Fragility of Surface States in Non-Wigner-Dyson Topological Insulators

Topological insulators and superconductors support extended surface states protected against the otherwise localizing effects of static disorder. Specifically, in the Wigner-Dyson insulators belonging to the symmetry classes A, AI, and AII, a band of extended surface states is continuously connected to a likewise extended set of bulk states forming a “bridge” between different surfaces via the mechanism of spectral flow. In this work we show that this mechanism is absent in the majority of non-Wigner-Dyson topological superconductors and chiral topological insulators. In these systems, there is precisely one point with granted extended states, the center of the band, $E=0$. Away from it, states are spatially localized, or can be made so by the addition of spatially local potentials. Considering the three-dimensional insulator in class AIII and winding number $\nu =1$ as a paradigmatic case study, we discuss the physical principles behind this phenomenon, and its methodological and applied consequences. In particular, we show that low-energy Dirac approximations in the description of surface states can be treacherous in that they tend to conceal the localizability phenomenon. We also identify markers defined in terms of Berry curvature as measures for the degree of state localization in lattice models, and back our analytical predictions by extensive numerical simulations. A main conclusion of this work is that the surface phenomenology of non-Wigner-Dyson topological insulators is a lot richer than that of their Wigner-Dyson siblings, extreme limits being spectrumwide quantum critical delocalization of all states versus full localization except at the $E=0$ critical point. As part of our study we identify possible experimental signatures distinguishing between these different alternatives in transport or tunnel spectroscopy.

Popular Summary

Topological insulators are materials whose interiors insulate while their surfaces conduct. In the iconic case of the quantum Hall effect, the conducting surface states spanning different surfaces are connected at high energies through the bulk of the material by delocalized “bridging states.” This surface-bulk connection is called spectral flow, and it underpins some of the most unusual properties of topological matter, such as the ability to pump electrons from one surface to the other through the bulk. In this work, we show that spectral flow is not as closely tied to the physics of topological matter as previously believed.

In particular, we demonstrate that most 3D topological phases do not possess spectral flow. We show how the surface states can be completely detached from the bulk and that there need not be delocalized bridging states in the bulk. The detachment can be accomplished by modifying the surface of the material without changing the topological phase.

The most remarkable physical consequences of these findings concern the response of the surface states to disorder, which is inevitable in real materials. We show that disorder can destroy the conducting properties of almost all surface states in most 3D topological classes. By contrast, all surface states are robustly protected from disorder in topological phases that possess spectral flow (such as the quantum Hall effect). At the same time, we show that surface states can conduct at all energies when the disorder satisfies a certain statistical symmetry, in accordance with previous work.

Our results thus showcase a far richer phenomenology of 3D topological phases than was previously understood, which is of particular relevance in the quest to experimentally identify bulk topological superconductors.

Figure 1

(a) A topological free-fermion phase with spectral flow possesses a robust continuous attachment of anomalous boundary states (red) to delocalized bulk states (black) residing in a Wannier nonlocalizable bulk band (gray). (b) Without a spectral flow principle, anomalous surface bands (red) may in principle be detached from the bulk spectrum by a spectral gap. In this case there is no obstruction to Wannier localization of all bulk states (gray). The figure shows a schematic one-dimensional boundary spectrum, assuming translation symmetry parallel to the boundary.

Figure 2

Schematic picture of surface (red) and bulk (black) spectra of the topological insulator Hamiltonian $H^{\prime }=H_{\partial }\oplus H_{\mathrm{bulk}}$, see Eq. (2), before (a) and after (b) a rescaling $H_{\partial }\to \lambda H_{\partial }$ to shrink the surface band to a width narrower than the bulk gap.

Figure 3

Schematic dispersions of the minimal edge Hamiltonian Eq. (8) (a) and of the nonminimal model Eq. (9), which has an additional band of localized states at the edge (b). Since it derives from a lattice model, the asymptotically flat band from the localized states must be continuously connected for $k\to \infty$ and $k\to -\infty$, to reflect the periodicity of the edge Brillouin zone. Unbounded bands are continuously connected to the bulk spectrum for $k\to \infty$ and $k\to -\infty$. The arrows indicate how the occupation of states is changed after insertion of a flux quantum through the bulk.

Figure 4

Schematic dispersions of the minimal edge Hamiltonian Eq. (10) (a) and the nonminimal model Eq. (11), which has two additional bands of localized states at the edge (b). The asymptotically flat band from the localized states must be continuously connected for $k\to \infty$ and $k\to -\infty$, to reflect the periodicity of the edge Brillouin zone. The ultraviolet divergent bands from the chiral edge states are continuously attached to the bulk spectrum for $k\to \infty$ and $k\to -\infty$. In the nonminimal model, the high-energy band is detached from the low-energy band containing the linear crossing at zero energy. The arrows indicate how the occupation of states is changed after insertion of a flux quantum through the bulk.

Figure 5

(a) Spectrum of the topological insulator Eq. (14) with $M=2$ for open boundary conditions in the $x$ direction. The bulk (surface) spectrum is shown in black (red). There is a single low-energy Dirac cone at $(k_y,k_z)=\mathrm{\Gamma }=(0,0)$. Surface and bulk bands merge at high energy. (b) The same as (a) but with the additional perturbation $U_{\mathrm{f}}$ of Eq. (25) with $u_{\mathrm{f},x}=0.5$ and $n=3$. (c) A minimal value of the strength $u_{\mathrm{f},x}$ of the fragmenting potential Eq. (25) required to detach surface and bulk bands, for a perturbation with support on the outermost surface layer only ($n=1$, solid line) and for a perturbation with support on the three outermost surface layers ($n=3$, dashed line). The vertical axis shows the minimal value of the indirect gap ${\mathrm{\Delta }}_{\mathrm{s}}$ between surface and bulk bands.

Figure 6

(a) Integrated Berry curvature $\theta (E)$ of the surface band of the model Eq. (14) versus energy $E$ for the case of zero fragmenting potential $u_{\mathrm{f},x}=0$ (red line), $u_{\mathrm{f},x}=-0.3$ (black, solid line), and $u_{\mathrm{f},x}=-1$ (black, dotted line). The fragmenting potential is added on the three outermost layers adjacent to the $x$ surface, $n=3$, and the vertical red line marks the position of the bulk gap. For $|u_{\mathrm{f},x}|>u_{\mathrm{f}}^{\mathrm{c}}$, surface and bulk bands are detached, so that $\theta (E)$ can be determined for the full surface band; for $u_{\mathrm{f},x}=0$, $\theta (E)$ can be calculated only for energy $E$ inside the bulk gap. (b) The same as (a) but for the minimal surface theory Eq. (10) (red line) and the nonminimal model Eq. (11) (black line) with $|\gamma |/{ϵ}_{\mathrm{c}}=0.05$. According to the criterion of Ref. [20], surface states at energy $E$ are delocalized if $\theta (E)=\pi \text{mod}2\pi$.

Figure 7

(a) Band structure for a slab geometry of the flattened version of the model Eq. (14) with open boundary conditions along $x$ and two symmetrically placed domain walls and periodic boundary conditions along $y$. The fragmenting potential is of the form $u_{\mathrm{f},x}(x,y)$, where $u_{\mathrm{f},x}(x,y)$ switches from 0.2 to $-0.2$ and back at the two domain walls in the $n=1$ outermost surface layers, and is zero otherwise. The bulk, surface, and domain-wall part of the spectrum are indicated in black, red, and green, respectively. Solid and dashed green curves are for the domain wall for which $u_{\mathrm{f},x}$ goes from negative to positive and from positive to negative upon increasing $y$, respectively. Note that the addition of the fragmenting potential to the surface raises part of the surface spectrum above the flattened bulk band. (b) In the presence of disorder, only surface states at $E=0$ are delocalized. In this case, the chiral domain-wall modes are expected to extend from the bulk spectrum all the way down to zero energy.

Figure 8

The scaled exponent, ${\mathrm{\Delta }}_q/q(1-q)$ for $q=-0.5$, 0.5, 0.75 (left to right), and system sizes $N_x=8$ and $N_y=N_z=L=24$ to $L=128$ of the model Eq. (14). Black data, no fragmenting potential $u_{\mathrm{f}}=0$ [Eq. (25)] and disorder strength $W=0.15$; blue data, a constant potential with $u_{\mathrm{f}}=0.3$ applied to the outermost surface layers, and disorder strength $W=0.2$; green data, random fragmenting potential with zero average and standard deviation $u_{\mathrm{f}}=0.3$ [Eq. (34)], and disorder strength $W=0.2$. The horizontal dashed lines mark the value 0.25 of quantum-Hall criticality, and the localization limit ${\tau }_q=0$ corresponds to ${\mathrm{\Delta }}_q/q(1-q)\to \infty ,4,8/3$ for $q=-0.5$, 0.5, 0.75. (The value $\infty$ reflects the formal divergence of the IPR in the localized limit for negative $q$.)

Figure 9

Distribution functions of $P_q$ at $q=0.5$ for $u_{\mathrm{f}}=0.3$ (left), $u_{\mathrm{f}}=0$ (middle), and $u_{\mathrm{f}}$ random (right) with rms $u_{\mathrm{f}}=0.5$. (Here, the topological control parameter is $M=2.0$ and the disorder strength is $W=0.2$.) The horizontal axis has been rescaled as discussed in the text. Inset: the convergence of the mean of the IPRs, scaled by the trivial scaling exponent of fully extended states for constant (blue), zero (yellow), and random (green) $u_{\mathrm{f}}$.

Figure 10

Distribution functions of $P_q$ at $E=0.1$, $u_{\mathrm{f}}=0$, $W=0.15$ and linear surface extension $L=64$ for different slab widths $N_x$ on a linear (main) and semilogarithmic scale (inset).