Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second order topological insulator on a torus

Introduction.—The quantum Hall effect (QHE), observed by Klitzing et al. [1] in 1980, is one of the fundamental phenomena in condensed matter physics. It has since been a driving force for the field of topological insulators (TI) with a plethora of new topological materials discovered in the last decades [2, 3, 4, 5, 6, 7]. As a consequence, many intriguing generalizations such as the anomalous QHE (in the absence of an external magnetic field) in 2D [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] and the 3D-QHE [23, 24, 25, 26, 27] have been studied. Moreover, the concept of TIs has been extended to higher-order TIs, which are governed by a new bulk-boundary correspondence [28, 29, 30, 31, 32, 33, 34, 35]. For example, a second (third) order TI with dimension $D$ features $D-2$ ($D-3$) dimensional hinge (corner) states. Recently, within 3D second-order TIs (3D-SOTI), an additional class of systems exhibiting the QHE has been proposed [31, 36, 37, 38, 39].

One way to realize a 3D-SOTI is to build on a 3D-TI [40, 6, 7, 12, 41, 42] and introduce anisotropic gaps on different surfaces (boundaries), e.g., with an effective Zeeman term by magnetic doping or using an external magnetic field [43, 38, 39, 44, 45, 46, 47, 48, 49]. At special positions where the effective gap closes and reverses sign, topological hinge or corner states emerge, in analogy to the Jackiw-Rebbi mode [50].

In this Letter, we will consider such a model and discuss a specific realization using a torus geometry (or smooth deformations thereof) as shown in Fig. 1, where the emerging hinge states are shown with blue and red arrows. This geometry allows us to insert a magnetic flux $\phi$ through the torus hole, and thus we construct a 3D analogy of the Corbino disc which is used in the Laughlin argument for the conventional 2D-QHE [51]. However, in contrast to the Corbino setup, the 3D-SOTI QHE discussed here is not induced by orbital effects but results purely from the Zeeman effect.

While there are several approaches known to deduce the conventional 2D quantized Hall conductance [51, 52, 53, 54], we will present a generalization of a recently developed boundary charge approach to the case of a 3D-SOTI, which, most importantly, is applicable to both clean and disordered systems. It is based on recent detailed studies of the boundary charge in one-dimensional (1D) modulated wires [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65], together with dimensional extensions to 2D-QHE models [56, 57] revealing the fundamental relationship between the linear dependence of the boundary charge on the phase of the potential in 1D models and the quantized Hall conductance in 2D models, a concept which can even be generalized to explain the fractional QHE in the presence of interactions [62].

Most importantly, we demonstrate for the 3D-SOTI model on a torus that the boundary charge, driven by the Aharonov-Bohm flux $\phi=f\phi_{0}$, where $\phi_{0}$ is the flux quantum, shows a linear relation with respect to $f$, with the slope being universal and quantized to integer numbers $\pm e$, where $e$ is the electron charge. Examining all states below the chemical potential, we reach a remarkable conclusion: the whole contribution to the boundary charge comes only from the top-most occupied valence band [shown in orange in Fig. 2(a)]. We further include disorder effects by considering a 3D tight-binding (tb) counterpart of our continuum model with broken rotational symmetry and show robustness of the quantized slope. The boundary charge concept, thus, provides a powerful tool in characterizing 3D-SOTIs and universally explains topological effects in condensed matter systems as well as their stability against random potential disorder and continuous surface deformations.

Model.—In this Letter we consider a continuum model in a finite three-dimensional region, which is confined to the interior of a torus [see Fig. 1(a))], to realize a second order TI with a minimal set of ingredients, given by band inversion, spin-orbit coupling, magnetic flux, and Zeeman term. The Hamiltonian (with $e=\hbar=1$) is given by

where p and $\textbf{S}=\frac{1}{2}\textbf{s}$ are 3D vectors of the momentum and the physical spin-$\frac{1}{2}$ operators, respectively. The parameters $m^{*}$ and $\alpha>0$ are the effective mass of the electron and the spin-orbit interaction, respectively, while $\varepsilon_{Z}$ denotes the Zeeman energy. Both the magnetization direction inducing the Zeeman term and the uniform magnetic field B generated by the vector potential A are assumed to point along the $z$-axis. We note that $\varepsilon_{Z}$ is an independent parameter, controllable by magnetic doping. The Pauli matrices $\sigma_{i}$ span the orbital degrees of freedom. The parameter $\delta>0$ is chosen positive in order to realize the band inversion at finite values of $\alpha$, $\varepsilon_{Z}$, and $B\equiv|\textbf{B}|$. The vector potential $\textbf{A}=\textbf{A}_{B}+\textbf{A}_{\phi}=\frac{B\rho}{2}\textbf{e}_{\varphi}+\frac{\phi}{2\pi\rho}\textbf{e}_{\varphi}\equiv\frac{B\rho}{2}\textbf{e}_{\varphi}+\frac{f}{\rho}\textbf{e}_{\varphi}$ contains the terms generating both the uniform magnetic field $B$ and the Aharonov-Bohm (AB) magnetic flux $\phi$. Here, we used the cylindrical coordinates $(\rho,\varphi,z)$ and introduced the number of flux quanta $f=\frac{\phi}{\phi_{0}}\stackrel{{\scriptstyle!}}{{=}}\frac{\phi}{2\pi}$.

Besides the spin-orbit length $l_{so}=1/(\alpha m^{*})$, we introduce two further typical scales $l_{B}$ and $l_{Z}$ associated with the orbital magnetic field and the Zeeman term, respectively, via $B=1/l_{B}^{2}$ and $\epsilon_{Z}=1/(2m^{*}l_{Z}^{2})$. In the main part of this Letter we will discuss the most critical case of strong orbital fields, where $l_{B}\sim l_{Z}$, and will see that the orbital effects do not influence the quantization of the Hall conductance. In the Supplemental Material (SM) [66] we will also discuss the case of zero orbital effects with the same conclusion. This shows that our results are expected to be stable for any orbital field.

In the torus geometry and in the absence of disorder the model (1) possesses the axial symmetry generated by the $z$-component of the total angular momentum operator $J_{z}=L_{z}+\frac{1}{2}s_{z}=-i\frac{\partial}{\partial\varphi}+\frac{1}{2}s_{z}$. Exploiting this symmetry we reduce the 3D model to an effective 2D model. It is described by the Hamiltonian (see SM [66] for details)

and defined on the disc of radius $r_{0}$ in the $(x>0,z)$ half-plane, which is displaced away from the $z$-axis by $l_{0}>r_{0}$ [see Fig. 1(b)]. We introduce the following continuous parameter $\tilde{j}_{z}=j_{z}+f$, where $j_{z}$ are half-integer eigenvalues of the operator $J_{z}$, and $f$ is restricted to fractional values, $f\in[0,1)$, interpolating between adjacent values of $j_{z}$. The parameter $\tilde{j}_{z}$ appears inside the function $J(x)=\tilde{j}_{z}+\frac{Bx^{2}}{2}$, where the last term accounts for the orbital $B$-field effect.

Hinge states.—Approximating (2) by its tight-binding (tb) lattice counterpart or by solving it directly in an appropriately chosen basis (see SM [66] for details), we calculate the band structure shown in Fig. 2(a). Among the valence and conduction bands $\varepsilon_{\lambda}(\tilde{j}_{z})$, labeled by the band indices $\lambda\leq-1$ and $\lambda\geq 1$ and lying below and above the chemical potential $\mu=0$, respectively, there is a distinguished band $\lambda=0$ (orange line) which crosses the chemical potential at the two points $\tilde{j}_{z;R,L}^{*}$ (with $\varepsilon^{\prime}_{0}(\tilde{j}_{z;R,L}^{*})\lessgtr 0$), see also in Fig. 4(a).

As shown in Fig. 2(b), the states on the negative linear slope near $\tilde{j}_{z,R}^{*}$ (indicated by the red star) form the right ($R$) hinge mode, since their density is localized near the disc right side $\theta=\frac{\pi}{2}$ (expressed in the polar coordinates $x=l_{0}+r\sin\theta$, $z=r\cos\theta$, with $\theta\in(-\pi,\pi]$ and $r<r_{0}$, associated with the disc). They propagate in the clockwise direction along the outer equator of the torus (looking from the top), see the red line in Fig. 1(a). The left ($L$) hinge mode with the positive slope resides near $\tilde{j}^{*}_{z,L}$ [indicated by the blue star in Fig. 2(a)], its states are localized near $\theta\approx-\frac{\pi}{2}$ (inner equator of the torus) and propagate in the counter-clockwise direction, see the blue line in Fig. 1(a).

To understand the properties of these hinge modes more deeply we rely on their low-energy description [66] which is valid in the regime $l_{0}\gg r_{0}\gg l_{so}$ and $\frac{r_{0}}{l_{so}}\gg\frac{l_{so}^{2}}{l_{Z}^{2}}\gg\frac{l_{so}}{r_{0}}$. It allows us to reveal the spinor structure of the hinge modes as well as to provide quantitative estimates of their properties.

In particular, we establish that the $R$ and $L$ hinge modes are stabilized by the Zeeman term, staying exponentially localized near the torus surface on the length scale of $l_{so}$. In the angular direction, their localization is gaussian with the variance $\sigma_{\theta}=\frac{l_{Z}}{\sqrt{r_{0}l_{so}/2}}$. The presence of $l_{Z}$ in the variance is indicative of the Zeeman mechanism of their onset. Moreover, we notice that their angular spread exceeds the radial localization length, that is $r_{0}\sigma_{\theta}\gg l_{so}$. Finally, we approximate the energy dispersion of the $R$ and $L$ hinge modes by $\varepsilon_{0}^{R,L}(\tilde{j}_{z})\approx\mp\frac{\alpha}{l_{0}}(\tilde{j}_{z}-\tilde{j}_{z;R,L}^{*})$, where the offsets $\tilde{j}_{z;R,L}^{*}\approx-\frac{B\cdot\pi(l_{0}\pm r_{0})^{2}}{2\pi}$ account the $B$-field flux quanta through the outer and inner equatorial rings of the torus, i.e. are generated by the orbital effects of the $B$-field.

Plateau region.—In the broad range $\tilde{j}_{z,R}^{*}<\tilde{j}_{z}<\tilde{j}_{z,L}^{*}$, there is a nearly degenerate plateau of weakly dispersing Landau surface states [between yellow and green stars in Fig. 2(a)]. Looking first at the states in the plateau’s middle (that is near the purple star in Fig. 2(a) at $\tilde{j}_{z}\approx-\frac{B\cdot\pi l_{0}^{2}}{2\pi}$), we observe that they are the linear combinations of the so called top and bottom Landau surface states that are localized near $\theta=0$ and $\theta=\pi$, respectively. Due to the finite torus size, the splitting between $\varepsilon_{0}$ and $\varepsilon_{-1}$ bands in the middle of the plateau is estimated by an exponentially small value $\propto\alpha\sqrt{B}e^{-2Br_{0}^{2}}$ [66]. The density of the state with higher energy, i.e. belonging to the band $\varepsilon_{0}$, is shown in the central inset of Fig. 2(b). This state is drastically different from the above discussed $R$ and $L$ hinge states, since its stabilization occurs due to the orbital $B$-field effect, while the Zeeman term produces only a perturbative contribution $-\varepsilon_{Z}$ to its energy. The low-energy analysis also predicts that its angular variance $\sigma_{\theta}=\frac{l_{B}}{r_{0}}$ is both independent of $\varepsilon_{Z}$ and smaller than the variance of the $R$ and $L$ hinges (while the radial localization length $l_{so}$ remains the same).

The other states belonging to $\varepsilon_{0}$ connect the top-bottom Landau surface states in the plateau middle with the $R$ and $L$ hinges on the linear branches of the band $\varepsilon_{0}$. Their density localization evolves along the disc circumference in the angular direction with the changing $\tilde{j}_{z}$ value [see Fig. 2(b)]. These states retain the same radial localization length $l_{so}$, while their angular variances increase from the top-bottom Landau surface states to the $R$ and $L$ hinges, with the stabilization mechanism accordingly changing from the orbital to the Zeeman effect.

Boundary charge and 3D-SOTI QHE.—A convenient way to introduce the Hall conductance is to define it via the charge localized within an appropriately chosen boundary region [55, 56, 57, 58]. In our model, the outer ($R$) and the inner ($L$) boundary regions $\mathcal{B}_{R,L}$ [shown in brown in Fig. 1(b)] are bounded by the cylindrical shells of the radii $\rho_{R,L}=l_{0}\pm(r_{0}-d)$, respectively. Choosing the width $d$ such that $r_{0}\gg d\gg l_{so},\frac{l_{Z}^{2}}{l_{so}}$, we ensure that the $R$ and $L$ hinge states are fully confined within $\mathcal{B}_{R,L}$, respectively. It is natural to define the boundary charges $Q_{B}^{R,L}(f)=\sum_{\lambda}Q_{B,\lambda}^{R,L}(f)$, with

where $\bar{Q}_{B,\lambda}^{R,L}(\tilde{j}_{z})=\int_{\mathcal{B}_{R,L}}dxdz|\Psi_{\lambda}(x,z;\tilde{j}_{z})|^{2}$ is a partial contribution of the state $(\lambda,j_{z})$ evaluated at the AB flux $f\cdot\phi_{0}$. To make the definition of $Q_{B,\lambda}^{R,L}(f)$ for each band insensitive to a specific choice of $d$, we subtract a large background contribution at zero flux.

The central result of this Letter, supported by the numerical result shown in Fig. 3, is that the dependence of $Q_{B}^{R,L}(f)$ on the flux $f$ is piece-wise linear,

with jumps at $f=f_{R,L}^{*}$, where $\tilde{j}_{z;R,L}^{*}-f$ is half-integer. The slope value is correlated with the jump size such that the periodicity $Q_{B}^{R,L}(f)=Q_{B}^{R,L}(f+1)$ is maintained. Changing the flux adiabatically in time, the immediate consequence of (4) is the quantization of the Hall conductance, relating the Hall current to the electric field via $I_{\rho}^{R,L}=\sigma_{\rho\varphi}^{R,L}E_{\varphi}^{R,L}$, see Fig. 1(b) and the discussion in Refs. [56, 57, 62]. Using Faraday’s law and the continuity equation we get $E_{\varphi}^{R,L}=-\dot{\phi}/(2\pi\rho_{R,L})$ and $I_{\rho}^{R,L}=\pm\dot{Q}_{B}^{R,L}/(2\pi\rho_{R,L})$. Together with $\dot{Q}_{B}^{R,L}\approx\frac{\partial Q_{B}^{R,L}}{\partial f}\dot{f}$ and $\dot{\phi}=2\pi\dot{f}$, we obtain the quantization of the Hall conductance $\sigma_{\rho\varphi}^{R,L}=\frac{1}{2\pi}\stackrel{{\scriptstyle!}}{{=}}\frac{e^{2}}{h}$ away from the jump points $f^{*}_{R,L}$.

The second important result is that the contribution $Q_{B,\lambda}^{R,L}$ from all valence bands with $\lambda\leq-1$ is negligible (see SM [66] for the numerical verification). This is an intuitively expected result, since the valence band $\varepsilon_{-1}$ is separated from the band $\varepsilon_{0}$ by the topologically trivial, though exponentially small, gap. The only contribution to the boundary charge is thus provided by the mode $\varepsilon_{0}$, which is shown in Fig. 4.

Let us provide the analytical arguments for the results presented above. Concerning the discrete jumps of $Q_{B}^{R,L}(f)$ at $f=f_{R,L}^{*}$, it is obvious that they appear at those flux values, where a half-integer $j_{z}$ exists with $j_{z}+f_{R,L}^{*}=\tilde{j}_{z;R,L}^{*}$. These are the points where the $R$/$L$ hinge modes of the $\lambda=0$ band cross the chemical potential, leading to a jump of $Q_{B,\lambda=0}^{R,L}(f)$ by $\pm 1$. In contrast, for all fully occupied bands $\lambda<0$, there is no jump.

The linear dependence on the flux follows essentially from the smooth behaviour of $\bar{Q}_{B,\lambda}^{R,L}(\tilde{j}_{z})$ as function of $\tilde{j}_{z}$, i.e., it varies typically on the scale $\Delta\tilde{j}_{z}\sim\tilde{j}_{z,L}^{*}-\tilde{j}_{z,R}^{*}$ of the plateau size, see Fig. 4(b) for $\lambda=0$ (the same applies for all $\lambda\neq 0$, except that the asymptotic values are coinciding). The plateau size is roughly related to the number $N_{B}\sim r_{0}l_{0}/l_{B}^{2}$ of flux quanta threaded through the torus surface from the orbital field (for $B=0$ another scale $\sim l_{0}$ is obtained, see SM [66]). Therefore, the $n$-th order derivative of $\bar{Q}_{B,\lambda}^{R,L}(\tilde{j}_{z})$ will scale with $1/N_{B}^{n}$ in the plateau region (outside this region the derivatives are very small), leading with an additional factor $N_{B}$ from the sum over $j_{z}$ to the scaling $(\frac{d}{df})^{n}Q_{B,\lambda}^{R,L}(f)\sim 1/N_{B}^{n-1}$. Thus, all higher-order derivatives are negligible for $n\geq 2$, provided that $N_{B}\gg 1$ is fulfilled. This can be achieved to any desired accuracy by increasing the torus radius $l_{0}$ (this result holds for any $B$, see SM [66]). Therefore, only the linear term in $f$ contributes significantly to $Q_{B,\lambda}^{R,L}(f)$ for each band. Since the slope value of the linear term of $Q_{B,\lambda}^{R,L}(f)$ is correlated with the discrete jumps due to the periodicity property $Q_{B,\lambda}^{R,L}(f)=Q_{B,\lambda}^{R,L}(f+1)$, we find that the slope is zero for all bands $\lambda<0$ and $\mp 1$ for $\lambda=0$. This shows that the contribution of all valence bands $\lambda<0$ is negligible and the universal linear slope of the total boundary charge is fully determined by the $\lambda=0$ band.

Disorder effects.—We add random disorder potentials to every site of the 3D tb lattice realization of the Hamiltonian (1). Disorder breaks the rotational symmetry, the quantum number $j_{z}$ is no longer conserved, and the band index $\lambda$ is not well defined either. Remarkably, the boundary charge $Q_{B}^{R,L}$, in which all the states below $\mu$ are included, is still well defined, as well as the relation between the boundary charge and the Hall conductance. In Fig. 5 we show that the linear slopes of $Q_{B}^{R,L}(f)$ remain very close to the universal value $\mp 1$, i.e. staying essentially unaffected by disorder. This demonstrates the fundamental significance of the boundary charge concept: It is capable of explaining the Hall conductance quantization even in the disordered case, when the simple clean-case explanation presented above is no longer applicable.

Discussion.— The proposed 3D-SOTI QHE is robust to disorder which is very promising for its experimental realization using standard 3D-TIs with Zeeman fields induced by external magnetic fields, magnetic doping, or nearby ferromagnets. It occurs specifically only in 3D systems (for system sizes exceeding the localization length of hinge states) and is insensitive to orbital effects (see SM [66]), providing various fingerprints for its unique observation. Moreover, it is robust to shape distortions and will even occur in a spherical model with an infinitesimally thin hollow tube along the vertical axis (to allow for the Aharonov-Bohm flux insertion). The obtained results for the boundary charge showcase the same qualitative features as in the torus model (see SM [66]) and its detailed study opens another avenue of future research.

Acknowledgments.—This work was supported by the Deutsche Forschungsgemeinschaft via RTG 1995, the Swiss National Science Foundation (SNSF) and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769. We acknowledge support from the Max Planck-New York City Center for Non-Equilibrium Quantum Phenomena. Simulations were performed with computing resources granted by RWTH Aachen University under projects rwth0752 and rwth0841, and at sciCORE (http://scicore.unibas.ch/) scientific computing center at University of Basel. Funding was received from the European Union’s Horizon 2020 research and innovation program (ERC Starting Grant, grant agreement No 757725).