solved with the help of confluent hypergeometric functions.</footnote>
<figure><img src="image_76.png" /><figcaption>Figure 7.12: Sketch of the edge states around $x = 0$: chiral (topological) edge states $n = 0$ (red line) and massive states for $n \neq 0$ (blue lines). The blue areas indicate the bulk states above the gap $\Delta$ in the conduction band and those below $-\Delta$ in the valence band.</figcaption></figure>

with $[\hat{a}, \hat{a}^{\dagger}] = 1$, in which case the Hamiltonian becomes$^{27}$

$$H_{\text{edge}}^{\xi=+} = \hbar v \begin{pmatrix} q_y & \sqrt{2} \frac{\hat{a}^{\dagger}}{\ell_S} \\ \sqrt{2} \frac{\hat{a}}{\ell_S} & -q_y \end{pmatrix} \qquad (7.94)$$

if the gap changes in the valley $\xi = +$ or else

$$H_{\text{edge}}^{\xi=-} = \hbar v \begin{pmatrix} q_y & -\sqrt{2} \frac{\hat{a}}{\ell_S} \\ \sqrt{2} \frac{\hat{a}^{\dagger}}{\ell_S} & -q_y \end{pmatrix} \qquad (7.95)$$

if $\xi = -$ is the active valley.

Remarkably, the change of the gap parameter across the interface manifests itself similarly to a vector potential that yields a magnetic field, and the characteristic length $\ell_S$ can be identified with the magnetic length $l_B$. Similarly to the latter, it determines the extension of the wave function, here

<footnote>$^{27}$For notational simplicity, we omit the prime at the Hamiltonian which we introduced in the context of the unitary transformation.</footnote>
in the x-direction. More precisely, it is the equivalent of the cyclotron radius $\sqrt{2n\ell_S}$ that determines this extension, which must naturally be smaller than the interface width,

$$ \sqrt{2n\ell_S} < \ell \Leftrightarrow \sqrt{2n} < \sqrt{\ell/\lambda_C}. \qquad (7.96) $$

If this condition were not fulfilled, the extension of the wave function would be larger than the region in which we used the linear approximation for the change of the gap parameter, i.e. we would be beyond the range of validity of the linear approximation. We shall see below that we may also interpret this condition as a condition for the energy of the bands that may be calculated in the same manner as the Landau-level spectrum of massive Dirac fermions in section 5.2.3. One obtains

$$ E_{\lambda,n}^{\xi}(q_y) = \lambda\hbar v \sqrt{q_y^2 + 2n/\ell_S^2} \qquad (7.97) $$

for the states $n \neq 0$ and

$$ E_{n=0}^{\xi}(q_y) = \xi \hbar v q_y \qquad (7.98) $$

for the special state $n = 0$, which happens to be the most important edge state. The spectrum of the edge states is sketched in figure 7.12 along with the bulk states. As one sees from the figure and the expression (7.98), the state $n = 0$ is chiral in the sense that the group velocity in the y-direction perpendicular to the interface is entirely determined by the valley $\xi$ in which the gap changes its sign across the interface at $x = 0$,

$$ v_y = \frac{1}{\hbar} \frac{\partial E_{n=0}^{\xi}}{\partial q_y} = \xi v. \qquad (7.99) $$

Moreover, this state depends energetically only on the intrinsic parameters of the material (here the velocity parameter $v$) and not on the manner how we model the interface. This is to say that it does not depend on the interface width $\ell$, in contrast to the massive edge states with $n \neq 0$, and it survives in the limit of an abrupt interface, $\ell \to 0$.<fnref n="28" /> Because of the robustness of the chiral state $n = 0$, whose presence is simply guaranteed by the gap inversion in the active valley $\xi$, and because it necessarily crosses the gap at the edge,

<footnote n="28">²⁸The persistence of the chiral state $n = 0$ can also be seen in the original treatment by Jackiw and Rebbi in 1976, Phys. Rev. D, **13**, 3398 (1976). Their approach is not based on a smooth interface right from the beginning and does not appeal to Landau quantisation.</footnote>
this state is also called *topological* or *topologically protected edge state*. It is this state that physicists have in mind when they speak about the *bulk-edge correspondence*, which may be rephrased in the following manner:

If one interfaces two different topological insulators (or materials), characterised by two different Chern numbers, this interface is itself described by a discrete change in the Chern number. This change requires a gap closing in one of the valleys $\xi$ across the interface while the other valley, $-\xi$, remains unaffected (spectator valley). No matter what the precise form of the interface may be, there exists a *topologically protected* and *chiral* edge state that necessarily crosses the gap between the bulk valence and conduction bands at this interface.

Let us briefly mention also the massive states with $n \neq 0$, which correspond to higher Landau levels within our analogy with massive 2D Dirac fermions in a magnetic field. Their presence is also caused by the topological interface between two topologically distinct materials and the resulting sign change in the gap parameter across the interface,<fnref n="29" /> but contrarily to the $n = 0$ chiral state their presence depends on the interface smoothness and the width parameter $\ell$. As we have already seen in equation (7.96), the interface width must indeed be larger than the intrinsic length $\lambda_C$ in order to observe them. This condition can also be understood on the basis of energy arguments. In order for a state with $n \neq 0$ to be visible, its minimal energy $\sqrt{2n\hbar v}/\ell_S$, at $q_y = 0$, must be smaller than the parameter $\Delta$ determining the bulk gap,

$$ \hbar(v/\ell_S)\sqrt{2n} < \Delta. \qquad (7.100) $$

Otherwise the edge state would be swamped by the bulk states. With the help of the expression (7.92) for the characteristic length $\ell_S = \sqrt{l\lambda_C}$, this yields the same condition

$$ 2n < \frac{\ell}{\lambda_C} = \frac{\ell\Delta}{\hbar v} \qquad (7.101) $$

for the number of visible states as that (7.96) based on arguments concerning the relevant length scales.<fnref n="30" />

<footnote n="29">²⁹These states had already been noticed and theoretically described by the two Soviet physicists Volkov and Pankratov in the 1980ies, long before the advent of topological materials. For this reason, these states are also called *Volkov-Pankratov states*.</footnote>

<footnote n="30">³⁰Notice that this condition is also in line with a more precise treatment of bound states in a model with a smooth interpolation of the gap parameter, as given by equation (7.88).</footnote>
In this section, we have concentrated on an interface between a topological
insulator with a Chern number $C = \pm 1$ and a trivial one with $C = 0$, and
we have seen that our treatment in terms of a massive Dirac fermion in the
active valley $\xi$ yields a single chiral edge state with $n = 0$. Notice that there
are other situations, where the Chern number could take a larger value. We
do not aim at an extensive treatment of this case, but only provide some
qualitative arguments about how the above picture needs to be modified.
There are two (actually equivalent) ways to treat an insulator with a Chern
number $C = \pm N$. First, we may consider a situation where the off-diagonal
elements of the low-energy Hamiltonian (7.89) are proportional to $(q_x \pm iq_y)^N$
that yield an $N$-fold covering of the Bloch sphere when the system explores
the $\mathbb{R}^2$ plane of wave vectors. In this case one may already appreciate that
one obtains ladder operators with a power of $(\hat{a}^{\dagger})^N$ in our picture. These
ladder operators then yield $N$ chiral edge states $E_n^{\xi}(q_y)$ with $n = 0, \dots, N-1$.
This can be seen better in the second equivalent argument, where the $N$-fold
zero at $q=0$ is split into $N$ single zeros, $q \to q+\kappa_n = (q_x+\kappa_{n,x})+i(q_y+\kappa_{n,y})$,
in which case the (active) valley is split into $N$ individual (massive) Dirac
fermions. All $N$ fermions have the same winding and therefore provide an
edge state of the same chirality at the interface. One thus obtains $N$ chiral
edge states, in the same manner as in the case of the quantum Hall effect for
$N$ completely filled Landau levels.

In conclusion, we have seen in this section that the analogy between
topological (Chern) insulators and the quantum Hall effect can indeed be
pushed quite far, here in the context of the bulk-edge correspondence. In
the case of the quantum Hall effect, we obtain as many chiral edge states as
completely filled Landau levels below the Fermi level. Here, we have seen
that the number of chiral edge states is determined by the Chern number of
the filled bulk bands. In the former case of the quantum Hall effect it is the
orientation of the magnetic field that imposes the chirality, i.e. the sense of
propagation of the electrons at the edge. Similarly we have seen that in the
case of the Chern insulator it is the Chern number that imposes a chirality
on the edge states via the active valley, where the gap changes its sign across
the interface [see equation (7.99)]. This analogy furthermore allows us to
understand the electric edge transport in a topological insulator, again in
the framework of the Landauer-Büttiker formalism, as we discuss in detail
in the remainder of this section.
<figure><img src="image_77.png" /><figcaption>Figure 7.13: (a) 2D topological insulator in the ribbon geometry, with edges at $x = 0$ and $x = -L$. The gap parameter (red curve) now varies both in the vicinity of $x = 0$ and $x = -L$, albeit with opposite sign so that it becomes positive outside the ribbon, $+\Delta_0$. (b) Dispersion of the edge states associated with an active valley $\xi = +$. The edge state at $x = 0$ has a positive slope $v$ (continuous blue line) and that at $x = -L$ has a negative group velocity $-v$ (dashed blue line) in the $y$-direction along the edge. Both states cross the gap between the conduction and valence bands (shown in light blue). (c) Addition of the edge states in the original band structure, the other bands indicating the bulk bands.</figcaption></figure>

### 7.4.2 Chiral edge states in a topological insulator without time-reversal symmetry – quantum anomalous Hall effect

We have just seen that the bulk-edge correspondence of topological materials, in the picture where a massive Dirac fermion (in the active valley $\xi$) changes the sign of its mass across the interface, yields a chiral metallic state at the interface (at $x = 0$) with the group velocity $v_y = \xi v$.<fnref n="31" /> One can easily convince oneself that, in the case of a ribbon of width $L$, there is a second chiral state at the other edge, at $x = -L$ [see figure 7.13(a)]. In order to appreciate its chirality, i.e. the sense of propagation, let us consider the gap parameter. In contrast to the edge at $x = 0$, it varies as $\Delta_{\xi}(x'/\ell) = -\Delta((x+$

<footnote n="31">³¹From now on, we are no longer interested in the massive interface states, with $n \neq 0$.</footnote>
$L$)/$ℓ$) there, the linearisation of which yields $-\Delta x'/ℓ$ (where $x' = x + L$). One thus obtains the same edge model as for $x = 0$ if we change the sign of the gap parameter. We have already understood that this yields a change of the relative signe between the real and the imaginary part of the off-diagonal terms in the edge Hamiltonian (7.91) and thus an exchange of the ladder operators $\hat{a}$ and $\hat{a}^†$ in the expressions (7.94) and (7.95). As a consequence the sign in the dispersion of the edge state also changes, and one thus obtains

$$E_{n=0,L}^{\xi}(q_y) = -\xi\hbar v q_y, \qquad (7.102)$$

where the index $L$ indicates that we are now at the edge situated at $x = -L$, while $\xi$ continues to denote the active valley. The dispersion of this second edge state is represented in figure 7.13(b) by the blue dashed line.

From the above argument, we see that the sense of propagation (the group velocity) is inverted at the second edge of the ribbon [see figure 7.13(a)], as we would have expected from the analogy with the quantum Hall effect at $\nu = 1$ (see discussion of section 5.3).<fnref n="32" /> Notice finally that the dispersion is often schematically added to the bulk bands of the 2D system, as indicated in figure 7.13(c) for the Haldane model. Strictly speaking, this form arises naturally when the associated tight-binding model is solved in the ribbon geometry, where the wave vector $q_y$ remains a good quantum number while $q_x$ becomes quantised due to the finite ribbon width.

## Experimental realisation of the quantum anomalous Hall effect and chiral edge states

The bulk-edge correspondence provides us with an edge-state picture that is identical to that of the quantum Hall effect at $\nu = 1$: the bulk of the sample is insulating while chiral edge states cross the Fermi level, which is situated between the bulk valence and conduction bands. We may therefore reasonably expect the same transport characteristics in a 2D topological (Chern) insulator as in the quantum Hall effect, dominated by ballistic chiral transport at the edges that may conveniently be described in the framework of the Landauer-Büttiker formalism, discussed in chapter 4. The predictive power of this formalism has indeed been proven, in 2015, within mesoscopic transport measurements in the ferromagnetic 2D compound (Cr$_{0.12}$Bi$_{0.26}$Sb$_{0.62}$)Te$_3$ (with a thickness of 10 nm, on an insulating

<footnote n="32">³²Because the dispersion of the massive edge states does not depend on the sign of the gap parameter, their spectrum remains the same as that at $x = 0$.</footnote>
<figure><img src="image_78.png" /></figure>

Figure 7.14: Measurement of the quantum anomalous Hall effect in a (ferromagnetic) 2D topological insulator based on (Cr<sub>0.12</sub>Bi<sub>0.26</sub>Sb<sub>0.62</sub>)Te<sub>3</sub>. (a) Image of the sample in the Hall-bar geometry. (b) Measurement of the longitudinal and transverse resistances as a function of the magnetic field used to orient the magnetisation of the system. The directions (*sweep*) indicate the evolution of the magnetic field that determines, via the observed hysteresis typical of ferromagnets, the history of the system. (c) Two-terminal resistance *R*<sub>16,16</sub> and four-terminal resistance *R*<sub>16,36</sub> as a function of the evolution of the magnetic field. The current is driven through the system via the contacts 1 (source) and 6 (drain). One obtains a quantised two-terminal resistance that is extremely close to *h/e*<sup>2</sup> indicating ballistic transport, at zero external magnetic field. The result for the four-terminal resistance depends on the history of the sample and thus the orientation of the magnetisation. In the case of a sweep from left to right (negative to positive magnetic fields), it is quantised in the same manner as the Hall resistance, *R*<sub>16,36</sub> = *h/e*<sup>2</sup>, while in the opposite sweep it vanishes as a consequence of the contacts 3 and 6 being at the same potential. After Bestwick *et al.*, Phys. Rev. Lett. **114**, 187201 (2015).
GaAs substrate) by the Goldhaber-Gordon group at Stanford University. As a consequence of the ferromagnetic order in the material, time-reversal symmetry is broken even in the absence of a magnetic field, and the system is thus reminiscent of that described by the Haldane model. However, in the experiment, an initial magnetic field is applied to the system to orient the magnetisation. The sample in the Hall-bar geometry is shown in figure 7.14(a), and the measurements of the longitudinal and Hall resistances are presented in figure 7.14(b), as a function of the magnetic field and the direction in which the latter is varied. “Sweep right” indicates a change of the magnetic field from negative to positive values and “sweep left” a change in the opposite direction. The most prominent feature one should notice in these measurements is the hysteresis in the resistances, which is ressembles the magnetisation of a ferromagnet in an external magnetic field. It is indeed a signature of the underlying ferromagnet, but here it is the magnetisation of the 2D compound that imposes the chirality of the edge channels in the otherwise insulating material.

The effect can be understood as follows. The insulating system is determined by a Chern number $±1$ the sign of which is fixed by the orientation of the magnetisation. If the magnetisation is oriented in the $+z$-direction, a quantum anomalous Hall effect described by a Chern number $C = +1$ is found, with a Hall resistance close to $h/e^2$,<fnref n="33" /> while a magnetisation in the $-z$-direction yields a Chern number $-1$ and consequently a Hall resistance of $-h/e^2$. As mentioned above, the external magnetic field is used to fix the magnetisation, and one sees that the magnetisation in the $+z$-direction persists within a left sweep down to a vanishing magnetic field and even to slightly negative values ($B \sim -150$ mT) before it switches abruptly to the $-z$-direction. Similarly this negative magnetisation persists in a right sweep.

The relevance of the edge-state transport is evidenced by the results of the two-terminal, $R_{16,16}$, and the four-terminal, $R_{16,36}$, resistances shown in figure 7.14(c), which we can easily understand with the help of the conductance

<footnote n="33">³³The precision of the quantisation is on the order of a percent, that is much lower than that in the quantum Hall effect. However, the present experiment shows one of the first measurements of the quantum anomalous Hall effect, and higher precisions are achieved today.</footnote>
matrix

$$ \mathcal{G}(C = -1) = \frac{e^2}{h} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & -1 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \end{pmatrix}. \qquad (7.103) $$

Recall that this 6 × 6 matrix reflects the fact that the sample is connected to six contacts in the present Hall geometry. This conductance matrix corresponds to a Chern insulator with $C = -1$ that is stabilised by sweeping the magnetic field from negative to positive values (sweep right) and that is accompanied by edge states propagating in a clockwise manner around the sample. As mentioned above, the conductance matrix for a Chern insulator with the opposite magnetisation (with a Chern number $C = +1$), stabilised in a left sweep, is obtained by simple matrix transposition,

$$ \mathcal{G}(C = +1) = {}^t \mathcal{G}(C = -1), \qquad (7.104) $$

since the edge states have now a chirality in a counter-clockwise sense.

Let us now investigate in detail the situation of a right sweep represented in the left panel of figure 7.14(c). We start with the determination of the potentials in the different contacts, namely in those where the current is zero, $I_j = 0$, that is in the contacts 2, 3, 4, and 5. The contacts 1 and 6 are the source and drain, respectively, by which the current is driven through the system. These potentials are readily calculated, since the Landauer-Büttiker equation, $I_j = \mathcal{G}_{jk}V_k$, simply reads

$$ 0 = \frac{e^2}{h} (-V_{k-1} + V_k), \qquad (7.105) $$

from which we see that the potentials in all contacts, 2 to 5, must be the same as in the source 1,

$$ V_2 = V_3 = V_4 = V_5 = V_1, \qquad (7.106) $$

exactly as in the situation encountered in the quantum Hall effect.<fnref n="34" /> The only potential that remains unfixed is that in the drain (contact 6). The

<footnote n="34">³⁴Remember that we may also set this potential to zero, as we can freely choose the reference potential.</footnote>
latter can be related to the potential $V_1$ by the Landauer-Büttiker formula for the source (or the drain). Indeed, in contact 1, it reads

$$I_1 = I = \frac{e^2}{h}(V_1 - V_6), \qquad (7.107)$$

and we can thus immediately read off the two-terminal resistance,

$$R_{16,16} = \frac{V_1 - V_6}{I} = \frac{h}{e^2}, \qquad (7.108)$$

as well as the four-terminal resistance,

$$R_{16,36} = \frac{V_3 - V_6}{I} = \frac{V_1 - V_6}{I} = \frac{h}{e^2}, \qquad (7.109)$$

in agreement with the measured values at $B = 0$ shown in the left panel of figure 7.14(c). In the last step we have used the fact that $V_3 = V_1$ [see equation (7.106)].

What changes if we consider the inverse magnetisation, i.e. a magnetic field that sweeps “left” from positive to negative values? As we have already discussed, the Chern number changes its sign, and we thus need to take the corresponding conductance matrix (7.104). The voltages in the contacts 2 to 5, with $I_j = 0$, are obtained in the same manner as before, but since the conductance matrix is transposed now, all these contacts are at equilibrium with $V_6$, and no longer with $V_1$,

$$V_2 = V_3 = V_4 = V_5 = V_6. \qquad (7.110)$$

The two-terminal resistance remains unaffected by this change of the potentials, and the balance in the source (7.107) is the same as before. The four-terminal resistance

$$R_{16,36} = \frac{V_3 - V_6}{I} = 0 \qquad (7.111)$$

thus vanishes due to the zero voltage drop between the contacts 3 and 6. This agrees indeed with the $B = 0$ value measured in the right panel of figure 7.14(c) (blue curve).
<figure><img src="image_79.png" /><figcaption>Figure 7.15: (a) Band structure of a topological insulator with time-reversal symmetry, $E_{\uparrow}(\mathbf{k}) = E_{\downarrow}(-\mathbf{k})$. There are two pairs of edge states. On the edge at $x = 0$ (continuous lines), the states associated with the two different spin orientations have opposite chirality (helical edge states). The chirality is inverted at the opposite edge at $x = -L$ (dashed lines). (b) Sketch of the orientation of the edge states in the sample.</figcaption></figure>

### 7.4.3 Helical edge states of a topological insulator with time-reversal symmetry – quantum spin Hall effect

We are now well prepared to understand the more complex situation of a topological insulator that respects time-reversal symmetry. Historically, this type of topological insulators was studied some years before the Chern insulator discussed in the previous section. We may again use the Kane-Mele model in the low-energy limit as a generic model for the edge states. As we have seen in section 7.3.1, the band structure is doubled as compared to that of the Haldane model, as a consequence of the spin-orbit coupling, $E_{\uparrow}(\mathbf{k}) = E_{\downarrow}(-\mathbf{k})$ (see figure 7.9). What does this doubling of the bulk bands imply for the edges? We know by now that if, within a topological phase transition, the gap parameter changes sign in the valley $\xi$ for spin-↑ electrons, time-reversal symmetry requires that this must be accompanied by an equivalent sign change in the valley $-\xi$ for spin-↓ electrons. If we apply again the picture of a gap inversion to the electronic behaviour at the edge, we see that the spin-↑ electrons, which are described by a Haldane model, yield a chiral edge state represented by the Hamiltonian $H_{\text{edge}}^{\xi}$ [see equation (7.94)]. As a consequence of time-reversal symmetry, spin-↓ electrons at the edge are then
represented by the edge Hamiltonian $H_{\text{edge}}^{-\xi}$. Let us consider, for the sake of simple illustration, that the spin-↑ electrons ($s = +$) experience a gap change in the valley $\xi = +$ and spin-↓ electrons ($s = -$) in the valley $\xi = -$ . Spin-↑ electrons at the edge $x = 0$ are then described by the Hamiltonian (7.94) and spin-↓ electrons by the Hamiltonian (7.95)

$$
H_{\text{edge}}^{s=+} = \hbar v \begin{pmatrix} q_y & \sqrt{2} \frac{\hat{a}^\dagger}{\ell_S} \\ \sqrt{2} \frac{\hat{a}}{\ell_S} & -q_y \end{pmatrix} \quad \text{and} \quad H_{\text{edge}}^{s=-} = \hbar v \begin{pmatrix} q_y & -\sqrt{2} \frac{\hat{a}}{\ell_S} \\ \sqrt{2} \frac{\hat{a}^\dagger}{\ell_S} & -q_y \end{pmatrix}, \tag{7.112}
$$

so that the chiral states $n = 0$ have a dispersion

$$
E_{n=0,x=0}^{s}(q_y) = s\hbar v q_y, \qquad (7.113)
$$

with a spin-dependent group velocity $v_s(x = 0) = sv$ in the $y$-direction. Our modeling of the interface therefore provides us with a *pair of edge states that propagate in opposite directions along the edge*, and their chirality (their sense of propagation) is locked to the electrons' spin orientation. This pair of edge states with opposite chirality is also called *helical edge states*. They are shown in figure 7.15, where the spin-dependent states at the edge $x = 0$ are represented by continuous lines (blue for spin-↑ and red for spin-↓ electrons). Notice that this could have been expected right from the beginning in view of time-reversal symmetry: in the absence of spin, there could be no chiral edge state since the direction of velocity would have been chosen spontaneously, thus breaking time-reversal symmetry. A chiral edge state therefore needs to be compensated by a counter-propagating state to restore time-reversal symmetry. Here, we see that this is naturally ensured within the Kane-Mele (and also within the BHZ) model, where the existence of the second edge state is provided by the second spin oriented in the opposite direction with respect to the first type of electrons that yields the first edge state.

Finally, if we consider the edge at $x = -L$, we can directly adopt the above picture and inspect the physical properties of the electrons there. As in the case of the Haldane model, the chiralities for the states $n = 0$ (for both spin orientations) need to be inverted with respect to the states at $x = 0$ because the gap parameter changes its sign in the opposite direction. As a consequence, the $n = 0$ edge states have the opposite dispersion

$$
E_{n=0, x=-L}^{s}(q_y) = -s\hbar v q_y, \qquad (7.114)
$$

as compared to those at the first edge at $x = 0$. This is indicated in figure 7.15 by the dashed lines (again blue for spin-↑ and red for spin-↓ electrons).
<figure><img src="image_80.png" /><figcaption>Figure 7.16: Measurement of the quantum spin Hall effect in a six-terminal configuration, in two CdTe/HgTe/CdTe quantum wells of different size. The resistances $R_{14,14}$ and $R_{14,23}$ are measured as a function of the gate voltage $V^*$, which allows one to tune the electronic density such that the Fermi level is in the bulk gap. After Roth *et al.*, Science **325**, 325 (2009).</figcaption></figure>

## Experimental realisation of the quantum spin Hall effect and helical edge states

Similarly to the quantum anomalous Hall effect, the helical edge states, which are at the origin of the quantum spin Hall effect, have a special transport signature. This was first observed in 2006 by the Molenkamp group at the University of Würzburg in the CdTe/HgTe/CdTe quantum well discussed in section 7.3.2. Figure 7.16 shows later experiments from 2009 by the same group. These experiments unveil happen to unveil the full power of the Landauer-Büttiker formalism in the analysis of helical edge transport, as we will see below. The figure shows the two-terminal $R_{14,14}$ and four-terminal $R_{14,23}$ resistances for two devices of different size in the Hall-bar geometry, as a function of the gate voltage $V^*$. The latter is used to control the position of the Fermi level in the bulk and thus to compensate possible intrinsic
doping due to impurities in the sample via the field effect. Naturally, the most interesting situation is that of a Fermi level in the bulk gap between the conduction and valence band so that one probes only the metallic edge transport. The geometry is sketched in the insets of figure 7.16, and the contacts 1 and 4 are the source and the drain, respectively.

In order to understand the measured resistances, we first need to set up the conduction matrix. Recall that the system consists of two Haldane models with opposite chirality depending on the spin orientation. The charge transport is therefore governed by the simple sum of the two models, and the conduction matrix is given by the sum of the two corresponding conduction matrices (7.103) and (7.104),

$$ \mathcal{G}_{\text{QSHE}} = \mathcal{G}(C = +1) + \mathcal{G}(C = -1) = \frac{e^2}{h} \begin{pmatrix} 2 & -1 & 0 & 0 & 0 & -1 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 \\ -1 & 0 & 0 & 0 & -1 & 2 \end{pmatrix}. \qquad (7.115) $$

The voltages in the different contacts are obtained in a similar manner as in the quantum anomalous Hall effect, by first investigating contacts with no current, $I_k = 0$, where one obtains

$$ 0 = \frac{e^2}{h} (-V_{k-1} + 2V_k - V_{k+1}). \qquad (7.116) $$

The voltage in the contact $k$ is therefore given by the average value of the voltages in the adjacent contacts,

$$ V_k = \frac{V_{k-1} + V_{k+1}}{2}, \qquad (7.117) $$

in agreement with Millman's theorem (4.27). Let us use $V_1 = 0$ as our reference potential and set $V_2 = V$. One then obtains, from the balance in contacts 2 and 3, $V_3 = 2V$ and $V_4 = 3V$. In order to obtain the voltages in the other contacts with $I_k = 0$, one may simply appeal to the mirror symmetry of the system around the axis connecting the contacts 1 and 4 that relates the potentials $V_2 = V_6 = V$ and $V_3 = V_5 = 2V$,<fnref n="35" /> so that one

<footnote n="35">³⁵This can also be seen explicitly if one defines the auxiliary potentials $V_6 = V'$. In</footnote>
eventually obtains

$$V_1 = 0, \quad V_2 = V, \quad V_3 = 2V, \quad V_4 = 3V, \quad V_5 = 2V, \quad V_6 = V, \qquad (7.118)$$

in terms of a single undetermined voltage $V$. One notices a remarkable difference with respect to the quantum Hall and the quantum anomalous Hall effects here: the potentials do not remain constant along the edge between the source and the drain. The presence of additional contacts, even if there is no current injected, alters the potentials. This is precisely due to the helical character of the edge states as one may see, e.g., when investigating contact 2. Indeed, this contact receives electrons not only from contact 1 but also from contact 3, which itself communicates with the drain (contact 4). The electrons arriving in contact 2 do not all have the same chemical potential, as it was the case in the quantum (anomalous) Hall effect, and the potential in the contact 2 is therefore the average between the potentials at which the electrons arrive, as it is stipulated by equation (7.117).

Now that we have identified the potentials in the different contacts, we need again to related the current to the voltage $V$, and this can again be achieved by the Landauer-Büttiker formula for the source or the drain. Choosing the source, we find

$$I_1 = I = \frac{e^2}{h}(-V_6 + 2V_1 - V_2) = -2\frac{e^2}{h}V \quad \Leftrightarrow \quad \frac{V}{I} = -\frac{1}{2}\frac{h}{e^2}. \qquad (7.119)$$

This allows us to obtain the two-terminal resistance

$$R_{14,14} = \frac{V_1 - V_4}{I} = \frac{-V_4}{I} = \frac{-3V}{I} = \frac{3}{2}\frac{h}{e^2} \qquad (7.120)$$

as well as the four-terminal resistance

$$R_{14,23} = \frac{V_2 - V_3}{I} = \frac{-V}{I} = \frac{1}{2}\frac{h}{e^2}, \qquad (7.121)$$

which are both in excellent agreement with the measured maximal values of figure 7.16, obtained for $V^* \simeq 0$ where the Fermi level is then in the gap between the conduction and valence bands.

<footnote>this case the balance equations in the contacts 6 and 5 yield $V_5 = 2V'$ and $V_4 = 3V'$, respectively. Since we already know that the potential in the drain is $V_4 = 3V$, one can immediately identify $V' = V$.</footnote>
Notice finally that it is hasardous, in the context of the quantum spin Hall effect, to speak generically of a transverse (or Hall) resistance as we usually do in the quantum (anomalous) Hall effect. The arguments based on the Berry curvature and the Chern number (see sections 7.1 and 7.3.1) indicate that the transverse resistivity associated with charge transport should be zero. While this is indeed the case for the voltage drop between the opposite contacts 2 and 6 (or 3 and 5),

$$R_{14,26} = R_{14,35} = 0, \qquad (7.122)$$

one obtains a different (non-zero) result for the four-terminal resistance

$$R_{14,25} = \frac{V_2 - V_5}{I} = \frac{-V}{I} = \frac{1}{2} \frac{h}{e^2} \qquad (7.123)$$

between the contacts 2 and 5 even if they are situated at opposite edges, as are the contacts 2 and 6 (or 3 and 5). In the case of the quantum (anomalous) Hall effect, the latter would be zero as for the other cases. As generally required by the Landauer-Büttiker formalism, one clearly sees here that one needs to specify precisely the contacts by which the voltage drop is measured, in addition to the source and the drain. The value of the resistances depends crucially on the arrangement of the contacts that connect electrically the sample to the outer world.

## Notions and concepts to retain

* time-reversal symmetry and its manifestation in Bloch bands and the Berry curvature

* quantum anomalous Hall effect and quantum spin Hall effect

* Haldane, Kane-Mele, and Bernevig-Hughes-Zhang models

* CdTe/HgTe/CdTe heterostructures

* bulk-edge correspondence

* chiral and helical edge states and related edge transport in the Landauer-Büttiker formalism
# Chapter 8

## Three-dimensional topological materials

In the preceding chapters, our interest was mainly focused on 2D materials and their topological classification with the help of the Berry curvature. The latter turns out to be useful also in the geometric description of certain 3D materials, but one needs then to take into account seriously its vectorial character encoded in equation (6.19). Remember that only the z-component of the Berry curvature matters in the classification of 2D systems because the wave vector is varied in the $xy$-plane, and the Berry curvature is perpendicular to this plane of variation. The aim of the present chapter is to provide a (non-exhaustive) glimpse at 3D topological materials, still with the help of the Dirac equation. We have noticed that this equation, and its corresponding Hamiltonian, is represented with the help of the 2 × 2 Pauli matrices. This is not the case in three spatial dimensions where 4 × 4 matrices are required to correctly represent the Dirac equation, as Dirac realised in his original work from the 1920ies.

We start this chapter with a brief recapitulation of the Dirac equation and the Dirac Hamiltonian in its historical form (section 8.1) before discussing its utility in the description of 3D topological insulators. Similarly to 2D systems, where the topological character of the band structure provides us with chiral edge states, we encounter chiral, topologically protected *surface states*. Here, chirality no longer refers to the sense of propagation of the edge states but rather to the winding of the electronic spin (or a pseudospin) on a closed path in reciprocal space around singular points. The appearance of such metallic surface states, even for bulk insulating materials, is again
enforced by the bulk-boundary correspondence, which was covered (in the form of the bulk-edge correspondence) in detail in section 7.4 of the preceding chapter. The chiral surface states have been observed first in photoemission and then in transport measurements.

Finally, we will encounter another unusual phase of 3D matter in this chapter: the so-called Weyl semimetals, which may somewhat grossly be described as 3D graphene with massless low-energy fermions. It is again an example of relativistic quantum mechanics applied to condensed-matter physics. Indeed, the German mathematician and theoretical physicist Weyl showed that the 3D Dirac equation for zero-mass particles possesses a chiral symmetry that allows for a description of these massless Weyl fermions in terms of a 2 × 2 Hamiltonian, i.e. in terms of Pauli matrices, similarly to 2D relativistic fermions. We provide a succinct introduction to Weyl semimetals at the end of this chapter in section 8.3.

## 8.1 A bit of history – the 3D Dirac equation

During your basic quantum-mechanics classes, you should have been confronted with two (equivalent) formulations of quantum mechanics: Heisenberg's matrix formulation (with the convenient Dirac notation in terms of bras and kets) and Schrödinger's wave mechanics. Usually one starts with Schrödinger's formulation in these basic classes because it is considered to be more accessible than Heisenberg's approach. This was also the case in the 1920ies when both formulations came to life: indeed, physicists at that moment were much more familiar with (partial) differential equations that are used in Schrödinger's approach than with matrix calculus required in Heisenberg's theory. Notice, however, that the latter is even a bit older (formulated in 1925) than Schrödinger's (1926), which was based on de Broglie's PhD work from 1924 where the concept of matter waves was first introduced, $\psi(\mathbf{r}, t) \propto \exp[i(Et - \mathbf{p} \cdot \mathbf{r})/\hbar]$, to describe the interferential power of electrons. As you know, in de Broglie's matter waves, energy is related to the wave frequency, $E = \hbar\omega$, and momentum to the wave vector, $\mathbf{p} = \hbar\mathbf{k}$, in terms of the (reduced) Planck constant $\hbar$. In order to obtain the Schrödinger equation, it is therefore sufficient to interpret the equation

$$E = \frac{\mathbf{p}^2}{2m} + V(\mathbf{r}) \qquad (8.1)$$
as an operator equation acting on the wave function, where the operators are introduced with the usual substitution

$$E \rightarrow i\hbar \frac{\partial}{\partial t} \quad \text{and} \quad \mathbf{p} \rightarrow -i\hbar \nabla. \qquad (8.2)$$

That does not seem to be too complicated! However, this apparent simplicity is based on a century of quantum calculus familiar to a scientist living in the 21st century, who makes use of quantum mechanics on an almost daily basis. Let us therefore be a bit more modest. You may nevertheless ask yourself why it took Schrödinger a rather long time between his acquaintance with de Broglie's work and the formulation of his equation. There may naturally be many reasons, but one of them is certainly that physicists, especially from the younger generation as Schrödinger's, were already acquainted with Einstein's theory of (at least special) relativity, according to which the correct energy-momentum relation for a particle of mass m is given by

$$E = \sqrt{m^2c^4 + \mathbf{p}^2c^2}, \qquad (8.3)$$

while equation (8.1) is valid only in the non-relativistic limit where the particle moves at a velocity that is small compared to the speed of light c.

Indeed, Schrödinger first tried to use the full dispersion (8.3) to build up his wave equation, but he was confronted with a formidable problem: the differential operator associated with the momentum is under the square root, i.e. the energy is not a simple power of the momentum operator as it is the case in the non-relativistic limit (8.1). There is an obvious solution to this problem proposed, by the way, both by Klein and Gordan on the one-hand side and by Schrödinger on the other hand: just take the square of equation (8.3).<fnref n="1" /> However, while this solves the problem of differential operators under the square root, it is the source of a new problem: instead of specifying only the wave function at an initial moment $t = 0$, $\psi(\mathbf{r}, t = 0)$, one is also forced to specify its time derivative $\partial_t\psi(\mathbf{r}, t = 0)$ as a boundary condition since it is a second-order differential equation in time.

It was finally the British physicist Dirac in 1928 who proposed the most widely used solution to the square-root problem, in a somewhat punk-style manner: if you are bothered by the square root, just omit it! Well, that gives

$$E = mc^2 + \mathbf{p}c \qquad (\text{utterly wrong!!!}). \qquad (8.4)$$

<footnote n="1">¹The resulting differential equation is known in relativistic quantum mechanics under the name Klein-Gordon or relativistic Schrödinger equation.</footnote>
An advice to students: even if Dirac got the Nobel Prize, don't do that in your exams! Naturally, this expression does not make any mathematical sense because one adds up a scalar and a vector on the right-hand side of the equation. Dirac was clearly aware of that problem. In order to heal it, he introduced some auxiliary mathematical objects, $\beta$ and $\alpha = (\alpha_x, \alpha_y, \alpha_z)$ in the equation,

$$E = \beta mc^2 + \alpha \cdot pc \quad (\text{already better}), \quad (8.5)$$

that allowed him to make use of the substitution (8.2) in order to obtain the differential equation, known as the Dirac equation,

$$i\hbar \frac{\partial}{\partial t} \psi(\mathbf{r}, t) = (\beta mc^2 - i\hbar c\alpha \cdot \nabla) \psi(\mathbf{r}, t), \quad (8.6)$$

which has again plane-wave solutions $\psi(\mathbf{r}, t) \propto \exp[i(\mathbf{p} \cdot \mathbf{r} - Et)/\hbar]$. There is nevertheless a price to pay that is given by the new quantities $\beta$ and $\alpha$, which need to be specified. From a dimensional point of view, they are purely mathematical objects with no physical dimension. Furthermore, we need to make sure that the square of the equation yields again the correct equation given by (the square of) equation (8.3). Indeed, the square of equation (8.5) yields

$$E^2 = \beta^2 m^2 c^4 + (\alpha_x^2 p_x^2 + \alpha_y^2 p_y^2 + \alpha_z^2 p_z^2)c^2 + \sum_{i=x,y,z} (\beta\alpha_i + \alpha_i\beta)p_i mc^3 + \sum_{i \neq j} (\alpha_i\alpha_j + \alpha_j\alpha_i)p_i p_j c^2, \quad (8.7)$$

with a large number of undesired (mixed) terms that are absent in equation (8.3). These undesired terms are those of the second line in the above equation, and one notices that they become zero if the new mathematical objects satisfy

$$\alpha_i \alpha_j + \alpha_j \alpha_i = 0 \quad \text{and} \quad \beta \alpha_j + \alpha_j \beta = 0, \quad (8.8)$$

for $i \neq j$, while they square up to one individually,

$$\alpha_j^2 = 1 \quad \text{and} \quad \beta^2 = 1. \quad (8.9)$$

The price to pay for Dirac's trick of omitting the square root is now clear: the newly introduced mathematical objects must *anti-commute* and therefore be represented in terms of *matrices*. Furthermore, these matrices must be
Hermitian in order for the Hamiltonian to be Hermitian itself, and the last equation (8.9) shows that they are also unitary then. If we rename $\beta = \alpha_0$, for notational economy, equations (8.8) and (8.9) may be rewritten in a compact manner as

$$ \{\alpha_{\mu}, \alpha_{\nu}\} = 2\delta_{\mu,\nu}, \qquad (8.10) $$

where $\mu = 0, 1, 2, 3$ (with the identification of the indices $x = 1, y = 2$ et $z = 3$). As usual, the curly brackets denote the anti-commutator $\{A, B\} = AB + BA$ between two matrices (or operators) A and B. Equation (8.10) represents a Clifford algebra of which we have already encountered an example in the form of the three Pauli matrices $\sigma_x, \sigma_y$, and $\sigma_z$. Because the Pauli matrices also form a basis (along with the $2 \times 2$ one matrix $I$) of Hermitian $2 \times 2$ matrices, it is not possible to construct a fourth Hermitian $2 \times 2$ matrix that anti-commutes with the three Pauli matrices. For this reasons, we may not use Pauli matrices to represent the Clifford algebra and thus the Dirac equation in three spatial dimensions. The latter are nevertheless sufficient to represent the Clifford algebra (8.10) in two spatial dimensions, identifying $\alpha_1 = \sigma_x, \alpha_2 = \sigma_y$, and $\alpha_0 = \beta = \sigma_z$.<fnref n="2" /> In spatial dimensions higher than $D=2$, we thus need to search for a representation in terms of larger matrices. It turns out that $3 \times 3$ matrices are not eligible because one needs $2n \times 2n$ matrices in order to represent the Clifford algebra, and the smallest matrix dimension in a 3D space is then $4 \times 4$.<fnref n="3" />

As a consequence of the enlarged matrix size, we have a great choice of equivalent representations of the Clifford algebra by $4 \times 4$ matrices. Quite generally, we may use a second set of Pauli matrices, $\tau_x, \tau_y$, and $\tau_z$ (plus the $2 \times 2$ one matrix $I$), along with the first set to generate a basis of $4 \times 4$ matrices by using a tensor product between the two sets, $\tau_\mu \otimes \sigma_\nu$.<fnref n="4" /> The so-

$$ H_{2D} = mc^2\sigma_z + c\sigma \cdot p, $$

which describes electrons in graphene ($m=0$) or boron nitride ($m = \Delta/c^2$), see equation (2.71) in section 2.3.

<footnote n="2">²Indeed, we retrieve, upon substitution of these matrices in equation (8.5), the 2D Dirac Hamiltonian</footnote>

<footnote n="3">³In order to prove this point, recall that the determinant of a product of two matrices does not depend on the order of the matrices in the product even if we deal with non-commutative matrices, i.e. $\det(\alpha_i\alpha_j) = \det(\alpha_j\alpha_i)$. Furthermore, we know that, because of the anti-commutation for $i \neq j$, $\det(\alpha_i\alpha_j) = \det(-\alpha_j\alpha_i) = (-1)^N \det(\alpha_j\alpha_i)$, where we consider $N \times N$ matrices in the last step. One therefore necessarily has $(-1)^N = 1$, i.e. $N = 2n$ must be an even integer.</footnote>

<footnote n="4">⁴In order to have a basis of Hermitian $4 \times 4$ matrices, one needs 16 matrices, one of</footnote>
called *standard representation* with $\beta = \tau_z \otimes \mathbb{I}$ and $\alpha_j = \tau_x \otimes \sigma_j$ is one such choice and yields the Hamiltonian

$$ H_{\text{st.r.}} = \begin{pmatrix} mc^2\mathbb{I} & c\mathbf{p} \cdot \boldsymbol{\sigma} \\ c\mathbf{p} \cdot \boldsymbol{\sigma} & -mc^2\mathbb{I} \end{pmatrix}. \qquad (8.11) $$

Alternatively, another commonly used representation, called the *Weyl representation*, with $\beta = -\tau_x \otimes \mathbb{I}$ and $\alpha_i = \tau_z \otimes \sigma_i$ gives a Hamiltonian of the form

$$ H_{\text{W.r.}} = \begin{pmatrix} c\mathbf{p} \cdot \boldsymbol{\sigma} & -mc^2\mathbb{I} \\ -mc^2\mathbb{I} & -c\mathbf{p} \cdot \boldsymbol{\sigma} \end{pmatrix}. \qquad (8.12) $$

The latter representation is advantageous in the description of 3D massless particles (with $m = 0$) because the Hamiltonian consists then of two independent $2 \times 2$ blocks that do not communicate among each other. It was indeed Weyl who pointed out first the resulting redundancy in the representation of massless particles and that massless fermions (called *Weyl fermions*) can then also be more easily represented in terms of the usual $2 \times 2$ Pauli matrices, whereas we have just seen that massive particles require a description in terms of $4 \times 4$ matrices. We shall see below that both cases are encountered, somewhat unexpectedly perhaps, in 3D topological materials. Topological insulators in 3D and the related 3D massive Dirac particles require indeed a description in terms of $4 \times 4$ matrices, while 3D Weyl semimetals, with band crossing points that are protected against gap opening, are conveniently described by $2 \times 2$ matrices.

Before discussing in detail their use in 3D topological materials, let us get back to the Dirac Hamiltonian (8.11) or (8.12) in relativistic quantum physics. Since they are $4 \times 4$ matrices, the associated wave functions or states that diagonalise the Dirac Hamiltonian are clearly *four-component spinors* the physical meaning of which is yet to be unveiled. Recall first that the diagonalisation yields two energy bands that are two-fold degenerate,

$$ E_{\lambda}(\mathbf{p}) = \lambda \sqrt{m^2 c^4 + \mathbf{p}^2 c^2}. \qquad (8.13) $$

Within Bloch's band theory in condensed matter, we are now used to positive ($\lambda = +$) and negative ($\lambda = -$) energy bands in the form of the conduction and valence bands, respectively. Again, this was not the case in 1928 when

<footnote>which is the $4 \times 4$ one matrix. The above tensor product between the two sets of three Pauli matrices yields already 9 such matrices, and the remaining 6 are obtained by using a tensor product between the Pauli matrices and the one matrix, $\mathbb{I} \otimes \sigma_j$ and $\tau_i \otimes \mathbb{I}$.</footnote>
Dirac proposed his equation along with such positive- and negative-energy solutions, and the physicists of that time had difficulty in accepting the concept of particles with a negative energy and a downward dispersion. Some of them did not take Dirac seriously and thought that this was simply an artefact of an anyhow crazy theory! It was then certainly not very helpful that Dirac showed a certain temerity in postulating that these negative-energy solutions be indeed physical and represent a new type of particle, the *anti-particle* (in the present case, it is the positron of charge $+e$ opposite to that, $-e$, of an electron). However, his interpretation turned out to be correct, and four years later the American physicist Andersen discovered the positron experimentally.<fnref n="5" /> Notice that this unusual concept of anti-particles was in resonance with Bloch's band theory (also in 1928), where positrons arise "naturally" in the form of *holes*<fnref n="6" /> in the valence band, which has a negative curvature (and thus a negative dispersion) as opposed to that of the conduction band.

We underline finally that each of the (negative and positive) energy bands are twofold degenerate. This twofold degeneracy is nothing other than the spin degeneracy, which is now intrinsic in the Dirac equation. Recall that the spin needed to be added "by hand" to the Schrödinger equation. Here, however, the Pauli matrices $\sigma$ are required (within a tensor product with others) to satisfy the Clifford algebra. *They represent precisely the physical spin-1/2 of the electrons that is intrinsic in the Dirac equation.* The twofold degeneracy is then nothing other than the Kramers' degeneracy,<fnref n="7" />

<footnote n="5">⁵Anderson used a cloud chamber to measure the charge-to-mass ratio of elementary particles created in the atmosphere by cosmic rays. Every time such a particle passes through the cloud chamber, which consists of a saturated vapor, it leaves behind a trace of ionised droplets that indicates the particle's trajectory. In order to measure the charge-to-mass ratio the particle's trajectory is bent by an external magnetic field. While the charge-to-mass ratio requires knowledge of the electric and magnetic field and can then be read off by measuring the radius $R_C$ of the bent trajectory, $q/m = E/B^2 R_C$, the sign of the charge $q$ is simply due to the deviation of the trajectory. The trace he attributed to the positrons had exactly the required property to be bent in the direction opposite to that of an electron, while it had the same charge-to-mass ratio.</footnote>

<footnote n="6">⁶The notion of the *hole* was originally introduced by Peierls in 1929, interestingly in a similar context as that of the positron discovery. Indeed, Peierls tried to understand, in the framework of Bloch's very recent band theory, the Hall response of certain metals, which indicated that the charge carriers had the opposite charge $+e$ as that of an electron. He understood that the responsible charge carrier was then a missing electron in the valence band that he called accordingly a hole.</footnote>

<footnote n="7">⁷Kramers formulated his theorem in 1930, two years after the advent of the Dirac</footnote>
which we discussed in the preceding chapter in section 7.1.1. That the Pauli matrices $\sigma$ describe indeed the physical spin, i.e. its particular coupling to external electric and magnetic fields, may be shown by restricting the Dirac equation in the presence of an electric and a magnetic field to the electron (positive-energy) band and treating the fields perturbatively. This perturbative treatment, which is beyond the scope of these lecture notes, yields then the usual term $\propto \sigma \cdot B$ responsible for the Zeeman effect as well as the spin-orbit coupling $\propto \sigma \cdot (\mathbf{p} \times \mathbf{E})$ [see equation (7.34) in section 7.1].

## 8.2 3D topological insulators and surface states

With the help of our understanding of 2D topological insulators, based on chapter 7, and that of the 3D Dirac equation introduced in the preceding section, we are now armed to approach the physics of 3D topological insulators. Similarly to a 2D topological insulator, its 3D counterpart is a bulk insulator accompanied by metallic states that emerge at the surface of the material or if it is interfaced with a trivial insulator (or more generally an insulator with a different topological invariant). We do not specify, here, the different 3D topological invariants, but we invoke again our picture of a topological phase transition (described in section 6.4.1) in terms of a gap inversion at the surface of the material (see section 7.4). Remember that a topological phase transition comes along with a gap closing that can conveniently be described in terms of a massive Dirac fermion, where the mass term (or gap parameter) changes its sign while vanishing at the transition. This feature is also present in 3D topological insulators. However, we need to take into account a complication that consists of the necessity to describe a 3D massive Dirac fermion in terms of a 4 × 4 matrix Hamiltonian, as we have just seen in the preceding section. Otherwise, the approach is similar and the surface state may be modeled by a mass term

$$H_{\text{masse}} = \beta m(z) c^2 \qquad (8.14)$$

that changes its sign across the interface, which we have chosen to be perpendicular to the z-axis here. As in the case of edge states in 2D topological insulators, the non-commutativity between the position z that enters in the argument of the gap parameter and the z-component $q_z$ of the wave vector

<footnote>equation.</footnote>
yields a quantisation of the motion perpendicular to the interface into discrete levels, indexed by the quantum numbers $(\lambda, n)$, while the electronic motion remains free in the $xy$-plane of the interface. We therefore obtain a family of 2D bands, $E_{\lambda,n}^{\xi}(q_x, q_y)$, that describes the surface states. Furthermore, as we shall see in detail below, the state with $n = 0$ is again a chiral metallic state that disperses through the bulk gap.

### 8.2.1 Angle-resolved photoemission experiments

Before discussing in mathematical detail the above-mentioned surface states and the fate of the bulk-boundary correspondence in 3D materials, let us first introduce some experimental observations of these surface states. Contrary to 2D topological insulators that had originally been studied in transport experiments, the first family of 3D topological insulators – binary $Bi_{(1-x)}Sb_x$ alloys – turned out to consist of strongly disordered materials so that they are no good bulk insulators. Indeed, disorder levels in the bulk gap yield electric short-circuits that cover the electric response from the surface states. It is therefore by spectroscopic measurements – more precisely by angular-resolved photoemission spectroscopy (ARPES) – that 3D topological insulators were experimentally discovered in 2008 [Hsieh et al., Nature 452, 970 (2008)]. The ARPES technique is based on the photoelectric effect, which allows one to detect an electron extracted from a material by an incident photon. Knowledge of the incident photon's energy and momentum and the measurement of the energy and momentum (via the angle) of the emitted electron provides direct access to the band structure $E_n(k)$ of a material as a consequence of the involved conservation laws.<fnref n="8" />

Shortly after this initial discovery, a second family based on bismuth and antimony – $Bi_2Se_3$, $Bi_2Te_3$, $Sb_2Se_3$, $Sb_2Te_3$ – has been identified as 3D topological insulators, still by the ARPES technique. The results obtained by Hsieh et al., Nature (London) 460, 1101 (2009) for the compound $Bi_2Se_3$ are depicted in figure 8.1. The first line (a) of the figure shows the evolution of the Fermi surface in reciprocal space upon change of the Fermi level by hole doping. Here, hole doping is achieved chemically with the help of $NO_2$ adsorption on the (111)-surface of the crystal. The second line (b) indicates the same evolution of the energy dispersion as a function of $k_x$ for a fixed

<footnote n="8">⁸For further information about ARPES, see e.g. the book by Henri Alloul used for the class “Electrons in Solids” (PHY552A).</footnote>
<figure><img src="image_81.png" /><figcaption>Figure 8.1: Measurement of the surface states in Bi₂Se₃ by APRES, on the (111) surface. Figure (a) shows the evolution of the Fermi surface (in the (kₓ, kᵧ)-plane) as a function of (hole) doping achieved by adsorption of NO₂ molecules. Figure (b) shows the energy of the surface state at kᵧ = 0 as a function of kₓ. Only occupied states are visible. Notice that there are also some bulk states in the conduction band (plain circle) visible on the left panel. After Hsieh *et al.*, Nature (London) **460**, 1101 (2009).</figcaption></figure>
value of $k_y = 0$. One clearly sees in these measurements the appearance
of a surface state that disperses as a 2D massless Dirac fermion. Since the
emitted electron originally (before its interaction with the photon) occupied
a state with energy $E(\mathbf{k}) < E_F$, ARPES gives access to the occupied parts
of the electronic bands below the Fermi level. Notice furthermore, on the
left panels, that one does not only see the occupied surface states [in the
form of the bright ring in the $(k_x, k_y)$-plane] but also some occupied parts
of the bulk conduction band (inner circle). In the absence of $NO_2$ doping,
the Fermi level is therefore in the bulk conduction band and the system is
metallic not only at the surface but also in the bulk. It is only upon hole
doping that the Fermi level is placed in between the bulk bands so that
one obtains an insulator where the only metallic states reside at the surface
(panels in the middle column). In the figures of the right column, the doping
is such that the Fermi level is situated precisely at the Dirac point where the
surface bands cross. Notice finally that the material is also characterised by
a strong spin-orbit coupling. Consequently, the spin in the bands depends
on the orientation of the wave vector, as it is indicated schematically by the
arrows in figure 8.1(b) pointing in opposite directions depending on the sign
of the component $k_x$ of the wave vector.

The observed surface state behaves as one might have expected qualita-
tively from the proposed modeling via a spatially dependent mass gap (8.14)
that vanishes at the surface: it consists of a massless Dirac fermion with a
conical dispersion in the vicinity of the band crossing. This is due to the
2D character of the surface state, while we have seen that the edge state of
a 2D topological insulator is itself 1D with a linear band dispersion. The
nature of this 2D surface state is discussed in more detail in the following
section, where we revisit the bulk-boundary correspondence. However before
doing so, notice that the above ARPES measurements agree with numeri-
cal band-structure calculations that are reproduced in figure 8.2. As for the
ARPES measurements they deal with a (111) surface of the Bi- and Sb-based
compounds, and one notices the appearance of states crossing the band gap.
Most saliently in the case of Sb<sub>2</sub>Te<sub>3</sub> and Bi<sub>2</sub>Se<sub>3</sub>, one clearly observes Dirac
cones.

8.2.2 Bulk-boundary correspondence in 3D

Let us now proceed to a more quantitative (albeit simplified) description of
3D topological insulators and their surface states with the help of the model
<figure><img src="image_82.png" /><figcaption>Figure 8.2: Numerical band-structure calculations for Sb₂Se₃ (a), Sb₂Te₃ (b), Bi₂Se₃ (c), and Bi₂Te₃ (d). Here, the 2D bands are associated with the (111)-surface of this family of compounds. The bars over the letters (on the x-axis), $\bar{\Gamma}$, $\bar{M}$, and $\bar{K}$, indicate the projection of the respective reciprocal-space points $\Gamma$, $M$, and $K$ of the first Brillouin zone onto the (111) surface. From Zhang *et al.*, Nat. Phys. **5**, 438 (2009).</figcaption></figure>
for 3D Dirac fermions or, more precisely, with the help of variants of the models presented in equations (8.11) and (8.12). With respect to the latter, one needs to recall that, in condensed matter, the components of the wave function and thus of the Bloch Hamiltonian denote the respective weight on the underlying atomic orbitals as well as the physical spin, while in the 3D Dirac equation the components other than the spin-1/2 copies are more abstract, i.e. they do not have a clear physical meaning and can to some extent be chosen arbitrarily, as it is stipulated by equations (8.11) and (8.12) in section 8.1. Furthermore, the Dirac Hamiltonian, which describes the low-energy properties of the systems under scrutiny must respect time-reversal symmetry that is not broken in our Bi- and Sb-based compounds. Such a Hamiltonian that respects these conditions has indeed been proposed by Zhang, Kane, and Mele [Phys. Rev. B 86, 081303(R) (2012)] to describe low-energy electrons in the model system Bi₂Se₃ around the Γ point where the direct gap is situated,

$$ H_{3DTI}(\mathbf{q}) = \Delta\tau_z \otimes \mathbb{I} + \hbar v q_z \tau_y \otimes \mathbb{I} + \hbar v \tau_x \otimes (q_y \sigma_x - q_x \sigma_y) \quad (8.15) \\ = \begin{pmatrix} \Delta & 0 & -i\hbar v q_z & \hbar v(q_y + i q_x) \\ 0 & \Delta & \hbar v(q_y - i q_x) & -i\hbar v q_z \\ i\hbar v q_z & \hbar v(q_y + i q_x) & -\Delta & 0 \\ \hbar v(q_y - i q_x) & i\hbar v q_z & 0 & -\Delta \end{pmatrix} $$

Here, the Pauli matrices $\tau_j$ represent (superpositions of) atomic orbitals<fnref n="9" /> and the matrices $\sigma_j$ represent the physical spin, which explicitly enters the model as a consequence of the prominent spin-orbit coupling in these materials. Instead of discussing the origin of this model, let us just check some essential properties to convince ourselves that it is a faithful representation of the low-energy electronic properties that are of interest here.

1. The model represents indeed a massive Dirac fermion (with mass $m = \Delta/v^2$) since all matrices $\tau_z \otimes \mathbb{I}$, $\tau_y \otimes \mathbb{I}$, $\tau_x \otimes \sigma_y$, and $\tau_x \otimes \sigma_x$ used to construct the Hamiltonian anti-commute mutually and therefore satisfy the Clifford algebra.

2. Time-reversal symmetry is respected since $H_{3DTI}(\mathbf{q}) = H_{3DTI}^*(-\mathbf{q})$. This can be seen directly from the action of this symmetry on the

<footnote n="9">⁹The relevant orbitals are the $p_z$ orbitals of Bi and S that are coupled, the eigenvalue $|+\rangle$ of $\tau_z$ representing the $p_z$ orbital of Se and $|-\rangle$ that of Bi.</footnote>
physical spin that is reversed [see equation (7.6) of the preceding chapter), i.e. $T\sigma_j T^{-1} = -\sigma_j$, for the spin components represented by the Pauli matrices $\sigma_j$. As for the Pauli matrices $\tau_j$ representing the orbital degrees of freedom, which are not affected by time-reversal symmetry, we have $T\tau_{x/z}T^{-1} = \tau_{x/z}$. However, we must take into account the purely imaginary character of the matrix $\tau_y$, which is thus transformed as $T\tau_yT^{-1} = -\tau_y$.

In the same manner as for 2D topological insulators, one may describe the surface states with the help of a gap inversion over an interval of width $\ell$, which we choose arbitrarily to be in the $z$-direction such that the surface is in the $xy$-plane. As in section 7.4, this allows us to model the surface (at $z=0$) by a gap

$$ \Delta(z/\ell) = \Delta \tanh(z/\ell) \qquad (8.16) $$

that varies between a negative value, $-\Delta$ for $z/\ell \ll 0$, and a positive one, $\Delta$ for $z/\ell \gg 0$. If we restrict the model to the surface $-\ell < z < \ell$, we may again linearise the gap function and obtain $\Delta(z/\ell) \simeq \Delta z/\ell$. The calculation of the spectrum is identical to that of the 2D case, and we recall the steps only in a succinct manner here. In order to obtain the canonical representation in terms of ladder operators and the effective length $\ell_S = \sqrt{\ell\hbar v/\Delta}$,

$$ \hat{a} = \frac{\ell_S}{\sqrt{2}}(q_z - iz/\ell_S^2) \quad \text{and} \quad \hat{a}^\dagger = \frac{\ell_S}{\sqrt{2}}(q_z + iz/\ell_S^2), \qquad (8.17) $$

we perform the unitary transformation $U = \exp(i\pi\tau_y \otimes \mathbb{I}/4)$, which exchanges the Pauli matrices $\tau_x$ and $\tau_z$. This yields the Hamiltonian

$$ H_{\text{surface}} = \hbar v \begin{pmatrix} 0 & q_y + i q_x & \frac{\sqrt{2}}{\ell_S} \hat{a} & 0 \\ q_y - i q_x & 0 & 0 & \frac{\sqrt{2}}{\ell_S} \hat{a} \\ \frac{\sqrt{2}}{\ell_S} \hat{a}^\dagger & 0 & 0 & -(q_y + i q_x) \\ 0 & \frac{\sqrt{2}}{\ell_S} \hat{a}^\dagger & -(q_y - i q_x) & 0 \end{pmatrix}, \qquad (8.18) $$

the spectrum of which consists of a family of 2D bands localised at the surface (around $z=0$),

$$ E_n(\mathbf{q}) = \hbar v \sqrt{|\mathbf{q}|^2 + 2n/\ell_S^2}, \qquad (8.19) $$

for $n \neq 0$, as we have seen in the preceding chapter. However, now, the wave vector in the surface is itself 2D, and we have set $\mathbf{q}_\parallel = (q_x, q_y)$. As in the 2D
<figure><img src="image_83.png" /><figcaption>Figure 8.3: Chirality of the massless Dirac fermion at the surface of a 3D topological insulator. As a consequence of the spin-orbit coupling, which is inherited from the bulk properties, the spin orientation of an electron is linked to its wave vector. At constant energy in the conduction band (say at the Fermi level), the spin turns in the same sense as the wave vector when the latter makes a full loop around the band touching point at $q_\parallel = 0$. If the Fermi level is in the valence band the sense of rotation is inverted. After Hasan and Kane, Rev. Mod. Phys. 82, 3045 (2010).</figcaption></figure>

case, the effective length $\ell_S$ determines the extension of these surface states in the z-direction, which is $\sim \sqrt{2n\ell_S}$.

Again, the state with $n = 0$ is special: the associated wave functions have only two non-zero components,

$$ \psi_{n=0}(\mathbf{q}_{\parallel}) = \begin{pmatrix} 0 \\ 0 \\ u_{\uparrow}(\mathbf{q}_{\parallel}) \\ u_{\downarrow}(\mathbf{q}_{\parallel}) \end{pmatrix}, \qquad (8.20) $$

and the *chiral surface state* is thus described in terms of the 2×2 Hamiltonian

$$ H_{\text{surface}}^{n=0} = -\hbar v(q_y \sigma_x - q_x \sigma_y). \qquad (8.21) $$

One clearly sees that one retrieves a Dirac Hamiltonian for 2D massless fermions, similarly to low-energy electrons in graphene discussed in chapter 2. However, in comparison to the latter, we need to emphasise a certain number of remarkable differences.
1. We find a *single* (massless) Dirac fermion at the surface and not a pair as in the case of graphene. This is due to the particular geometry chosen here with a semi-infinite space.

2. If we had a second surface delimiting the topological insulator at $z = -L$, such as in a slab geometry of width $L$, one would obtain a second Dirac fermion with opposite chirality with respect to the original surface. Again, this can be seen from the sign change in the gap function at the second surface, which yields, as in the 2D case discussed in section 7.4, an exchange of the operators $\hat{a}$ an $\hat{a}^\dagger$ in Hamiltonian (8.18). As a consequence, the first two components of the associated $n=0$ wave function are now non-zero. The Hamiltonian in the $n=0$ subspace is the same as that of the original surface (8.21) but with an opposite global sign (reversed spin).

3. The surface states with $n=0$ are said to be chiral. However, contrary to the 2D topological insulators discussed in the preceding chapter with its 1D edge channels, *chirality* does not refer here to the sense of propagation. Chirality now means the orientation of the spin and its change when an electron moves along a closed path around the band contact point at $q_\| = 0$. In the conduction band of positive energy, the spin turns in the same sense as the wave vector (see figure 8.3), while it turns in the opposite sense in the surface valence band.

4. As for 2D topological insulators, the existence of chiral surface states does not depend on the precise manner how the surface is modeled. They persist even in the limit of an abrupt interface, $\ell \to 0$, in contrast to the *massive surface states*, with $n \neq 0$, the energy of which is pushed into that of the bulk continuum, as one may see from equation (8.19).

### 8.2.3 Relativistic quantum Hall effect at the surface

Beyond spectroscopic measurements, there is experimental evidence for chiral surface states in quite remarkable transport measurements, with the observation of another form of the relativistic quantum Hall effect similar to graphene. In order to appreciate these measurements, let us start with its conceptual discussion. Imagine for the moment that we could create a Hall bar on a single surface of a 3D topological insulator. What would you expect from a Hall measurement based on our knowledge of the quantum Hall effect
in graphene (section 5.2.3)? In the same manner as in graphene, the low-energy physical properties are described in terms of a 2D Dirac equation for massless fermions, with the noticeable difference that we have now a single fermion at the surface where the Hall measurement is carried out. Remember that in graphene we are confronted with a quadruplet of massless Dirac fermions, as a consequence of the double spin and the double valley degeneracy. At the surface of the 3D topological insulator, there are no valleys and there is no spin degeneracy due to the above-mentioned spin-orbit coupling. Therefore the associated Landau levels, formed by the magnetic field perpendicular to the surface, are only degenerate due to the usual orbital degeneracy, in terms of the number of flux quanta threading the surface. As a consequence, the quantum Hall effect should manifest itself at the filling factors

$$ \nu = \pm(n + 1/2), \qquad (8.22) $$

which would result in a half-quantised Hall resistance

$$ R_{xy}^{\text{single surface}} = \pm \frac{h}{e^2} \left(n + \frac{1}{2}\right). \qquad (8.23) $$

However, this gedankenexperiment is problematic since the Hall bar is based on a single conducting surface, but there is also necessarily a conducting surface at the other side of the sample. Moreover, this opposite surface is connected to the original one, used to define the Hall bar, by lateral surfaces that are themselves necessarily conducting. Therefore, a Hall measurement is sensitive also to the response of the opposite surface, which contributes to the conductances and resistances. This yields an effective double degeneracy of the Landau levels where we take into account the response from the two surfaces threaded by the magnetic field and connected electrically by the lateral (non-quantised) surfaces. Thus, the Hall resistance should rather be quantised as

$$ R_{xy}^{\text{3DTI-surface}} = \pm \frac{h}{e^2(2n + 1)}, \qquad (8.24) $$

i.e. with an odd instead of a half-integer denominator.

The latter result has indeed been obtained experimentally, and in figure 8.4 we reproduce the results of an experiment by Yoshimi et al. from 2015 on a topological-insulator film (Bi$_{(1-x)}$Sb$_x$)$_2$Te$_3$ of 8 nm width. The figures 8.4(a) and (b) show the longitudinal and Hall resistances, respectively, for a film with an Sb concentration of $x = 0.84$. Similarly, the figures 8.4(e)
<figure><img src="image_84.png" /></figure>

Figure 8.4: Measurement of the quantum Hall effect on the surface of a 3D topological insulator. (a) and (b) show the longitudinal and the Hall resistances, respectively, as a function of the gate voltage, of a $(\text{Bi}_{1-x}\text{Sb}_x)_2\text{Te}_3$ film with $x = 0.84$, and (c) and (d) those of a film with a concentration $x = 0.88$. (e) and (f) depict the conductivities derived from the resistances (a) and (b), while (g) and (h) show those associated with the resistances (e) and (f). The sketch (i) represents the surface Landau levels when the charge neutrality point of the two surfaces is at the same energy, while the charge neutrality points (and the associated $n = 0$ Landau levels) are at different energies at the opposite surfaces in figure (j), as a consequence of a non-zero gate voltage. After Yoshimi *et al.*, Nat. Comm. **6**, 6627 (2015).
and (f) show the corresponding results for $x = 0.88$. Here, the magnetic
field is $B = 14$ T, and the carrier concentration at the surface is controlled
by a gate voltage $V_G$, with a metallic gate at 23 nm below the film. The
reference voltage $V_{CNP}$ is that corresponding to a Fermi level at the contact
point between the conduction and the valence band of the chiral surface state
(charge neutrality point). The figures (c), (d), (g), and (h) show the values
of the conductances derived from the corresponding resistances [(a), (b), (e),
and (f)]. The quantisation of the Hall conductance, visible in figure 8.4(c),
corresponds to the filling factors $\nu = \pm 1$, in agreement with equation (8.24)
when both surfaces contribute equally to the Hall transport. However, the
gate has a stronger effect on the bottom surface of the film, as a consequence
of its closer proximity, than on the top surface. This is shown in the second
sample in which the charge neutrality point (the Dirac point) and conse-
quently the $n = 0$ Landau level of the bottom surface is not at the same
energy as those of the top surface. The double degeneracy is therefore lifted
for the two families of Landau levels [as sketched in Figure 8.4(j)], and this
leads to the formation of another Hall plateau at $\nu = 0$, as shown in figure
8.4(g) with an associated Hall conductivity $\sigma_{xy} = 0$.

8.3 Weyl semimetals

In the preceding sections, we have presented a description of 3D topological
insulators in terms of massive Dirac fermions represented by a 4 × 4 Hamilto-
nian. As we have seen in the discussion of the “historical” Dirac equation in
three spatial dimensions (section 8.1), a special situation occurs in the case
of 3D massless Weyl fermions in 3D that can be represented in terms of a
2 × 2 Hamiltonian. Also in condensed-matter physics, this type of fermions
turns out to be relevant in the treatment of so-called Weyl semimetals that
are the subject of the present section. These materials are intensely studied
since the first experimental discovery of a Weyl semimetal (TaAs) in 2015,
i.e. even after that of 2D and 3D topological insulators. The present section
is meant to provide a succinct introduction to Weyl semimetals concentrating
on the most fundamental aspects, aiming at a coherent presentation in view
of the preceding discussion of topological materials.

The Weyl Hamiltonian may be obtained directly from the Dirac Hamil-
tonian in the Weyl representation (8.12) for *m* = 0 in which case the latter
becomes block-diagonal, and one may limit the discussion to a single 2 × 2
block,

$$H_W = \hbar(v_x q_x \sigma_x + v_y q_y \sigma_y + v_z q_z \sigma_z). \qquad (8.25)$$

Here, we have explicitly taken into account the possibility that the (Fermi) velocities in the different spatial directions are not necessarily the same. Notice that this may arise naturally in condensed-matter physics, where the Dirac or Weyl Hamiltonian represent the low-energy electronic properties of a system that may have low crystal symmetries so that the dispersion is naturally anisotropic.<fnref n="10" /> This needs to be contrasted to high-energy physics, where particles described by a Weyl or Dirac Hamiltonian live in a homogeneous and isotropic space so that there is no a priori reason to consider a direction-dependent speed of light.

The spectrum associated with the Weyl Hamiltonian (8.25) consists of two bands ($\lambda = \pm$),

$$E_{\lambda}(\mathbf{q}) = \lambda \hbar \sqrt{v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2}, \qquad (8.26)$$

with a band contact at $\mathbf{q} = 0$, which we call Weyl node or Weyl point. Recall that the wave vector is defined in the continuum limit with respect to a point $\mathbf{k}_w$ that is situated somewhere in the first Brillouin zone and that is not necessarily a point of high crystalline symmetry. However, as we have already mentioned in the general context of the 3D Dirac equation (in

$$H(\mathbf{k}) = \hbar(\mathbf{k}) \cdot \vec{\sigma}$$

possesses a band contact point at a wave vector $\mathbf{k}_W$. In the framework of a series expansion of this Bloch Hamiltonian to linear order around $\mathbf{k}_w$, one obtains

$$H_W = \sum_j \hbar \mathbf{v}_j \cdot \mathbf{q} \sigma_j,$$

where $\hbar \mathbf{v}_j = \nabla_{\mathbf{k}} h_j(\mathbf{k})|_{\mathbf{k}_w}$ and $\mathbf{q} = \mathbf{k} - \mathbf{k}_w$. In order to retrieve the form (8.25) of the Weyl Hamiltonian, one needs to perform a rotation of the frame of reference (accompanied by a unitary transformation that redefines the “spin” quantisation axis) such that the axes coincide with the principle axes of the velocity tensor $\mathbf{v}_j = (v_{j1}, v_{j2}, v_{j3})$, similarly to the inertia tensor used in the description of the rotations of a rigid body in mechanics. Notice finally that it is also possible, at linear order in $\mathbf{q}$, to add a term $\hbar \mathbf{v}_0 \cdot \mathbf{q}$ to the Hamiltonian that breaks particle-hole symmetry and that “tilts” the conical bands. The discussion of this particularity is beyond the scope of the present lecture notes even if it turns out to be an extremely rich field for the exploration of this class of materials, Weyl semimetals but also topological insulators, in terms of relativistic quantum mechanics.

<footnote n="10">¹⁰A yet more general form of this Hamiltonian may be given if a two-band Bloch Hamiltonian</footnote>
section 8.1), the band contact point – i.e. the fact that it exists, but not its position in reciprocal space – is protected against the opening of a band gap, contrarily to the 2D case. In order to appreciate this point, let us consider a general term

$$ \delta H = \Delta_x \sigma_x + \Delta_y \sigma_y + \Delta_z \sigma_z \qquad (8.27) $$

that perturbs the Weyl Hamiltonian and let us study its effect on the spectrum. The latter becomes

$$ E'_{\lambda}(\mathbf{q}) = \lambda \hbar \sqrt{v_x^2(q_x - q_x^0)^2 + v_y^2(q_y - q_y^0)^2 + v_z^2(q_z - q_z^0)^2}, \qquad (8.28) $$

where we have defined $q_j^0 = -\Delta_j/\hbar v_j$, for $j = x, y, z$. The only effect of the perturbative term (8.27) is thus to displace the Weyl node from $\mathbf{q} = 0$ to a new position $\mathbf{q}_0$, but it does not affect the low-energy description of its electronic properties. This conclusion remains valid also if the parameters $\Delta_j$ become functions of the wave vector $\mathbf{q}$, as one may easily see from a series expansion.<fnref n="11" />

### 8.3.1 Berry curvature of a Weyl fermion and magnetic monopoles

In order to characterise Weyl fermions from a topological point of view, we consider a surface of constant energy in reciprocal space, as e.g. the Fermi surface (in the form of an ellipsoid) when it does not coincide precisely with the Weyl node. This surface then encloses the Weyl node, which represents a reciprocal-space singularity. The Berry flux through this surface may be calculated with the help of

$$ C_W = \oint \frac{d^2 q}{2\pi} \vec{n}_q \cdot \mathcal{B}_{\lambda}(\mathbf{q}), \qquad (8.29) $$

where $\vec{n}_q$ is the vector normal to the surface element around the point $\mathbf{q}$. Naturally, the calculation is more involved than in the 2D case since the Berry curvature is now a true 3D vector, and all three components are generally non-zero, while only the $z$-component of the Berry curvature matters in 2D systems. However, we may use the result from our calculation of the 2D Berry curvature for a Dirac fermion presented in section 6.4 [see equation (6.56)],

<footnote n="11">¹¹The linear-order term may nevertheless renormalise the original velocity parameters, but it does not affect the general structure of the Weyl Hamiltonian.</footnote>
if we concentrate, say, on the z-component of the 3D Berry curvature. The latter may be obtained from equation (6.56) by replacing the gap parameter by the dispersion in the z-direction, $\Delta \to \hbar v_z q_z$,

$$ \mathcal{B}_{\lambda}^{z}(\mathbf{q}) = -\frac{\lambda \xi}{2} \frac{|v_x v_y| v_z q_z}{(v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2)^{3/2}}, \quad (8.30) $$

where $\xi$ no longer denotes the valley but the sign of the product of the two velocities $v_x$ and $v_y$, $\xi = \operatorname{sgn}(v_x v_y)$. The other vector components of the Berry curvature are obtained in the same manner, with no additional calculation, by symmetry: since the choice of the z-quantisation axis of the “spin” $\vec{\sigma}$ is arbitrary, one might have chosen the matrix $\sigma_x$ (or $\sigma_y$) to be diagonal instead of $\sigma_z$. This amounts to a simple exchange of the indices denoting the vector components, and the j-th component of the Berry curvature thus reads

$$ \mathcal{B}_{\lambda}^{j}(\mathbf{q}) = -\lambda \frac{v_x v_y v_z}{2} \frac{q_j}{(v_x^2 q_x^2 + v_y^2 q_y^2 + v_z^2 q_z^2)^{3/2}}. \quad (8.31) $$

The Berry curvature may be written in an even more compact manner with the help of a scale transformation of the wave vectors,

$$ \mathbf{q}' = (|v_x|q_x, |v_y|q_y, |v_z|q_z), \quad (8.32) $$

such that<fnref n="12" />

$$ \vec{\mathcal{B}}_{\lambda}(\mathbf{q}) = -\frac{\lambda \operatorname{sgn}(v_x v_y v_z)}{2} \frac{\mathbf{q}'}{|\mathbf{q}'|^3}. \quad (8.33) $$

This result is remarkable! In comparison with electromagnetism, the Berry curvature has the form of a vector field generated by a monopole, as one would expect from a point-like charge that generates the corresponding electric field. Recall, however, that the Berry curvature plays the role of a reciprocal-space magnetic field rather than an electric one, and we are thus confronted with a magnetic monopole here. This monopole, which we have already encountered in the introduction to topological concepts in section 6.1, is quantised and given by the Berry flux (8.29) through the closed surface of constant energy,<fnref n="13" />

$$ C_W = -\lambda \operatorname{sgn}(v_x v_y v_z). \quad (8.34) $$

<footnote n="12">¹²The corresponding transformation of the integration element in equation (8.29) is implicitly understood.</footnote>

<footnote n="13">¹³In the context of the generalised Weyl Hamiltonian obtained by a series expansion</footnote>
This integer-quantised monopole plays the role of the topological charge in the context of Weyl semimetals. It can be viewed as a robust topological defect at $\mathbf{q} = 0$ in the Berry curvature, which has the form of a 3D vector field, as it may be seen from equation (8.33). Following the classification of topological defects, which we have introduced in section 6.1, it is a point-like defect (of dimension $d' = 0$) that lives in a 3D space ($d = 3$). In order to characterise, i.e. to encircle the defect, we need a “lasso” of dimension $r = 2$, according to the lasso rule (6.1). This lasso is precisely the constant-energy surface over which the Berry curvature is integrated in equation (8.29). The quantisation of the monopole is then simply given by the winding (or wrapping) number, which represents the number of times that one covers the sphere $S^2$ associated with the Berry curvature when one explores the sphere $S^2$ (or the ellipsoid before the scale transformation of the wave vectors) in reciprocal space at constant energy. In the same manner as charges in electromagnetism, the result (8.34) of the integration remains the same under a continuous deformation of the surface of integration in equation (8.29), as long as the singularity remains enclosed by the surface.

### 8.3.2 Weyl semimetals and Fermi arcs

In the same manner as for 2D Dirac fermions, it is not possible to create a single Weyl node in 3D crystals due to its associated monopole,<fnref n="14" /> and one therefore requires at least another Weyl node with opposite topological charge. Whether just a second Weyl node is sufficient or even more of them depends again on time-reversal symmetry. If the latter is respected, the functions $h_x(\mathbf{k})$ and $h_z(\mathbf{k})$, which occur in the two-band Bloch Hamiltonian $H(\mathbf{k}) = \vec{h}(\mathbf{k}) \cdot \vec{\sigma}$, are necessarily even functions around a TRIM, whereas

$$H_W = \sum_j \hbar \mathbf{v}_j \cdot \mathbf{q} \sigma_j,$$

this topological charge is given by the sign of the mixed product of the three velocities,

$$C_W = -\lambda \operatorname{sgn}(\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)).$$

<footnote>around a band contact point (see preceding footnote),</footnote>

<footnote n="14">¹⁴One can show that the singularity in the wave function yields a phase field that is at odds with the single-valuedness of the wave function and the periodicity of reciprocal space.</footnote>
$h_y(\mathbf{k})$ must be odd. These requirements already exclude the existence of Weyl fermions, with its linear (and therefore odd) dispersion, at a TRIM. In addition, time-reversal symmetry then tells us that if we have a Weyl node at a wave vector $\mathbf{k}_w$, there is necessarily also one situated at $-\mathbf{k}_w$. As a consequence of the parity of the functions $h_j(\mathbf{k})$, one finds that the velocities $v_x$ and $v_z$ have an opposite sign at the two different Weyl nodes, while $v_y$ has the same sign. Consequently, the mixed product $v_x v_y v_z$ has the same sign in the two Weyl nodes so that the topological charge is the same, $C_W(\mathbf{k}_w) = C_W(-\mathbf{k}_w)$. In other words, the second Weyl node does not annihilate the twist in the wave function generated by the first one, and a *second pair* of Weyl nodes is therefore required to obtain a Weyl semimetal that respects time-reversal symmetry. *Time-reversal-symmetric Weyl semimetals are described by at least one quartet of Weyl fermions.*

The situation becomes even more complicated when local (point) symmetries, e.g. in the form of rotations around a point, need to be taken into account, as it is the case for most materials. As an example, one might consider tantalum arsenide (TaAs), which was the first material to be identified as a Weyl semimetal in 2015 [Xu *et al.*, Science **349**, 613 (2015)] and that possesses a total of 24 Weyl nodes each of which is described by a Weyl Hamiltonian in the vicinity of the Fermi level [see figure 8.5(E)].

One of the most salient features of Weyl semimetals is the appearence of so-called *Fermi arcs* on the surface. Such Fermi arcs are quite special objects because we are usually confronted with Fermi surfaces that are closed lines in 2D systems (as the present surface states) because they separate the occupied from the unoccupied electronic states in a material. Fermi arcs, however, are open lines that connect two different Weyl nodes (of opposite topological charge) projected to the surface. This has been observed in ARPES experiments, and figure 8.5 reproduces the measurements by Xu *et al.* on the (001)-surface of TaAs. Figure 8.5(E) shows the first Brillouin zone of the bulk material and the position of the Weyl nodes, while panel (G) shows the projection of these points onto the (001)-surface, as well as the numerically calculated Fermi arcs (red lines). The corresponding ARPES measurements are shown in panel (H), and they agree quite well with the numerically obtained ones.

How can we understand the occurence of these Fermi arcs within a minimal model? We know already that a model hosting two Weyl nodes with opposite topological charge is required. Such a model is readily constructed if we omit time-reversal symmetry (simply to avoid the above-mentioned
<figure><img src="image_85.png" /><figcaption>Figure 8.5: ARPES measurement of Fermi arcs on the (001) surface of the Weyl semimetal TaAs. (E) Position of the Weyl nodes in the (bulk) first Brillouin zone. (G) First Brillouin zone projected to the (001)-surface of the crystal and numerical results for the Fermi arcs (red lines). (H) ARPES results on this surface showing the expected Fermi arcs. After Xu *et al.*, Science **349**, 613 (2015).</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>Figure 8.6: Sketch of the energy bands obtained from Hamiltonian (8.35). The dispersion is isotropic in the $(k_x, k_y)$-plane, which is represented by a single axis $k_\parallel$ in the plane. In the vicinity of the two band contact points, one obtains Weyl fermions with opposite topological charge. If one considers the band structure as that of a 2D system (as a function of $k_x$ and $k_y$), with the $k_z$-component being a simple parameter, the gap parameter is *inverted* between the two nodes (red line).</figcaption></figure>

second pair of Weyl nodes),

$$H_{2W} = \begin{pmatrix} \Delta - \frac{\hbar^2 k_z^2}{2m_0} & \hbar v(k_x - i k_y) \\ \hbar v(k_x + i k_y) & -\Delta + \frac{\hbar^2 k_z^2}{2m_0} \end{pmatrix}. \qquad (8.35)$$

For $\Delta > 0$ and $m_0 > 0$, this Hamiltonian gives a band structure with two band contact points situated along the $z$-axis at $\mathbf{k}_W = (0, 0, \sqrt{2\Delta m_0}/\hbar)$ and $-\mathbf{k}_W$ (see figure 8.6).<fnref n="15" /> Expansion around the two contact points $\xi \mathbf{k}_W$ yields two copies of the Weyl Hamiltonian (8.25), one for each node or valley $\xi = \pm$, with a $\xi$-dependent velocity

$$v_z = -\xi \sqrt{2\Delta/m_0} \qquad (8.36)$$

in the $z$-direction. Since the velocities in the $x$- and $y$-directions do not

<footnote n="15">¹⁵It is clear from the above discussion that time-reversal symmetry is broken if $\mathbf{k} = 0$ is a TRIM. As mentioned above, $h_x(\mathbf{k})$ should then be an even function in the wave vector, but this is evidently not the case. Furthermore, if $\mathbf{k} = 0$ were not a TRIM, time-reversal symmetry would then needed to be defined with respect to another wave vector $\mathbf{k}_0$, and inversion with respect to this point would generate a second pair of Weyl fermions, in agreement with the general considerations established above.</footnote>
depend on the Weyl nodes – they are given by $v$ – the topological charges

$$C_W^\xi = \lambda\xi \operatorname{sgn}(v) \qquad (8.37)$$

are opposite in the two valleys, as expected.

We have therefore obtained a model that hosts two separate Weyl nodes, and we just need to show how that leads to a Fermi arc at the surface. In order to have two distinct projected nodes, the surface must not be normal to the $z$ direction. Let us choose arbitrarily the surface normal to the $x$-axis. In order to appreciate the formation of this particular surface state, we may appeal to our knowledge acquired from the study of 2D topological insulators and the bulk-edge correspondance discussed in section 7.4. Indeed, at each value of $k_z$, the Hamiltonian (8.35) may be interpreted as a 2D massive Dirac fermion, with a 2D wave vector $(k_x, k_y)$ and a $k_z$-dependent gap $\Delta_{k_z} = \Delta - \hbar^2 k_z^2 / 2m_0$. Within this picture, $k_z$ is a parameter that defines a family of 2D Hamiltonians. One already notices that this gap changes its sign as a function of $k_z$: it is positive in the interval $-\sqrt{2\Delta m_0}/\hbar < k_z < \sqrt{2\Delta m_0}/\hbar$ and negative outside ($|k_z| > \sqrt{2\Delta m_0}/\hbar$). From a topological point of view, there are thus two distinct regimes described by the 2D Berry curvature, which we can read off immediately from equation (6.56),

$$\mathcal{B}_{\lambda, k_z}(k_{\|}) = -\frac{\lambda}{2} \frac{\hbar^2 v^2 \Delta_{k_z}}{(\Delta_{k_z}^2 + \hbar^2 v^2 k_{\|}^2)^{3/2}}. \qquad (8.38)$$

This 2D Berry curvature follows the sign change of $\Delta_{k_z}$ across the Weyl nodes $\pm \mathbf{k}_W$, and so do the associated half-Chern numbers

$$C_{\lambda, k_z} = -\frac{\lambda}{2} \operatorname{sgn}(\Delta_{k_z}), \qquad (8.39)$$

which we have precisely used in the identification of a topological phase transition in 2D systems. In view of the discussion of section 6.4.1, this phase transition takes place upon varying the “parameter” $k_z$ when crossing the Weyl nodes. Consider the topological regime being associated with a positive half-Chern number for illustration. In this case, we can interpret the $k_z$-parametrised family of Hamiltonians (8.35) as representing 2D topological insulators inside the interval $-\sqrt{2\Delta m_0}/\hbar < k_z < \sqrt{2\Delta m_0}/\hbar$ and trivial insulators outside. As we have seen in section 7.4, a 2D topological insulator comes along with a chiral edge state dispersing as a function of $k_y$,

$$E_{k_z}(k_y) = \hbar v k_y, \qquad (8.40)$$
if we consider – as we do here – the surface to be normal to the x-axis. These “edge states” naturally do no longer exist outside the interval, for $|k_z| > \sqrt{2\Delta m_0/\hbar}$, where the system is considered to be a trivial insulator. If we restore the wave-vector character of $k_z$, we see that equation (8.40) describes a 2D sheet in the space spanned by $k_x, k_y$, and $E$, where there are electronic quantum states. This sheet connects the two Weyl cones centered at $\pm \mathbf{k}_W$, as it is depicted in figure 8.7(a) by the light red surface (the Weyl cones are shown in light blue). It is important to point out that the figure may be a bit misleading. The light red sheet is a surface state, with a 2D dispersion (even if, within our treatment, the energy does not vary with $k_z$ either), while the light blue cones represent the bulk Weyl fermions that disperse in all three spatial dimensions. In principle, $k_y$ represents $\mathbf{k}_\parallel = (k_x, k_y)$ in the latter case. However, the picture becomes correct if we interpret the bands of the Weyl fermions to be projected onto the same surface where the surface states (8.40) live.

Now that we have understood the figure 8.7(a) as an effective dispersion of the electronic states at the surface normal to $k_x$, we can also understand why these states are visible in the form of Fermi arcs. Indeed, we need to account for the Fermi level, which determines the occupation of the electronic states, here at the surface. Consider the Fermi level to be in the conduction band ($E_F > 0$). As we have seen the Fermi surface in the bulk is then given by an ellipsoid. However, at the surface normal to the x-axis, the Fermi level does not only cross the conical Weyl fermions, where it yields two elliptical Fermi surfaces [blue lines in figure 8.7(b)], but also the sheet connecting the Weyl nodes, $E_{k_z}(k_y) = E_F$. There, the crossing yields a line of surface states represented by the red line in figure 8.7(b), which may be viewed as the simplest form of the Fermi arc.

One notices that this treatment is too simplistic to account for the experimentally observed form of the Fermi arcs, e.g. those shown in figure 8.5. Clearly they are not straight lines, and they show that the energy sheets also disperse in the $k_z$-direction connecting the two Weyl nodes. From a merely topological point of view, this can be achieved by a continuous deformation of the red sheet in figure 8.7(a) without changing the topological class. From a more physical point of view, such a dispersion may, e.g., be obtained by the application of a magnetic field inside the surface or – this turns out to be analogous – by a rotation of the band structure inside the surface if the latter is sufficiently smooth. Yet another possibility would be a $k_z$-dependence in the anisotropy of the velocity parameters due to microscopic band effects the
<figure><img src="image_87.png" /><figcaption>Figure 8.7: Formation of a Fermi arc. The system is restricted to the $(k_y, k_z)$-plane perpendicular to the x-axis, and the Weyl nodes are separated by $\Delta k_0 = 2\sqrt{2\Delta m_0}/\hbar$ along the z-axis. (a) Energy dispersion projected onto the surface normal to the x-axis. Within the 2D picture of topological insulators, the edge states disperse in the direction $k_y$ and are parametrised by the component $k_z$ so that they form a sheet of states given by the light red plane. The conical dispersion of the Weyl nodes (projected onto the same surface) is represented in light blue. The Fermi arc arises when the red sheet is cut by the Fermi level [dashed line in panel (a) and red line in panel (b)], and it connects the projected Fermi surfaces of the usual Weyl nodes [dashed line in panel (a) and blue circles in panel (b)]. If the surface is smooth, with a slow variation of the gap parameter at the surface, there may be additional surface states at higher energies (light green and light orange) that correspond to quantum numbers $n \neq 0$, as in the case of topological insulators. After Mukherjee et al., Phys. Rev. B **100**, 195412 (2019).</figcaption></figure>
discussion of which is far beyond the scope of the present lecture notes.

### 8.3.3 Chiral anomaly

Beyond spectroscopy, Weyl semimetals display also unusual transport properties, at least according to theoretical expectations. These transport properties, called *chiral anomaly* are the subject of this last section. We have already seen that a Weyl fermion is characterised by a topological charge, its chirality. In the presence of a magnetic field, this chirality turns out to provide us with a chirality in the direction of propagation with respect to the magnetic field.

In order to appreciate this point, and its special consequences for transport, let us consider again the model (8.35) and apply a magnetic field in the z-direction connecting the two Weyl nodes of opposite chirality $\xi$. The Landau-level spectrum is readily obtained with the help of the usual Peierls substitution, and the Hamiltonian (8.35) reads

$$H_{2W}^{B} = \begin{pmatrix} \Delta - \frac{\hbar^2 k_z^2}{2m_0} & \sqrt{2}\hbar \frac{v}{l_B} \hat{a} \\ \sqrt{2}\hbar \frac{v}{l_B} \hat{a}^{\dagger} & -\Delta + \frac{\hbar^2 k_z^2}{2m_0} \end{pmatrix}, \quad (8.41)$$

as we have extensively discussed in section 5.2 of chapter 5: the wave vector $(k_x, k_y)$ in the plane perpendicular to the magnetic field gets quantised and provides us with the usual Landau levels, whereas the wave vector $k_z$ in the field direction remains a good, conserved quantum number. The spectrum reads

$$E_{\lambda, n \neq 0}(k_z) = \lambda \sqrt{\Delta_{k_z}^2 + 2\hbar^2 \frac{v^2}{l_B^2} n}, \quad (8.42)$$

for the levels $n \neq 0$, while we have

$$E_{n=0}(k_z) = -\Delta_{k_z} = -\Delta + \frac{\hbar^2 k_z^2}{2m_0}, \quad (8.43)$$

for the $n = 0$ level, which crosses the Fermi level that we consider to be situated at the Weyl node ($E_F = 0$).

Let us now consider the low-energy limit, in which $k_B T$ and all other energy scales (such as $e\Delta V$ in the case of an applied voltage) are small with respect to the Landau-level gap $\hbar v/l_B$. In this case, we could expand $\Delta_{k_z}$
around the two Weyl nodes at $\xi \mathbf{k}_W$ so that we would have obtained two low-energy Hamiltonians

$$H_{\xi}^{B} = \begin{pmatrix} -\xi \hbar v_z k_z & \sqrt{2} \hbar \frac{v}{l_B} \hat{a} \\ \sqrt{2} \hbar \frac{v}{l_B} \hat{a}^{\dagger} & \xi \hbar v_z k_z \end{pmatrix}, \qquad (8.44)$$

where now $v_z = \sqrt{2\Delta/m_0}$ is defined as the modulus of the velocity in the field direction $z$ (in contrast to the discussion above about the Fermi arcs). We now see that the $n = 0$ Landau levels associated with the two chiralities $\xi$ are simply given by

$$E_{\xi,n=0}(k_z) = \xi \hbar v_z k_z, \qquad (8.45)$$

i.e. their sense of propagation in the magnetic-field direction is determined by the chirality, $\xi v_z$. Naturally, this is also in line with the full $n = 0$ band (8.43) of the two-node Hamiltonian (8.41) because the band consists of a parabola crossing the Fermi level at the two Weyl nodes $\xi \mathbf{k}_W$. Equation (8.45) simply represents this band in the linear approximation around the two nodes. The two linear $n = 0$ bands are reminiscent of edge states of a 2D topological insulator that breaks time-reversal symmetry, as we have discussed in section 7.4 of the preceding chapter. Notice, however, that in contrast to the latter the states (8.45) describe bulk states: there is still the orbital degeneracy, the number $N_B = n_B A = AB/(h/e)$ of flux quanta threading the surface $A$ perpendicular to the magnetic field, and there are thus as many 1D channels (8.45) as states inside the Landau level $n = 0$, as we have discussed in section (5.2) in the framework of the quantum Hall effect.

The most salient result of the last paragraph is that the sense of propagation in the direction of the magnetic field of a Weyl fermion is given by its topological charge (chirality). Let us now – first on a qualitative level – discuss the extremely weird consequences of this observation. Consider, for the moment, that there is only a single type of chiral fermion with chirality $\xi$ available in the system. We have already understood that they would drift all in the same direction imposed by the orientation of the magnetic field. This seems weird because the particles would disappear from the system or appear ex nihilo! In order to see this point, let us calculate the density of chiral fermions

$$n_{\xi} = \frac{1}{2} \int d\epsilon \rho(\epsilon) f[\epsilon = E_{\xi,n=0}(k_z), \mu] = \frac{n_B}{2} \int d\epsilon \rho_{1D}(\epsilon) f(\epsilon, \mu), \qquad (8.46)$$
where we have used that the 3D density of states in the present case is the product of the 1D density of states<fnref n="16" /> $\rho(\epsilon)_{1D} = 2g/2\pi\hbar v_z$ and the flux density $n_B$ that characterises the orbital degeneracy of the Landau levels. The global factor of 1/2 is again due to the fact that the carriers have well-defined chirality, while by definition the density of states $\rho_{1D}(\epsilon)$ counts states with $k_z$ and $-k_z$. So far the expression of the density of Weyl fermions makes sense: the number of carriers is given by the density of states and the chemical potentials that enters the expression of the Fermi-Dirac occupation factor $f(\epsilon, \mu)$. But let us now investigate the time evolution of this density in the presence of an electric field $\mathbf{E} = E_z \mathbf{u}_z$ applied in the same direction $z$ as the magnetic field. This may be readily taken into account, as discussed in section 2.2.3, with the help of the substitution

$$k_z \rightarrow k_z - \frac{e}{\hbar} E_z t, \qquad (8.47)$$

which yields for the density, within a series expansion to linear order,

$$n_{\xi} = \frac{n_B}{2} \int d\epsilon \rho_{1D}(\epsilon) f \left[ \epsilon = E_{\xi, n=0} \left( k_z - \frac{e}{\hbar} E_z t, \mu \right) \right] \approx n_{\xi}^{(E_z=0)} + \frac{\partial n_{\xi}}{\partial t} t, \quad (8.48)$$

where $n_{\xi}^{(E_z=0)}$ is the original density (8.46) of Weyl fermions in the absence of an electric field and

$$\begin{aligned} \frac{\partial n_{\xi}}{\partial t} &= \frac{e}{\hbar} \frac{n_B}{2} \int d\epsilon \rho_{1D}(\epsilon) \frac{\partial E_{\xi, n=0}}{\partial k_z} \left( -\frac{\partial f}{\partial \epsilon} \right) E_z \\ &= \xi \frac{e^2}{\hbar^2} g B E_z, \end{aligned} \qquad (8.49)$$

where we have used $\partial\epsilon/\partial k_z = \xi v_z$ and the above expression for the 1D density of states $\rho_{1D}(\epsilon)$. Furthermore, we have made use of the low-temperature limit in which the $-\partial f/\partial\epsilon \simeq \delta(\epsilon - E_F)$ so that the integral vanishes. This last result shows indeed that the number of particles of type $\xi$ is not conserved if an electric field is applied. Furthermore it means that the electric charge associated with these particles is not conserved, and the current associated with the charge varying in time is thus given by

$$I_{\xi} = -e \int d^3 r \frac{\partial n_{\xi}}{\partial t} = -\xi \frac{e^3}{\hbar^2} g V \mathbf{B} \cdot \mathbf{E}, \qquad (8.50)$$

<footnote n="16">¹⁶The degeneracy $g$ counts the number of Weyl nodes (of like chirality) and may be set to $g = 1$ for our discussion here.</footnote>
where V is the total volume of the system over which the charge density is integrated. This weird effect is called *chiral anomaly*, and the above current is called *chiral current*. It was first discussed in the 1960ies in the framework of high-energy physics, and one must indeed emphasise the anomalous behaviour. In principle, charge should be a conserved quantity,<fnref n="17" /> but here the combined application of an electric and a magnetic field makes charge disappear from the system.

How can we heal the present result? Naturally, we have made a highly unphysical assumption – at least in the present case of condensed-matter physics – by taking into account Weyl fermions of a *single* chirality. However, as we have seen, there is always a compensation of Weyl nodes so that the total topological charge (once integrated over the entire first Brillouin zone) is zero.<fnref n="18" /> Indeed, if we take into account the second class of Weyl nodes, we see that the chiral current (8.50) is compensated by the second type of carriers,

$$I_{\text{tot}} = \sum_{\xi} I_{\xi} = 0, \qquad (8.51)$$

so that the total charge is conserved as it should.

The anomaly can be interpreted, in the framework of condensed-matter physics or a lattice model, as arising from a truncation of the spectrum and a somewhat problematic disconnection of two actually connected parts of reciprocal space. In order to illustrate this point, take a look at figure 8.8, where the low-energy part of the Landau-level spectrum is sketched around the two Weyl nodes at $\xi k_W$. The relevant Landau levels are those with the index $n = 0$ that cross the Fermi level and that carry the current. The chiral current (8.50), which we have just calculated, can be represented by the red arrow indicating that charge carriers with chirality $\xi = +$ have disappeared. This is compensated by the appearance of charge carriers with chirality $\xi = -$. In other words we have transformed charge carriers of one chirality into charge carriers of another one. This seems strange if we think in terms of chirality as a conserved quantum number as, e.g., spin. However, keep in mind that chirality is only an *emergent* quantum number at low energies. The two “chiral” $n = 0$ Landau levels are indeed connected at higher energies

<footnote n="17">¹⁷This is a consequence of the gauge invariance of electromagnetism.</footnote>

<footnote n="18">¹⁸Let us emphasise that our argument applies only to condensed matter systems. In high-energy physics, there is no necessary compensation between Weyl fermions with different chirality so that the situation is truly *anomalous* as it is stipulated by the term “anomaly”.</footnote>
<figure><img src="image_88.png" /><figcaption>Figure 8.8: Sketch of the chiral anomaly. The full black lines indicate the low-energy Landau levels of a Weyl semimetal around the Weyl nodes $\xi \mathbf{k}_W$. Each chirality $\xi = \pm$ is associated with a particular direction of propagation for the $n = 0$ Landau level. The $n = 0$ Landau levels of the two nodes are connected at energies (far) away from the Fermi level and eventually form a single 1D band. The chiral current can then be understood as a charge transfer from one node to the other across this 1D band. Adapted from Hosur and Qi, Comptes Rendus Physique **14**, 857 (2013).</figcaption></figure>

(or here energies far below the Fermi level, as shown in figure 8.8) and form a single 1D band. In contrast to a true $\mathbb{Z}_2$ variable that can only take two distinct values, we may thus change the chirality continuously by exploring adiabatically the high-energy parts of the full $n = 0$ band connecting the two Weyl nodes. Naturally, this would have been impossible if we had started, as in high-energy physics, from the two disconnected Weyl Hamiltonians (8.44) in which case the chirality $\xi$ is a true $\mathbb{Z}_2$ quantum number at all energies.

We may thus revisit the chiral anomaly in terms of a more conventional quantum-mesoscopic transport description, as in section 3.3 of chapter 3. Let us first consider ballistic transport in which chirality is conserved within a transport process, i.e. there are no backscattering events. Indeed, we recall that the total current of a 1D band can be written as

$$I = I_{\rightarrow} + I_{\leftarrow} = I_{\xi=-} + I_{\xi=+}, \qquad (8.52)$$

where we have made use of the link between chirality and sense of propagation: charges of chirality $\xi = -$ move in the direction of the applied magnetic field ($\rightarrow$), while those of chirality $\xi = +$ move in the opposite direction ($\leftarrow$). This current is generated by a voltage difference between the lower and the upper surface, and naturally a charge of chirality $\xi = -$, moving from the lower to the upper layer, is injected into the system at a voltage $V_l$ whereas a charge of chirality $\xi = +$ is at thermodynamic equilibrium with a contact at the upper layer, at a voltage $V_u$. We may thus adapt the equations (3.68)
and (3.69), used in the derivation of the Landauer formula, to the present problem, keeping in mind that we need to consider the 3D density of states in the present case,

$$ 
\begin{aligned}
I &= I_{\rightarrow} + I_{\leftarrow} \\
&= -g \frac{e}{h} A n_B \int d\epsilon \mathcal{T}(\epsilon) [-f(\epsilon, E_F - eV_l) + f(\epsilon, E_F - eV_u)] \\
&= -g \frac{e^3}{h^2} A B \int d\epsilon \left(-\frac{\partial f}{\partial \epsilon}\right) (V_l - V_u) \\
&= -g \frac{e^3}{h^2} A B (V_l - V_u),
\end{aligned} 
\quad (8.53) $$

where we have used $\mathcal{T}(\epsilon) = 1$ (ballistic regime) and again $-\partial f / \partial \epsilon \simeq \delta(\epsilon - E_F)$. We therefore find a magnetic-field-dependent conductance of

$$ G = -g \frac{e^3}{h^2} A B. \quad (8.54) $$

It is proportional to the surface A (cross section) of the system and the magnetic field B, which determines the number of 1D channels. We thus obtain a result similar to the chiral current (8.50) if we consider that the potential difference between the lower and the upper layer is generated by an electric field $E_z = -(V_l - V_u)/L_z$, where $L_z$ is the thickness of the system in the z-direction. Notice, however, that these two currents are only partially comparable. Indeed, equation (8.53) represents a true current that is generated by a potential difference between lower and upper surface of the sample, while equation (8.50) represents an anomalous current due to the non-conservation of chiral charges, the total current being zero. This is due, in part, to the fact that the relation between a voltage difference and an electric field is problematic in the framework of ballistic transport. As discussed extensively in section (3.3), the voltage difference does not change if one increases the size $L_z$ because the voltage drop takes place only at the contact placed on the lower and upper surfaces.

To finish this section on transport in Weyl semimetals, we discuss the fate of the chiral anomaly in the context of diffusive transport given by a relaxation time $\tau$ that characterises a scattering event in which the chirality (via the sense of propagation) of Weyl fermion is changed. In orders of magnitude, this requires short-range scatterers with a characteristic range given by the inverse of the Weyl-node separation $\ell \sim 1/|\mathbf{k}_W|$. One may
already inspect the result based on our discussion of diffusive transport within the Drude model discussed in section 3.2: the temporal variation of the wave vector in reciprocal space $k_z(t) = k_z - eE_z t/\hbar$ is cut off at the relaxation time, and the system size $L_z$ in the z-direction is then replaced by the mean free path $l_e = v_z \tau$. Following the statistical calculations of section 3.2, we calculate first the current density associated with our $N_B = An_B$ chiral 1D channels,

$$ 
\begin{aligned}
I_{\xi} = j_z^{\xi} &= -g \frac{e}{L_z} An_B \sum_{k_z>0} \xi v_z f \left[ E_{\xi,n=0} \left( k_z - \frac{e}{\hbar} E_z \tau \right), \mu \right] \\
&\approx -\xi \frac{e^3}{h} AB \int \frac{d\epsilon}{2} \rho_{1D}(\epsilon) v_z^2 \left( -\frac{\partial f}{\partial \epsilon} \right) E_z \\
&\approx -\xi \frac{e^3}{h^2} g(\tau v_z) A E_z B = -\xi \frac{e^3}{h^2} g l_e A \mathbf{E} \cdot \mathbf{B},
\end{aligned}
\quad (8.55) $$

where we have restored the vectorial character of the fields in the last line and used $l_e = v_z \tau$. This is indeed the expected result where the volume $V$ of the ballistic case is replaced by an effective (reduced) volume $V_{\text{eff}} = Al_e$ given by the mean free path in the z-direction associated with a backscattering event that does not conserve the chirality of the carriers.

## Notions and concepts to retain

* 3D topological insulators, chiral surface states (= 2D massless Dirac fermions)

* 3D massive Dirac fermions vs. 3D Weyl fermions

* Weyl semimetals, the role of time-reversal symmetry: fermion doubling vs. fermion quadruplets

* magnetic monopoles and topological charges

* Fermi arcs

* chiral anomaly and chiral currents