Many-Body Order Parameters for Multipoles in Solids

We propose many-body order parameters for bulk multipoles in crystalline systems, which originate from the non-trivial topology of the nested Wilson loop spectrum. The many-body order parameters are designed to measure multipolar charge distribution in a crystalline unit cell, and they match the localized corner charge originating from the multipoles. We provide analytic arguments and numerical proof for the order parameters. Furthermore, we show that the many-body order parameters faithfully measure the physical multipole moments even when the symmetries quantizing multipoles are lost and thus the nested Wilson loop spectrum does not exactly reproduce the physical multipole moments.

[^1]

Theory of macroscopic polarization in a crystal lies at the heart of the modern development of topological band insulators @RevModPhys.66.899. The theory provides a classic example of how the non-trivial topology of bulk ground states can determine the quintessential properties of the quantum phases such as symmetry-enforced boundary states @RevModPhys.82.3045 [@RevModPhys.83.1057]. Furthermore, it nicely illustrates how the associated topology can and cannot be diagnosed; the topology cannot be measured by a local operator, but only through a gauge-invariant number defined in a momentum space referred as the Zak phase @PhysRevLett.62.2747, or through a non-local many-body order parameter @PhysRevLett.80.1800. In particular, the discovery of the many-body order parameters, or equivalently "invariants", for various topological insulators drastically increased our understanding of topological states and expanded the range of the search for the topological states beyond free-fermion limits, e.g., strongly-correlated systems and bosonic models @PhysRevLett.82.370 [@PhysRevB.88.155121; @Spin; @PhysRevB.91.195142; @PhysRevLett.118.216402].

Recently, a new class of topological states, namely higher-order topological insulators, has been discovered @Benalcazar61 [@PhysRevB.96.245115; @HOTI_1; @HOTI_2; @HOTI_3; @HOTI_4; @HOTI_12; @HOTI_15; @PhysRevB.98.201114]. Interestingly, these higher-order states are constructed out of the electric multipoles, which materialize as topologically-protected modes at the corners and hinges of the physical boundary. To characterize such higher-order topology, the so-called "nested Wilson loop" approach @PhysRevB.96.245115 has been put forward and used to detect the quantized multipoles. However, the nested Wilson loop indices successfully work only when there exist symmetries which quantize the multipole moments @PhysRevB.96.245115 and currently there is no generic bulk measure for the multipoles working even in the absence of the symmetries. Also, a new class of higher-order topological insulator has been introduced under the name an anomalous topological wallacebrandi@example.net whose non-trivial multipole moments cannot be diagnosed by a naive application of the nested Wilson loop approach. Moreover, the nested Wilson loop is genuinely a free-fermion quantity and thus cannot be a generic diagnosis for the higher-order topology in a general many-body condensed matter system. This opens a question how to measure multipole moments going beyond the nested Wilson loop picture.

Motivated from this, we propose a set of many-body order parameters for the multipoles in crystalline systems, which succsefully measure the multipole moments even when the quantizing symmetries are lost so that the nested Wilson loop indices cannot be used to detect the true bulk moments. In stark contrast to the nested Wilson loop approaches, we utilize real-space charge distribution, which can be evaluated even for strongly-correlated systems. To this end, we extend Resta's general definition of polarization @PhysRevLett.80.1800 to the multipoles: $$\begin{aligned}
&P_{x} = \frac{1}{2\pi} \text{Im}\left[ \log \langle \hat{U}_1 \rangle \right], ~~ \hat{U}_{1} = e^{2\pi i \sum_{x} \hat{p}_{x}}, 
\label{Analytic0}\end{aligned}$$ where $\hat{p}_x = \frac{x \hat{n}(x)}{L_x}$ is the polarization density (relative to $x=0$) in the system of the length $L_x$ with $\hat{n}(x)$ being the electron number at site $x$, and the sum is running over the whole system. Here the expectation value $\langle \hat{U}_1 \rangle$ is over the many-body ground state subject to the periodic boundary condition, i.e., $x \sim x+L_x$. Compared to the Berry-phase expression @PhysRevLett.62.2747 defined in momentum space, the above formula Eq.  provides a clearer picture of the macroscopic polarization $P_x$ by relating it to the local microscopic polarization density $\hat{p}_x$. It also has played an important role in developing theories of related topological states@Oshikawa-Senthil [@lu2017filling] and other strongly-correlated systems, which is known as generalized Hastings-Oshikawa-Lieb-Schultz-Mattis theorem@Oshikawa [@Affleck1986; @parameswaran2013topological].

Here, we define the macroscopic quadrupole moment in a crystal as following: $$\begin{aligned}
&Q_{xy} = \frac{1}{2\pi} \text{Im}\left[ \log \langle \hat{U}_2 \rangle \right], ~~ \hat{U}_{2} = e^{2\pi i \sum_{\bm{r}} \hat{q}_{xy}(\bm{r})}.
\label{Analytic1}\end{aligned}$$ Here $\hat{q}_{xy} = \frac{xy}{L_x L_y} \hat{n}(\bm{r})$ is the quadrupole moment density (relative to $x=y=0$) per unit cell at the site $\bm{r}$. Similar to Eq. , $L_x$ and $L_y$ are the sizes of the system, and the sum is over $(x,y) \in (0, L_x] \times (0, L_y]$. Analogously, the macroscopic octupole moment is defined as: $$\begin{aligned}
&O_{xyz} = \frac{1}{2\pi} \text{Im}\left[ \log \langle \hat{U}_3 \rangle \right], ~~ \hat{U}_{3} = e^{2\pi i \sum_{\bm{r}} \hat{o}_{xyz}(\bm{r})}. 
\label{Analytic2}\end{aligned}$$ Here $\hat{o}_{xyz} = \frac{xyz}{L_x L_y L_z} \hat{n}(\bm{r})$ is the octupole moment density per unit cell; the generalizations to higher-order poles follows immediately. As in Resta's formula, we use a many-body ground state defined on the torus with periodic boundary conditions and each multipole moment density is summed over the whole space. Our real-space formula provides a complementary view to the momentum space approach, the nested Wilson loop indices @PhysRevB.96.245115. In this paper, we prove that these many-body order parameters correctly measure the multipole moments which originates from the topology of the nested Wilson loop spectrum. @PhysRevB.96.245115

A few remarks are in order. First, the multipole moments defined as above are ambiguous up to "mod 1" because of the periodicity of complex phases; this ambiguity reproduces the observation made in the previous studies @Benalcazar61 [@PhysRevB.96.245115]. Second, we emphasize that the above expressions are truly many-body order quantities because, when the exponetials are expanded, they require to measure $\sim \sum_{\bm{r}_1, \bm{r}_2, \cdots, \bm{r}_J}\langle \hat{n}(\bm{r}_1) \hat{n}(\bm{r}_2) \cdots \hat{n}(\bm{r}_{J}) \rangle$, and this does not reduce to the product of the few-particle observables. To evaluate Eq.  and Eq. , we need the full knowledge of the many-body ground state. On the other hand, the expressions are gauge-invariant, i.e., it is independent of the basis choice, and thus they measure physical quantities.

We can also show that the expressions Eq.  and Eq.  are invariant under the superposition of the trivial atomic insulator to the system. For example, we take Eq.  and $L_x \cdot L_y$ being odd. Let us first consider a ground state $|GS_{\text{triv}}\rangle = \otimes_x |\hat{n}_x = 1 \rangle$ of an atomic trivial insulator, which has $\langle \hat{U}_2 \rangle = 1$. When such a ground state is superposed with the topological state $|GS _{\text{top}}\rangle$, i.e., $|GS_{\text{new}}\rangle = |GS_{\text{top}}\rangle \otimes |GS_{\text{triv}}\rangle$, the quadrupole moment remains the same as before $$\begin{aligned}
\text{Im} \log \langle GS_{\text{top}} |\hat{U}_2 |GS_{\text{top}} \rangle = \text{Im} \log \langle GS_{\text{new}} |\hat{U}_2 |GS_{\text{new}} \rangle, \nonumber \end{aligned}$$ and hence showing the stability against addition of trivial bands (without turning on the coupling to the topological states). @comment1 This shows that our invariants Eq.  and Eq.  measure certain topological properties of the insulators. To further identify them as the "measure" for the multipoles, we need a few more steps.

**1. Analytic Arguments**: Now we argue that Eq.  and Eq.  *are* the physical multipole moments. We will concentrate on the quadrupole moment Eq.  for clarity. Below we choose our system to be on a torus $(x,y) \in (0, L_x] \times (0, L_y]$ and subject to the periodic boundary conditions $x+ L_x \sim x$ and $y+ L_y \sim y$. The effect of other coordinate parametrization is discussed in the supplemental material.@suppl_ref

First, we note that the $Q_{xy}$ defined as above transforms under the crystalline symmetries as the physical quadrupole moment. For example, under the mirrors $\{\mathcal{M}_x, \mathcal{M}_y\}$, we see that $\hat{U}_2 \to \hat{U}_2^*$, where asterisk denotes the complex conjugation, and thus $Q_{xy} \to -Q_{xy}$. (We defer a more careful analysis of the symmetry actions including $C_4$ symmetry to the supplemental material@suppl_ref.)

Second, we can also intuitively understand why the polarization must vanish for the quadrupole moment to be well-defined. To this end, we perform $x \to x+ L_x$ globally, which maps $\hat{U}_2 \to \hat{U}_2' = \hat{U}_2 \hat{U}_{1;y} $ where $\hat{U}_{1;y} = \exp [ 2\pi i/L_y  \sum_{y} y\hat{n}(\bm{r}) ] = \exp (2\pi i \hat{\mathcal{P}}_y )$. Although $\langle \hat{U}_2 \rangle \neq \langle \hat{U}_2' \rangle$ in general (because the ground state is not generally an eigenstate of $\hat{U}_{1;y}$), we find $$\begin{aligned}
\langle \hat{U}_2' \rangle = \langle \hat{U}_2 \rangle \langle \hat{U}_{1;y} \rangle + \mathcal{O}(\frac{1}{E_{\text{gap}}}), \nonumber \end{aligned}$$ where $E_{\text{gap}}$ is the excitation gap, which is inversely proportional to the correlation length @PhysRevLett.82.370 [@Watanabe-Oshikawa; @suppl_ref]. In the ideal limit $E_{\text{gap}} \to \infty$, we can see that the quadrupole moment, which is the imaginary part of $\langle \hat{U}_2 \rangle$, is well-defined when the polarization vanishes. Similarly, the total polarization $\hat{\mathcal{P}}_x$ along $x$ must vanish to have a well-defined quadrupole moment. By passing, we note that this invariance of $\langle \hat{U}_2 \rangle$ up to the perturbative correction is *weaker* than the original Resta's formula Eq.  where $x \to x+ L_x$ is an exact symmetry@suppl_ref. Though we have shown that the polarization at $E_{\text{gap}} \to \infty$ must vanish to have a well-defined quadrupole moment, the polarization is invariant under the adiabatic change of the gap when the quantizing symmetries present and thus the same conclusion should apply to the finite-gap systems.

Finally, we can gain an important intuition from the phenomenological effective theory of the multipole moments. Using the standard field theory technique, we note $$\begin{aligned}
\langle GS| \hat{U}_a | GS \rangle = \text{Tr} \bigg[ \frac{e^{-\beta H}}{\mathcal{Z}_0} \hat{U}_a \bigg]\Big|_{\beta \to \infty} \propto e^{iS_{eff}[A_{0;a} (\bm{r})]},  
\label{Eff}\end{aligned}$$ in which $a = 1, 2, 3$ depends on the particular multipole that we are interested in, $A_{0;a}(\bm{r})$ is the gauge potential configuration generated by the unitary $\hat{U}_a$ @suppl_ref, and $\mathcal{Z}_0$ is the corresponding partition function. For the dipole case, we have $$\begin{aligned}
A_{0;a=1} (x, \tau) = 2\pi \frac{x}{L_x} \delta(\tau), ~~S_{eff} = \int^{L_x}_0 dx\int^{\beta}_0 d\tau ~ P_{x} \cdot E_x. \nonumber   \end{aligned}$$ Here we used the semi-classical electromagnetism response of the polarization $S_{eff}= \int \vec{P}\cdot \vec{E}$. Remarkably, despite of its semi-classical origin, we note that it is equivalent to the fully-quantum theory@Jackiw [@Qi_Hughes_Zhang] of one-dimensional topological insulator. Explicitly evaluating the value of the effective action, we find $\langle \hat{U}_1 \rangle \propto e^{2\pi i P_x}$. This can be further identified with the edge charge, which we derive by solving the equation of motion $q_{\text{edge}} = \int dx ~\frac{\delta \mathcal{L}_{eff}}{\delta A_0} = P_x$.

We generalize this to the quadrupole case, whose effective response is@suppl_ref $$\begin{aligned}
S_{eff} = \int^{L_x}_0 dx \int^{L_y}_0 dy \int^{\beta}_0 d\tau ~ \frac{Q_{xy}}{2} (\partial_y E_x + \partial_x E_y), \end{aligned}$$ where $Q_{xy}$ is the physical quadrupole moment density and $E_{x,y}$ are electric fields. This effective theory captures the defining characteristics of the quadrupole insulator, including the corner charge and the charge current pattern generated from an adiabatic Thouless pumping@suppl_ref. Implementing $A_{0, a=2} = \frac{2\pi}{L_x L_y} xy \delta (\tau) $, we find $
\langle \hat{U}_2 \rangle \propto e^{2\pi i Q_{xy}}. 
$ It is straightforward to show that this quadrupole moment $Q_{xy}$ is precisely the corner charge $q_c$, i.e., $Q_{xy} =q_c$, by solving the equation of motion for the probe gauge field $\delta A_0$ as done in the polarization case. Similar reasoning can be applied to the octupole case, too. In summary, our many-body order parameters agree with the corner charges $q_c$, i.e., $\langle \hat{U}_a \rangle = \exp(2\pi i q_c ), a= 1,2,3$, which are the physical manifestation of the multipole moments even in the absence of quantizing symmetries.

- **Generalizations**: With the effective theory in hand, we can now generalize our order parameters to arbitrary boundary condition, since the above relation Eq.  holds independently of which boundary condition being used. Furthermore, we can generalize the unitary $\hat{U}_a$ to other unitaries saturating the effective action. For example, we can use $$\begin{aligned}
\hat{V}_1 (l) & = 
\begin{cases}
\exp\Big[\frac{2\pi i}{l} \sum_{x} x \hat{n}(x)\Big] \quad \text{ for } x \in (0, l] \\ 
1 \qquad \qquad \qquad \qquad \quad \text{ for } x \in (l, L_x],
\end{cases}
\label{Analytic_G1}\end{aligned}$$ to measure the wallacebrandi@example.net. Here the whole system may be subject to the open boundary condition or to the periodic boundary condition. Similarly, we can define the generalized measure for the quadrupole moment as following: $$\begin{aligned}
\hat{V}_2 (l) = 
\begin{cases}
\exp\Big[\frac{2\pi i}{l^2}  \sum_{\bm{r}} xy \hat{n}(\bm{r})\Big] \quad \text{ for } \bm{r} \in (0, l] \times (0, l] \\ 
1 \qquad \qquad \qquad \qquad \quad \text{ otherwise.}
\end{cases}
\label{Analytic_G2}\end{aligned}$$ Again, the whole system may be subject to the periodic or open boundary conditions. We will see numerically that these operators work as good as the original formula.

![(a) The phase diagram of the quadrupole insulator Eq.  when $\delta=0$ and $\lambda_x = \lambda_y = 1$. The shaded region denotes topologically nontrivial phase (quadrupole moment equals 0.5) and black dots denote bulk energy gap closing points. (b) and (c) are evaluation of $Q_{xy}$ Eq.  for the quadrupole insulator Eq. . We set $\lambda_x = \lambda_y = 1$ and $\delta = 0$, and evaluate $Q_{xy}$ (b) along the cut $\gamma_x = \gamma_y$ and (c) along the cut $\gamma_x = 0.5$. We see in both cases there is a sharp change at $\gamma_y = 1$, which is consistent with the phase diagram (a). (d) Anomalous topological quadrupole insulator.@PhysRevB.98.201114 Here, the ground state is known to be topological if $\frac{V_z}{\sqrt{\Delta^2 + \mu'{}^2}}>1$ and trivial if $\frac{V_z}{\sqrt{\Delta^2 + \mu'{}^2}}<1$, and this is well captured by $\hat{U}_2$. See the supplemental material for details on the model and the numerical values of the parameters@suppl_ref. <span data-label="U2_TQI_fig"></span>](Figure_1.pdf){width="1.0\columnwidth"}

- **Corner Charge, Nested Wilson Loop Index, and Multipole Moment**: Here we discuss a few related but distinct quantities by restricting ourselves to the quadrupolar case. (1) corner charge $q_{\text{c}}$, which is the total electric charge near the corner when the system is subject to the open boundary conditions. (2) Wannier-sector polarizations @PhysRevB.96.245115 $\{ p_x^\omega,  p_y^\omega \}$ (when they can be defined) and associated quadrupole moment $Q^{\omega}_{xy} = 2 p_x^{\omega}p_y^{\omega}$, which are defined by nested Wilson loops, and (3) physical quadrupole moment $Q_{xy}^{\text{ph}}$. In general, all these three can be different from each other.

However, if the bulk and boundary-alone polarizations are absent (so that the quadrupole moments are well-defined) and if the quadrupolar charge distribution per unit cell is the only source of the corner charge, then the corner charge $q_c$ would agree with a sensible quadrupole moment $Q_{xy}^{\text{ph}}$.@PhysRevB.96.245115 On the other hand, the nested Wilson loop index $Q^\omega_{xy}$ does not agree with $q_{\text{c}}$ generally @PhysRevB.96.245115. Hence, when the bulk-boundary correspondence @PhysRevB.48.4442 [@PhysRevB.95.035421] holds, i.e., the boundary polarization in the systems of the open boundary condition is fully determined by the bulk quadrupole moments @PhysRevB.96.245115, we expect generically $$\begin{aligned}
q_{\text{c}} \equiv Q^{\text{ph}}_{xy} \neq Q^\omega_{xy}, \nonumber\end{aligned}$$ in the absence of quantizing symmetries.

In this regard, our key claim is that, if the gap in the nested Wilson loop spectrum is present (i.e., $Q^{\omega}_{xy}$ can be defined),@PhysRevB.96.245115 our many-body order parameter $Q_{xy}$ in Eq.  measures the physical observable $Q^{\text{ph}}_{xy} = q_c$, i.e., $Q_{xy} = Q^{\text{ph}}_{xy}=q_c$, which may not agree with the nested Wilson loop $Q^\omega_{xy}$ *in the absence of quantizing symmetries*.

Note that, when the gap in the nested Wilson loop spectrum is absent and thus the quadrupole moments cannot be defined from the nested Wilson loop method, e.g., some $C_4$-symmetric models in Ref. , $Q_{xy}$ does not show stable values against the change in parameters nor agree with the corner charge.@suppl_ref

**2. Numerical Proof**: We now proceed to the numerical proof for our many-body order parameters by testing the formula on the *non-interacting* topological quadrupolar states. We emphasize again here that, though the models are non-interacting, the quantity that we are computing is intrinsically a many-body quantity which requires the full knowledge of the many-body ground state. In the momentum basis, the Hamiltonian for the topological quadrupole insulator is $$\begin{aligned}
h(\bm{k})& =  \big(\gamma_x + \lambda_x \cos(k_x) \big) \Gamma_4 + \lambda_x \sin(k_x) \Gamma_3 \nonumber \\
&+ \big( \gamma_y + \lambda_y \cos(k_y) \big) \Gamma_2 + \lambda_y \sin(k_y) \Gamma_1 + \delta~\Gamma_0. 
\label{TQI_k_space}\end{aligned}$$ Here, $\Gamma_{a=0,\cdots,4}$ is a proper set of gamma matrices @suppl_ref. When $\delta = 0$, there are two anti-commuting mirror symmetries $\hat{M}_x$ and $\hat{M}_y$ which quantize the quadrupole moment $q_{xy}$ to either 0 or 1/2 mod 1. If $\lambda_x = \lambda_y$ and $\gamma_x = \gamma_y$ are imposed (while keeping $\delta = 0$), there is a $C_4$ symmetry which also quantizes the quadrupole moment $q_{xy}$ to either 0 or 1/2 mod 1. $\delta \ne 0$ breaks both the mirror symmetries and $C_4$ symmetry, which quantize the quadrupoles, but $C_2$ symmetry remains intact which enforce the total polarization to vanish.

![Comparison between three different physical quantities, complex phase of $\langle \hat{U}_2 \rangle$, the quadrupole index $Q_{xy}^\omega$, and corner charge $q_\textrm{c}$ for (a) isotropic Thouless pumping Eq.  and (b) anisotropic Thouless pumping Eq. . While the complex phase of $\langle \hat{U}_2 \rangle$ and the corner charge $q_\textrm{c}$ agree with each other with almost no discernible differences, the quadrupole index $Q_{xy}^\omega$ given in terms of the Wannier-sector polarizations agrees with the two only at $\theta = \pi/2, 3\pi/2$, i.e., $\delta = 0$ and the quadrupole moment is quantized.<span data-label="Thouless_pumping"></span>](Figure_2.pdf){width="1.0\columnwidth"}

In FIG. \[U2\_TQI\_fig\], we present how the complex phase of $\langle \hat{U}_2 \rangle $ changes along different cuts in the parameter space of Eq. . The many-body order parameter Eq.  reproduces the quantized quadrupole moments as well as the sharp phase transitions between topological and trivial quadrupole insulator, even when the bulk energy gap does not close at the transition point.

Given the success of our many-body formula in reproducing the phase diagram of Eq. , we applied it to the anomalous topological quadrupole wallacebrandi@example.net, where the naive application of the nested Wilson loop approach fails to predict the non-zero quantized quadrupole moments and the related phase transition. Here we find remarkably \[see FIG. \[U2\_TQI\_fig\] (d)\] that the expected quadrupole moment and phase transition are successfully reproduced with our many-body order parameter $\hat{U}_2$.

To define the quadrupole moment using the nested Wilson loop, the existence of the Wannier gap is crucial as we can then separate a Wannier band to evaluate the Wanner-sector polarization. In the case of Eq. (\[TQI\_k\_space\]) with $\delta=0$, the phase transition between topological and trivial quadrupole insulator happens when the Wannier gap (instead of physical energy gap) closes and re-opens. On the other hand, $|\langle \hat{U}_2 \rangle|$ always vanishes at phase boundary, where the Wannier gap closes i.e., $|\lambda_x/\gamma_x| = 1$ *or* $|\lambda_y/\gamma_y| = 1$ (Note that the physical gap closes only when $|\lambda_x/\gamma_x| = 1$ *and* $|\lambda_y/\gamma_y| = 1$). This highlights the relation between the Wannier gap and $|\langle \hat{U}_2 \rangle|$.

Having seen the success of $\hat{U}_2 $ operator in detecting quantized quadrupole moment, we next ask what would happen if we break symmetries that protect the quantization of the quadrupole moment. In this case, the quantization of the quadrupole moment is lost and the quadrupole moment can take any value. However, due to the bulk-boundary correspondence @PhysRevB.48.4442 [@PhysRevB.95.035421; @PhysRevB.96.245115], we expect that the corner charge is determined by the bulk quadrupole moment and so should be identical to the phase of $\langle \hat{U}_2 \rangle$. For numerical confirmations, we employ two Thouless pumping processes, one we call isotropic Thouless process and the other we call anisotropic Thouless process, where we relate three different quantities; a) the complex phase factor of $\langle \hat{U}_2 \rangle$ evaluated under full periodic boundary condition, b) the quadrupole index @PhysRevB.96.245115 $Q_{xy}^\omega = 2 p_x^\omega p_y^\omega$ obtained from the nested Wilson loop, and c) the corner charge $q_\textrm{c}$ localized at one corner under full open boundary condition. We note that the previous study @PhysRevB.96.245115 found numerically that b) and c) are not equivalent in the absence of the quantizing symmetries.

For the isotropic Thouless pumping case, we change the parameters in Eq. (\[TQI\_k\_space\]) according to $$\begin{aligned}
\label{Isotropic}
\begin{cases}
\gamma_x = \gamma_y = 1-0.6 \sin(\theta) \\
\lambda_x = \lambda_y = 1+0.6 \sin(\theta) \\
\delta = 0.6 \cos(\theta)
\end{cases} ,\end{aligned}$$ and for the anisotropic Thouless pumping process, we change the parameters according to $$\begin{aligned}
\label{Anisotropic}
\begin{cases}
\gamma_x = 1-0.6 \sin(\theta), \quad \gamma_y = 1-0.5 \sin(\theta) \\
\lambda_x = 1+0.6 \sin(\theta), \quad \lambda_y = 1+0.5 \sin(\theta) \\
\delta = 0.6 \cos(\theta)
\end{cases} ,\end{aligned}$$ where $\theta \in [0, 2\pi]$ is the pumping parameter. In FIG. \[Thouless\_pumping\], we have plotted the three fundamentally different physical quantities for (a) isotropic and (b) anisotropic Thouless pumping processes. Apparent from the plots, we see almost no discernible difference between the complex phase factor of $\langle \hat{U}_2 \rangle$ and the corner charge $q_\textrm{c}$, indicating that the many-body order parameter faithfully represents the physical quadrupole moment. On the other hand, the Wilson loop index $Q_{xy}^\omega$ differs from the two in general and agrees with the two only at quantized values. This is the numerical proof that $\hat{U}_2$ can measure the physical quadrupole moment density in a crystal.

![Evaluation of the phase of $\langle \hat{V}_{a=1,2} (l) \rangle $ of Eq.  and Eq.  for various values of $l$ in the case of (a) Su-Schrieffer-Heeger chain and (b) Quadrupole insulator Eq. (\[TQI\_k\_space\]). We set $\gamma=1.0$ ($\gamma_x = \gamma_y = 1.0$) and $\delta = 0.2$ and change $\lambda$ ($=\lambda_x = \lambda_y$).<span data-label="Partial"></span>](Figure_3.pdf){width="1.0\columnwidth"}

Next, we perform the numerical check for the generalized many-body order parameters Eq.  and Eq.  in FIG. \[Partial\]. We compute the phases of these invariants for various $l$'s for the Su-Schrieffer-Heeger chain @PhysRevLett.42.1698 and topological quadrupolar insulators Eq. (\[TQI\_k\_space\]), which are both subject to the nonzero $\delta$ that breaks the quantization of polarization and quadrupole moments. For the Su-Schrieffer-Heeger chain, we have used $$h_\textrm{SSH} (k) = \left(
\begin{array}{cc}
\delta & \gamma+\lambda e^{-ik} \\
\gamma + \lambda e^{ik} & -\delta
\end{array} \right) \nonumber .$$ Surprisingly, the overall trend is insensitive to which boundary condition is used and insensitive even when $l$ is less than a half of the system size. Generically, we find that as $l$ approaches the full system size and as the system size becomes large enough, these invariants collapse to the corner charge as we predicted. To benchmark our formula for the interacting models, we also confirmed that our many-body order parameter can diagnose the non-trivial topology of the exactly-solvable bosonic higher-order topological insulator@Hughes with the corner spin.@suppl_ref

Finally, we also have applied our invariant Eq.  to another model, the "boundary polarization model" @PhysRevB.96.245115 supporting quantized corner charges $q_c = \pm 1/2$ yet having vanishing quadrupole moment (with suitable perturbations). Our order parameter correctly captures the vanishing quadrupole moment despite of the nonzero corner charges coming from boundary-alone polarizations.@suppl_ref We also find several interesting numerical observations on $\hat{U}_a$ including the dependence on coordinate parametrizations, which we discuss in detail in supplemental materials.@suppl_ref

**3. Conclusions**: We have proposed many-body order parameters which work for both the quantized and non-quantized multipoles in crystalline many-body quantum states. Using analytical arguments, we have demonstrated that they can be identified with the corner charges originating from the multipole moments. Based on the reasonings of the effective field theories, we have provided an extension of our formulas, which work even for the case of open boundary condition. Finally, we provided numerical proof of our order parameters by presenting explicit computations in toy models and confirmed that they can measure the multipole moments. Compared to the nested Wilson loop indices @PhysRevB.96.245115, our order parameters can measure the multipole moments even in the absence of quantizing symmetries and can be applied to interacting systems as well. On the other hand, in the presence of the quantizing symmetries, we have consistent results with the nested Wilson loop indices and provide complementary real-space formulation.

We thank Akira Furusaki, Jung Hoon Han, Hyun Woong Kwon, Joel E. Moore, Masaki Oshikawa, Kwon Park, Shinsei Ryu, and Mike Zaletel for helpful discussions and comments. KS is supported by PRESTO, JST (JPMJPR18L4) and GYC is supported by BK 21 plus program at POSTECH in Korea. GYC thanks KIAS in Korea and RIKEN in Japan for their hospitality where some part of this work has been done. We thank Taylor Hughes for informing us about their parallel independent work@WWH and sharing it with us before uploading to arXiv. When their work and our work overlap, they agree with each other. We also thank Andrei Bernevig and Ben Wieder for their comments and explanation on nested Wilson loop indices.

[^1]: Electronic Address: wallacebrandi@example.net

{'source': 'arXiv_out/1812'}