Quantization in chiral higher order topological insulators: circular dichroism and local Chern marker

TL;DR

This paper investigates the unique quantization properties of chiral higher order topological insulators, revealing how circular dichroism and local Chern markers behave, which helps identify and distinguish these exotic phases.

Contribution

It introduces novel quantization features of circular dichroism and local Chern markers in chiral HOTIs, highlighting their potential for experimental detection and differentiation from other topological phases.

Findings

Circular dichroism is quantized depending on sample orientation.

The surface-averaged local Chern marker is nonuniversal for finite surfaces.

Hinge states contribute to nonuniversal surface properties.

Abstract

The robust quantization of observables in units of universal constants is a hallmark of topological phases. We show that chiral higher order topological insulators (HOTIs), bulk insulators with chiral hinge states, present two unusual features related to quantization. First, we show that circular dichroism is quantized to an integer or zero depending on the orientation of the sample. This probe locates the hinge states, and can be used to distinguish different types of chiral HOTIs. Second, we find that the average of the local Chern marker over a single surface, an observable related to the surface Hall conductivity known to be quantized in the infinite slab geometry, is nonuniversal for a finite surface. This is due to a nonuniversal contribution of the hinge states, previously unaccounted for, that distinguishes surfaces of chiral HOTIs from Chern insulators. Our findings are relevant to establish higher order topology in systems such as the axion insulator candidate EuIn$_2$As$_2$, and cold atomic realizations.

Figures

Equations

$$\dfrac{\Gamma_{\pm}(\omega)}{2\pi E^{2}}=\sum_{n,m}\big{|}\bra{m}\left(\hat{x}\pm i\hat{y}\right)\ket{n}\big{|}^{2}\delta^{(t)}(E_{m}-E_{n}-\hbar\omega),$$

$$\Delta \Gamma ({\omega }_{\max })\equiv {\int }_0^{{\omega }_{\max }}d\omega \left[{\Gamma }_{+}(\omega )-{\Gamma }_{-}(\omega )\right]/2.$$

$$\Delta\Gamma(\infty)=-2\pi iE^{2}\ \text{Tr}\Big{\{}\big{[}\hat{Q}\hat{x},\hat{P}\hat{y}\big{]}\Big{\}}\equiv E^{2}\sum_{\text{all }\textbf{r}_{i}}C_{xy}(\textbf{r}_{i}).$$

$$\Delta {\Gamma }_{b\to b}=C{E}^{2}A,$$

$$\Delta \Gamma (\infty )=\sum _{n,m\in \{e,b\}}\Delta {\Gamma }_{n\to m}=0.$$

$${\rho }_n({\text{r}}_i)=|⟨{\text{r}}_i⟩n{|}^2,$$

$${\hat{H}}_{\pm }={\hat{H}}_0+2E\left[\cos (\omega t)\hat{x}\pm \sin (\omega t)\hat{y}\right],$$

$$P_{\pm }(t)=1-|⟨\Psi (t)⟩{\Psi }_0{|}^2,$$

$$\Delta\Gamma(\infty)=-2i\pi E^{2} \text{Tr}\Big{\{}\big{[}\hat{Q}\hat{x},\hat{P}\hat{y}\big{]}\Big{\}}\equiv E^{2}\sum_{\text{all }\textbf{r}_{i}}C_{xy}(\textbf{r}_{i})\ ,$$

$$\Delta {\Gamma }_{b\to b}=C{E}^{2}A.$$

$$H=\left(M-t\sum _{i=1}^2\cos \left(k_{i}a\right)\right){\sigma }_3+\lambda \sum _{i=1}^2\sin (k_{i}a){\sigma }_i,$$

$${\rho }_n({\text{r}}_i,{\text{r}}_j)=⟨{\text{r}}_i⟩n⟨n⟩{\text{r}}_j,$$

$$C_{xy}({\text{r}}_i)=-4\pi \sum _{{\text{r}}_j,{\text{r}}_k}\sum _{\begin{array}{c}{E}_l<0\\ E_m>0\\ E_n<0\end{array}}\text{Im}\left[{\rho }_l({\text{r}}_i,{\text{r}}_j){\rho }_m({\text{r}}_j,{\text{r}}_k){\rho }_n({\text{r}}_k,{\text{r}}_i)\right]x_j{y}_k.$$

$$0=\sum _{{\text{r}}_i}{C}_{xy}({\text{r}}_i)=2C_b+4C_h+4C_v.$$

Full text

Introduction.—The quantization of global observables is a quantum mechanical signature that distinguishes trivial and topological matter. For example, two-dimensional (2D) insulators without time-reversal symmetry present a Hall conductance that is quantized to $Ce^{2}/h$ where $C$ is an integer known as the Chern number. In contrast, experiments that show a quantized observable in three dimensions (3D) remain scarce; the sole examples are the observation of a quantized optical rotation in 3D time-reversal invariant topological insulators Wu et al. (2016); Dziom et al. (2017), and the quantization of a photocurrent in chiral topological semimetals Rees et al. (2019).

Our first result is that 3D chiral higher order topological insulators (HOTIs) have a quantized circular dichroism. HOTIs Benalcazar et al. (2017a, b); Schindler et al. (2018a); Song et al. (2017); Sitte et al. (2012); Zhang et al. (2013); Langbehn et al. (2017); Liu and Wakabayashi (2017); Ezawa (2018a, b); Ahn and Yang (2019); Wang et al. (2018); Wieder and Bernevig (2018); Lin and Hughes (2018); Khalaf (2018); Varnava and Vanderbilt (2018); Kooi et al. (2018); van Miert and Ortix (2018); Franca et al. (2018); Trifunovic and Brouwer (2019); Matsugatani and Watanabe (2018); Song et al. (2018); Schindler et al. (2018b); Sessi et al. (2016) are a subclass of topological insulators Hasan and Kane (2010); Qi and Zhang (2011) for which a $d$-dimensional bulk topological invariant protects metallic boundary modes in dimension $d-2$ or lower. More specifically, 3D chiral HOTIs Sitte et al. (2012); Zhang et al. (2013); Schindler et al. (2018a); Langbehn et al. (2017); Ezawa (2018b); Ahn and Yang (2019); Wieder and Bernevig (2018); Khalaf (2018); Kooi et al. (2018); Varnava and Vanderbilt (2018); van Miert and Ortix (2018); Matsugatani and Watanabe (2018) are fully gapped with only conducting chiral hinge states. The protection is guaranteed by combinations of crystalline and discrete symmetries, but symmetry alone does not necessarily dictate which hinges have a boundary mode.

Circular dichroism is an optical probe that is sensitive to topological properties Wang et al. (2011); Goldman et al. (2012); Souza and Vanderbilt (2008); Wang and Gedik (2013); Schüler and Werner (2017); de Juan et al. (2017); Liu et al. (2018); Tran et al. (2017, 2018); Repellin and Goldman (2019); Schüler et al. (2019). It is the differential absorption of left and right circularly polarized light and its frequency integral is determined by the dc Hall conductivity through a sum rule Bennett and Stern (1965); Souza and Vanderbilt (2008). For 2D Hall insulators this integral is quantized to the Chern number for periodic boundary conditions Tran et al. (2017); Repellin and Goldman (2019), and vanishes upon including the edge state contribution Tran et al. (2018, 2017); Souza and Vanderbilt (2008). To filter out the edge contribution, it is possible to use optical selection rules in Landau levels Tran et al. (2018), or, in periodically driven cold-atomic gases, a trap-release protocol Tran et al. (2017) or a harmonic trap as was demonstrated experimentally Asteria et al. (2019). In 3D we show that the circular dichroic absorption of chiral HOTIs restricted to a frequency range within the energy gap but across the whole sample is determined by an integer. The integer depends on the orientation of the sample and thus may be used to distinguish different types of chiral HOTIs (see Fig. 1).

In addition to bulk properties, the topological character of topological insulators can emerge as a quantized surface property. The surface Hall conductivity of a chiral HOTI is quantized in units of $e^{2}/2h$, half of the conductance quantum Varnava and Vanderbilt (2018); Coh et al. (2011). This quantization is inherited from the Hall conductivity of a gapped topological insulator surface state Essin et al. (2009); Qi et al. (2008) and can be derived in the infinite slab geometry Varnava and Vanderbilt (2018); Rauch et al. (2018); each surface Dirac cone contributes with $\pm e^{2}/2h$ to the total surface Hall conductivity.

Our second result revises such quantization of surface observables; we show that for finite samples, the hinge states modify the surface quantization of the Hall conductivity by a nonuniversal factor. We use a real space version of the Berry curvature, known as the local Chern marker Bianco and Resta (2011). It measures the local (space resolved) contributions to the Hall conductivity. In 2D insulators, the bulk average of the local Chern marker is quantized and equal to the Chern number $C$ of the corresponding periodic system. For open boundary conditions, its average over the whole sample vanishes. It means that the total contribution from edge states is equal and opposite in sign to the bulk contribution Bianco and Resta (2011); Tran et al. (2018, 2017). We find that this does not happen for a finite surface of chiral HOTIs due to the spatial distribution of the local Chern marker in the hinge states, which distinguishes surfaces of HOTIs from purely 2D Chern insulators, despite their spectral similarities Matsugatani and Watanabe (2018).

Models.—We consider a generic tight-binding model of chiral HOTIs on the cubic lattice Schindler et al. (2018a); Khalaf (2018); Ezawa (2018b) {IEEEeqnarray}rCl

^H_0 & = ( M+J∑_i=1^3cos(k_ia) ) τ_3σ_0 + ∑_i=1^3λ_isin(k_ia)  τ_1σ_i +

• D( cos(k_1a) - cos(k_2a) ) τ_2σ_0 + τ_0B⋅σ   ,

where $k_{i}$ are the cartesian momentum components, $a$ is the lattice spacing, $\sigma_{i},\tau_{i}$ are two sets of Pauli matrices acting on the spin and orbital degrees of freedom and $\sigma_{0},\tau_{0}$ are $2\times 2$ identity matrices. At half-filling the first two terms describe a time-reversal invariant 3D topological insulator, with anisotropic Fermi velocities if $\lambda_{i}$ are chosen to be different. The third term, proportional to $D$, breaks $C_{4}^{z}$ and time-reversal ($\mathcal{T}$) symmetries, but preserves their combination. The last term proportional to $\mathbf{B}$ is a Zeeman term that breaks $\mathcal{T}$ but preserves inversion symmetry ($\mathcal{I}$). The Hamiltonian Eq. (1) interpolates between a topological insulator ($D=|\mathbf{B}|=0$), a chiral HOTI protected by the rotoinversion symmetry $C_{4}^{z}\mathcal{I}$ [Fig. 1 (a)] and a chiral HOTI protected by $\mathcal{I}$ [Fig. 1 (b)]. We choose the parameters that maximize the bulk gap and therefore the localization of hinge states. Specifically, $M/J=2$, $\lambda_{i}/J=1$ (all three velocities are the same unless otherwise stated) and $|\mathbf{B}|/J=0.5$, with $D/J=0$ ($D/J=1$) and $\mathbf{B}\parallel(1,1,1)$ ($\parallel(0,0,1)$), for $\mathcal{I}$- ($C^{z}_{4}\mathcal{I}$-) symmetric HOTIs (see also Sup sec. A).

Quantized circular dichroism of chiral HOTIs.—To first order in perturbation theory, the absorption rate $\Gamma(\omega)$ of a circular periodic excitation of intensity $E^{2}$ polarized in the $xy$ plane at frequency $\omega$ is (see Sup sec. B for details)

[TABLE]

where $\ket{n}$ ($\ket{m}$) are occupied (unoccupied) states with energy $E_{n}$ ($E_{m}$) and $\delta^{(t)}(\epsilon)\equiv(2\hbar/\pi t)\sin^{2}(\epsilon t/2\hbar)/\epsilon^{2}$ (analogous definitions hold for other cartesian planes). We assume half-filling and discuss changing the chemical potential in Sup sec B. In the long time limit $\delta^{(t)}(\epsilon)$ is the Dirac delta function, so that Eq. (1) becomes the familiar Fermi’s Golden Rule. The integrated differential rate of absorption is defined as

[TABLE]

In the limit where $\omega_{\mathrm{max}}\to\infty$ it simplifies to Souza and Vanderbilt (2008); Tran et al. (2017)

[TABLE]

This equality—valid in two and three dimensions—expresses the dichroic signal as the sum of the local Chern marker $C_{xy}(\textbf{r}_{i})$ defined as a trace of the complete set of Wannier states $\left\{\ket{\textbf{r}_{i}}\right\}$. $\hat{P}$ ($\hat{Q}$) projects onto the occupied (unoccupied) eigenstates, and $\hat{x}$, $\hat{y}$ are position operators. In a 2D insulator with periodic boundary conditions the average of the local Chern marker is equal to the Chern number Bianco and Resta (2011); Tran et al. (2015). Generically, for a finite system with open boundary conditions the average of $C_{xy}(\textbf{r}_{i})$ over the whole sample vanishes due to the trivial character of the fiber bundle Bianco and Resta (2011), leading to $\Delta\Gamma(\infty)=0$. To show this mathematically recall that the trace of the commutator in Eq. (3) vanishes in a finite vector space.

Our main result is that the integrated in-gap circular dichroism is finite and quantized; by choosing the frequency cutoff in Eq. (2) to be $\hbar\omega_{\mathrm{max}}=E_{g}$, where $E_{g}$ is the gap of the infinite slab geometry, $\Delta\Gamma(E_{g})$ isolates the quantized hinge state contribution.

To reach this result, consider first periodic boundary conditions in the polarization plane, in which case only bulk to bulk transitions contribute and $\Delta\Gamma(\infty)=\Delta\Gamma_{b\rightarrow b}$. From Eq. (3) $\Delta\Gamma_{b\rightarrow b}$ is proportional to the integral of the Chern marker, which is not an integer in general Tran et al. (2017). For a 2D insulator with area $A$, $\Delta\Gamma_{b\rightarrow b}=CE^{2}A$ where $C$ is the Chern number Tran et al. (2017). For a 3D chiral HOTI slab with surface area $A$ only the two surfaces parallel to the polarization plane contribute to $\Delta\Gamma_{b\rightarrow b}$; their surface Dirac cones contribute with $\pm 1/2$ each, resulting in a total of

[TABLE]

where $C$ is the sum of these contributions, and thus an integer Varnava and Vanderbilt (2018). We have checked this expectation numerically (see Sup sec. B).

In the more realistic case of open boundary conditions there are four types of $n\to m$ transitions contributing to Eq. (3): from bulk to bulk ($b\to b$), bulk to edge ($b\to e$), edge to bulk ($e\to b$) and edge to edge ($e\to e$) states. The sum of all transitions must satisfy

[TABLE]

We have evaluated the relative sizes of the different contributions numerically using Eq. (1) in the long time limit for the $\mathcal{I}$- and $C_{4}\mathcal{I}$-symmetric HOTIs [see Fig. 2 (c)]. In all cases the edge-bulk and bulk-edge transitions become negligible as we increase the system size; this is related to the exponentially small overlap between the surface and bulk states (see Sup sec. B). In virtue of Eq. (5) the bulk-bulk and edge-edge transitions are equal and opposite, and due to Eq. (4) their respective thermodynamic values are $\pm CE^{2}A$. Thus, restricting the integration of the circular dichroic signal to in-gap absorption singles out edge-edge transitions, which guarantees the quantization of $\Delta\Gamma(E_{g})$.

In Figs. 2 (a) and (b) we show the circular dichroism calculated by finite time evolution in the frequency range $\omega\in(0,E_{g})$ for $\mathcal{I}$- and $C_{4}^{z}\mathcal{I}$-symmetric HOTIs. By evaluating $\Delta\Gamma(E_{g})$ for $N_{\mathrm{tot}}=N^{3}$ sites with $N=17$ for $xy$ polarization we obtain $C=-0.73$ and $-0.86$ for the $\mathcal{I}$- and $C_{4}^{z}\mathcal{I}$-symmetric HOTIs, respectively. As $N$ grows, $C$ approaches exact quantization as seen in Fig. 2 (c). For $yz$ polarization, we obtain $C=-0.73$ and [math] for the $\mathcal{I}$- and $C_{4}^{z}\mathcal{I}$-symmetric HOTIs, respectively.

An experimentally useful property of circular dichroism is that it depends on the type of chiral HOTI and the polarization plane. This follows from Fig. 2 and is depicted schematically in Fig. 1. For the $\mathcal{I}$-symmetric HOTI, the circular dichroic signal is always nonzero, and quantized in the thermodynamic limit. However, for the $C^{z}_{4}\mathcal{I}$-symmetric HOTI, $C$ vanishes for $xz$ and $yz$ polarized light, but it remains finite for $xy$ polarization. Indeed, in the $C^{z}_{4}\mathcal{I}$ HOTI the projection of the hinge states onto the $xy$ plane results in an orientable closed path akin to the edge state of a Chern insulator. This path contributes an integer $C\neq 0$ to the circular dichroic signal, in contrast to the $xz$ and $yz$ projections, where no such path exists. On the contrary, in the $\mathcal{I}$-symmetric HOTI, all cartesian polarizations lead to orientable closed paths, and the circular dichroic signal is quantized. In general, the quantization is determined by the projection of the 3D hinge state onto the polarization plane, which might be non-Cartesian. In this plane, each orientable closed path contributes an integer $C\neq 0$ to the circular dichroic signal normalized by its area, and with a sign dictated by the orientation of the path.

Our characterization scheme relies on the in-gap ($\omega<E_{g}$) integration of the dichroic signal. Although ab initio methods may be able to approximate $E_{g}$, it is desirable to determine it experimentally. This is possible by measuring $\Delta\Gamma(\omega_{\mathrm{max}})$ as a function of $\omega_{\mathrm{max}}$, as illustrated Fig. 2 (d). We expect a singularity at $\hbar\omega_{\mathrm{max}}=E_{g}$, caused by the activation of bulk-to-bulk transitions. The additional singularities at $\hbar\omega_{\mathrm{max}}<E_{g}$ in Fig. 2 (d) are due to the discretization of hinge states in finite size, and will not affect the extraction of the gap in the thermodynamic limit.

Local Chern marker of chiral HOTIs.—So far we have taken advantage of the energy resolution of circular dichroism to predict a quantized dichroic signal. Can sacrificing energy resolution in favor of a local measurement lead to a universal quantized observable? In infinite slabs the answer is affirmative: the Hall conductivity of the surface of a chiral HOTI is quantized in units of $\pm e^{2}/2h$ Essin et al. (2009); Mong et al. (2010); Coh et al. (2011). It stems from the quantization to $\pm 1/2$ Varnava and Vanderbilt (2018); Rauch et al. (2018) of the local Chern marker $C_{xy}(\textbf{r}_{i})$ Bianco and Resta (2011), the dimensionless operator defined in Eq. (3), see also Sup sec. B and C. We show next that the situation is different for finite surfaces: the hinge contribution renders the surface average of the local Chern marker nonuniversal. Our results carry over to other choices of position operators in Eq. (3).

We have calculated the local Chern marker using a recent approach Varjas et al. (2019a) based on the kernel polynomial method Weiße et al. (2006a) implemented using the Kwant package Groth et al. (2014). It allows us to reach sizes beyond exact diagonalization, and up to $N_{\mathrm{tot}}=N^{3}$ sites with $N=28$. For concreteness, we focus on the distribution of $C_{xy}(\textbf{r}_{i})$ for the $C_{4}^{z}\mathcal{I}$-symmetric HOTI, shown in Fig. 3 (a). $C_{xy}(\textbf{r}_{i})$ is close to zero deep in the bulk of the cube. The largest contribution to $C_{xy}(\textbf{r}_{i})$ comes from the $xy$ surfaces and decays fast with increasing distance to the surface.

Consider the top $xy$ surface of the HOTI, as represented in Fig. 3 (b). Far from the hinges, we expect the local Chern marker to approach $-1/2$, since the Hall conductivity is quantized to $-e^{2}/2h$ in the infinite slab geometry Essin et al. (2009); Mong et al. (2010); Coh et al. (2011); Varnava and Vanderbilt (2018) (see also Sup sec. B and C). Our numerical results indeed show this convergence as a function of system size at the central points of the surface (see Fig. 3 a) and b). Remarkably, the average of the local Chern marker over a single surface of the HOTI does not vanish (see Fig. 3 d)). This is in stark contrast with 2D Chern insulators (see Fig. 3 c)) and highlights the 3D character of the HOTI.

The average of the local Chern marker over the surface of a HOTI is finite, but is it still universal? Note that the sum of all hinge contributions must exactly compensate the bulk, a fact we have numerically checked. However, $C^{z}_{4}\mathcal{I}$ symmetry only imposes that hinges related by this symmetry have equal weight. In Fig. 3 d) we show that the Chern marker can be distributed inhomogeneously over the horizontal and vertical hinges by choosing $\lambda_{x}=\lambda_{y}\equiv\lambda\neq\lambda_{z}$. These parameters introduce hopping anisotropy but do not break the $C_{4}^{z}\mathcal{I}$ symmetry that protects the topological state. As $N$ increases, and up to numerical precision (see also Sup sec. C), different parameters lead to a different distribution of the local Chern marker along the hinges. This means that although the total hinge contribution is fixed and equal to one, it can be unevenly distributed over the horizontal and vertical hinges. This in turn, implies that the surface average of the Chern marker is a non-universal number.

Discussion. We have shown that the in-gap circular dichroism, a spatially averaged, frequency integrated quantity is quantized and can differentiate different types of chiral HOTIs. Although experimentalists currently measure polarization rotations Wu et al. (2016); Dziom et al. (2017), we propose integrated circular dichroism as an advantageous alternative to measure quantized optical responses; importantly, it does not depend on the substrate’s details as long as it preserves time-reversal symmetry. Specifically, it could probe higher-order topology in EuIn2As2 Xu et al. (2019), CrI3/Bi2Se3/MnBi2Se4 heterostructures Hou and Wu (2019), and MnBi2Te4 Zhang et al. (2019); Otrokov et al. (2018); Gong et al. (2018), all of which are predicted to host chiral hinge states. In particular, a quantized signal can be expected for a sample of EuIn2As2, illuminated with light polarized in a plane perpendicular to any of the crystal axis, and integrated up to the gap energy ($E_{g}\sim 10m$eV from ab-initio calculations Xu et al. (2019)). These energies are comparable to those already used to measure quantized optical rotation Wu et al. (2016). Moreover, the quantization of circular dichroism applies to interacting systems Repellin and Goldman (2019), which suggests the possibility to distinguish different interacting HOTIs Tiwari et al. (2019). Lastly, quantized circular dichroism Tran et al. (2017) has been measured in ultra-cold atomic Chern insulators Asteria et al. (2019) opening up the possibility to measure quantization in synthetic chiral HOTIs.

We have also found that the surface average of the local Chern marker, a spatially resolved quantity integrated over all energies, is non-universal in chiral HOTIs due to the hinge states, in sharp contrast to the infinite slab geometry and to a 2D Chern insulator. As a consequence, an experiment aiming to characterize a chiral HOTI based on its surface properties must distinguish bulk and hinge contributions. In solid state systems the local Chern marker is related to the orbital magnetization Bianco and Resta (2013). In cold atom experiments, it could be measured using a quantum gas microscope Caio et al. (2019); Ardila et al. (2018). Mechanical Mitchell et al. (2018) and photonic topological metamaterials Schine et al. (2019) may offer additional avenues to measure this quantity.

We expect our work to motivate new experiments that can push the ongoing effort to measure quantization in 3D Wu et al. (2016); Okada et al. (2016); Dziom et al. (2017); Rees et al. (2019), and topology in real space Caio et al. (2019); Mitchell et al. (2018). It calls to revisit 3D topological markers Olsen et al. (2017); Malashevich et al. (2010); Taherinejad and Vanderbilt (2015) to complete our understanding of topological matter in real space.

Acknowledgements We are thankful to B. A. Bernevig, D. Benoît, F. Jean-Jacques, F. de Juan, T. Neupert, P. San Jose, I. Souza, D. Varjas, and M. A. H. Vozmediano for support and useful conversations. We are thankful to N. Goldman for his critical reading of the manuscript. A. G. G. is supported by the ANR under the grant ANR-18-CE30-0001-01 and the European Union’s Horizon 2020 research and innovation programme under grant agreement No 829044. C.R. is supported by the Marie Sklodowska-Curie program under EC Grant agreement No. 751859. O.P. is supported by an FPU predoctoral contract from MINECO No. FPU16/05460 and the Spanish MECD Grant No. FIS2014-57432-P.

Supplemental Material

I A. Spectral properties of chiral HOTIs

The chiral HOTIs we study in the main text, and also Chern insulators, present one dimensional chiral metallic states at their hinges and edges. These states are fingerprints of their non–trivial topology. Through exact diagonalization, one can solve the tight-binding model (Eq. (1) in the main text) and compute the eigenstates $\{\ket{n}\}$ in the Wannier basis $\{\ket{\textbf{r}_{i}}\}$. These metallic states are identified with the eigenstates energetically well–located inside the insulating gap, and the local density

[TABLE]

can be used to check their dimensionality and position. The hinge states of the $\mathcal{I}$- and $C_{4}^{z}\mathcal{I}$ symmetric HOTIs studied in the main text are shown in Fig. 4.

The slab geometry is achieved assuming periodic boundary conditions in two of the three spatial directions, leading to three different slab geometries. There are no hinges in the slab geometry, therefore the energy spectrum is fully gapped. The slab gap $E_{g}$ used in the main text is taken to be the minimum gap between the three possible slab geometries.

II B. Further details on the computation of the circular dichroism

II.1 Computation of the absorption rate

In the main text we presented the expression of the absorption rate, $\Gamma_{\pm}(\omega)$ (Eq. (1)), to analyze how circular dichroism is quantized in chiral HOTIs. Here we explain the necessary steps to determine such expression.

Consider a chiral HOTI subjected to an optical signal at frequency $\omega$ polarized in the $xy$ plane. Then the full Hamiltonian at time $t$ is given by

[TABLE]

where $\hat{H}_{0}$ is the HOTI Hamiltonian defined in Eq. (1) of the main text, $E$ is the amplitude of the electric field, $\pm$ indicates left- or right handed polarization and $\hat{x},\hat{y}$ are position operators. Part of this light is absorbed by the electrons of the system, bringing them from the initial ground state $\ket{\Psi_{0}}$ to an excited state $\ket{\Psi(t)}$. The probability of excitation is

[TABLE]

If the electric field amplitude is small enough, the wave function $\ket{\Psi(t)}$ can be computed using time-dependent perturbation theory up to first order. The resulting absorption rate $\Gamma_{\pm}(\omega)=P_{\pm}(t)/t$ is Eq. (2) in the main text. For long enough times this absorption rate is given by Fermi’s Golden Rule {IEEEeqnarray}l Γ±(ω)2πE2 = ∑_n,m — ⟨m—( ^x±i^y ) —n⟩ —^2 δ(E_m-E_n-ℏω),

where $\ket{n}$ ($\ket{m}$) are occupied (unoccupied) states and $E_{n}$ ($E_{m}$) their energies. The total circular dichroism is obtained by taking the difference $(\Gamma_{+}-\Gamma_{-})/2$ and integrating over all frequencies as defined in the main text. This difference can be expressed as

[TABLE]

where $\hat{P}$ $(\hat{Q})$ is the projector over occupied (unoccupied) states. Independently of dimensionality, the total absorption rate is proportional to the total Hall conductivity $\sigma_{\alpha\beta}$ of the sample Tran et al. (2017) where $\alpha\beta$ is the polarization plane of the optical signal.

II.2 Layer-resolved Chern number

In the case of periodic boundary conditions in the polarization plane $\alpha\beta$, let us resolve the right hand side of Eq. (9) along the direction $\gamma$ perpendicular to the polarization plane. We show this layer-resolved Chern number $C_{\alpha\beta}(\gamma)$ in Fig. 5 (a) and (c) for the $C^{z}_{4}\mathcal{I}$-symmetric HOTI. In the infinite slab geometry, the Hall conductivity can be calculated following Varnava and Vanderbilt (2018), by summing the layer-resolved Chern number over the direction of propagation $\gamma$. Fig. 5 (a) and (c) show that the only sizable contributions are due to the surface states which contribute $\pm 1/2$, and extend as much as two layers from the surface. This justifies our choice in Fig. 3 (a) in the main text where we chose the two last layers as our surface. Summing over the whole slab we find that the contributions of opposite surfaces either cancel, leading to $C=0$ [see Fig. 5 (b)], or add up to a quantized value $C=-1$ [see Fig. 5 (d)].

The results in Fig. 5, and using Eq. (9) show that for periodic boundary conditions, where only bulk-bulk transitions are present, the circular dichroism is quantized to

[TABLE]

as we use in the main text.

II.3 Edge-bulk and bulk-edge contributions in Chern insulators

The quantization of bulk responses is only ensured in the thermodynamic limit. It is in such case that bulk states of a finite system are independent of boundary conditions and resemble those of periodic boundary conditions. As a consequence, our finite size results do not show an exact quantization of circular dichroism but rather a trend as we approach the thermodynamic limit. Due to the 2D character of Chern insulators, it is possible to calculate the absorption rate of larger systems with exact diagonalization Varjas et al. (2019a, b). This allows us to show the quantization of edge-bulk, bulk-bulk and edge-edge contributions in the thermodynamic limit to larger precision (see Fig. 6). The Chern insulator model we have considered is

[TABLE]

where $M/t=3/2$ and $\lambda/t=-1/2$.

II.4 Effect of a finite chemical potential

In the main text we have assumed half-filling, and thus the chemical potential $\mu$ sits exactly at the center of the gap. This assumption made the edge-bulk and bulk-edge contributions equivalent in Fig. 2 b) of the main text. In general these two types of transitions can be different. However, our results remain unchanged so long as the chemical potential remains in the gap. Out of all possible transitions, the bulk-bulk transitions are still quantized if the chemical potential is within the gap. The edge-bulk and bulk-edge contributions vanish, independent of the chemical potential. Therefore, the sum of all edge-edge transitions must exactly compensate the sum of bulk-bulk transitions with an equal and opposite Chern number. Summing over all edge-edge transitions implies integrating $\Delta\Gamma(E_{g})$, as for the case where $\mu=0$.

III C. Further details on the computation of the Local Chern marker

In Fig. 3 of the main text, we see that $C_{xy}(\textbf{r}_{i})$ reaches its thermodynamic value much faster in the bulk than in the hinges. This is related to the contrasting scaling of correlation functions in these two regions: states are respectively exponentially and algebraically localized in the bulk and the hinges. As a result, at a given bulk point, the main contribution to the local Chern marker is determined by the nearest points Resta (2006, 2011); Irsigler et al. (2019), while it involves larger non-local contributions for the hinges. A more formal way to understand this is to consider the expression of the local Chern marker in terms of the density correlator

[TABLE]

which reads Irsigler et al. (2019)

[TABLE]

Since the magnitude of $\rho_{n}(\textbf{r}_{i},\textbf{r}_{j})$ is proportional to the modulus of the density of the state $\ket{n}$ at $\textbf{r}_{i}$ and at $\textbf{r}_{j}$, bulk-edge and edge-bulk contributions are suppressed here as in the case of circular dichroism. Because of the long-range density correlations within the hinge, the local Chern marker at a hinge point $\textbf{r}_{i}$ receives significant contributions from other hinge points $\left(\textbf{r}_{j},\textbf{r}_{k}\right)$, even though they are not spatially close.

III.1 Convergence of the hinge contribution

As shown in Fig. 3 (a) and (b) in the main text at a point $\textbf{r}_{i}$ in the top and bottom surface of the $C_{4}^{z}\mathcal{I}$-symmetric HOTI $C_{xy}(\textbf{r}_{i})$ is quantized to $1/2$, similar to how a bulk point in a Chern insulator defined by Eq. (11) is quantized to $\pm 1$ Bianco (2014). In contrast, the value of the local Chern marker at a hinge point need not be quantized. We analyze the non-negligible local Chern marker contributions, which come from the surface bulk ($b$), the horizontal ($h$) and the vertical ($v$) hinges (see Fig. 3 of the main text). Note that, due to the $C_{4}^{z}\mathcal{I}$-symmetry, the contribution of all vertices is the same and we can democratically associate one vertex to each hinge. In an finite system, these contributions verify the following sum rule

[TABLE]

$C_{b}$ is quantized to $\pm 1/2N_{s}$ where $N_{s}$ is the number of surface area points defined in blue in Fig. 3 (a) of the main text (in the thermodynamic limit $N_{s}\simeq N^{2}$). The sum $C_{h}+C_{v}$ is thus quantized (in units of $N_{s}$), but $C_{h}$ and $C_{v}$ are not forced by $C_{4}^{z}\mathcal{I}$ symmetry to be quantized independently. We note this is unlike the Chern insulator Eq. (11), which is invariant under $C_{4}^{z}$, implying that the Chern marker at each edge is forced to be $C_{\mathrm{edge}}=-C/4$.

Finally, for Fig. 3 (d) we computed the local Chern marker using the efficient numerical approximation based on the Kernel Polynomial Method Weiße et al. (2006b) implemented via the Kwant Python package Groth et al. (2014). It allows us to reach larger system sizes than the ones achievable with exact diagonalization. The projector $\hat{P}$ is approximated as a polynomial in powers of $\hat{H}$ of degree inversely proportional to the broadening parameter indicated in the main text (see Varjas et al. (2019a) for details). We have checked convergence by observing that a broadening of $10^{-3}$ reproduces the local Chern marker results from exact diagonalization for small $N$, and that our results change by less than $0.1\%$ when the broadening is reduced by a factor of $10$.