Approximations of interface topological invariants

Abstract

This paper concerns the asymmetric transport observed along interfaces separating two-dimensional bulk topological insulators modeled by (continuous) differential Hamiltonians and how such asymmetry persists after numerical discretization. We first demonstrate that a relevant edge current observable is quantized and robust to perturbations for a large class of elliptic Hamiltonians. We then establish a bulk edge correspondence stating that the observable equals an integer-valued bulk difference invariant depending solely on the bulk phases. We next show how to extend such results to periodized Hamiltonians amenable to standard numerical discretizations. A form of no-go theorem implies that the asymmetric transport of periodized Hamiltonians necessarily vanishes. We introduce a filtered version of the edge current observable and show that it is approximately stable against perturbations and converges to its quantized limit as the size of the computational domain increases. To illustrate the theoretical results, we finally present numerical simulations that approximate the infinite domain edge current with high accuracy, and show that it is approximately quantized even in the presence of perturbations.

Figures (5)

• Figure 1: Left panel: illustration of the spatial filters $P$ and $Q$. The function $P$ is smooth and constant within the vertical slabs $x_1 \le x \le x_2$ and $x_3 \le x \le x_4$, while $Q$ is equal to $1$ in the shaded region and rapidly (but smoothly) vanishes outside this region. Right panel: illustration of the coefficients of the periodic operator $H_\kappa$. For example, a coefficient can transition smoothly between $-1$ and $1$.
• Figure 2: Eigenfunctions of the periodic $2 \times 2$ Dirac Hamiltonian (top) and $p$-wave superconductor (bottom) models. First three columns: edge states localized at the $y=0$ and $y = \pm L_y/2$ interfaces; far right: bulk states that do not contribute to the edge current. The plotted vector components and corresponding eigenvalues are labeled.
• Figure 3: Perturbed eigenfunctions of $2 \times 2$ Dirac (top) $p$-wave superconductor (bottom) models. For all plots, $r = 2$ and $a = 7.5$. We see a mix of propagating and evanescent modes. Despite qualitative differences between perturbed and unperturbed eigenfunctions, the edge current remains approximately the same.
• Figure 4: The left panel demonstrates the numerical stability of $\tilde{\sigma}_I$ for the $2 \times 2$ Dirac (solid line) and $p$-wave superconductor (dashed line) models. For the solid line, we fixed $a = 7.5$ (with $r$ varying) and the dashed line corresponds to $r = 10$ (with $a$ varying). The center panel shows the edge current for the $2 \times 2$ Dirac (solid line) and $p$-wave superconductor (dashed line) models as a function of the center of $Q_Y$. As expected, we get $-1,-2$ when the filter selects the increasing domain wall and $1,2$ when the filter selects the decreasing domain wall, with a sharp transition in between. The right panel plots the edge current for the $3 \times 3$ model as a function of perturbation strength, for perturbations in four distinct matrix elements. The edge current is stable with respect to perturbations of the other five matrix elements; see text.
• Figure 5: Branches of continuous spectrum for the $3 \times 3$ equatorial wave Hamiltonian, for different values of regularization parameter $\mu$. The top line corresponds to $f(y) = \mathop{\mathrm{sgn}}\nolimits (y)$ while the bottom line corresponds to $f(y) = \tanh (\beta y)$. The solid curves represent the nontrivial increasing branches (and the two flat bands at $\pm 1$ when $\mu = 0$ and $f(y) = \mathop{\mathrm{sgn}}\nolimits (y)$). For $\mu = 0$, we omit many eigenvalues approximately equal to $0$, as they correspond to essential spectrum for the continuous problem. When $\mu > 0$ ($\mu < 0$), these eigenvalues populate the region $\{E(\xi) > 0\}$ ($\{E(\xi) < 0\}$), making it difficult to identify branches of spectrum there.

Theorems & Definitions (75)

• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4: Bulk-Edge Correspondence
• Corollary 2.5: Fedosov-Hörmander formula
• proof : Proof of Corollary \ref{['cor:bic']}
• proof : Proof of Corollary \ref{['cor:FH']}
• Proposition 2.6
• Proposition 2.7
• Proposition 2.8