Inversion-symmetric topological insulators

We analyze translationally invariant insulators with inversion symmetry that fall outside the current established classification of topological insulators. These insulators exhibit no edge or surface modes in the energy spectrum and hence they are not edge metals when the Fermi level is in the bulk gap. However, they do exhibit protected modes in the entanglement spectrum localized on the cut between two entangled regions. Their entanglement entropy cannot be made to vanish adiabatically, and hence the insulators can be called topological. There is a direct connection between the inversion eigenvalues of the Hamiltonian band structure and the midgap states in the entanglement spectrum. The classification of protected entanglement levels is given by an integer $\mathcal{N}$, which is the difference between the negative inversion eigenvalues at inversion symmetric points in the Brillouin zone, taken in sets of 2. When the Hamiltonian describes a Chern insulator or a nontrivial time-reversal invariant topological insulator, the entirety of the entanglement spectrum exhibits spectral flow. If the Chern number is zero for the former, or time reversal is broken in the latter, the entanglement spectrum does not have spectral flow, but, depending on the inversion eigenvalues, can still exhibit protected midgap bands similar to impurity bands in normal semiconductors. Although spectral flow is broken (implying the absence of real edge or surface modes in the original Hamiltonian), the midgap entanglement bands cannot be adiabatically removed, and the insulator is “topological.” We analyze the linear response of these insulators and provide proofs and examples of when the inversion eigenvalues determine a nontrivial charge polarization, a quantum Hall effect, an anisotropic three-dimensional (3D) quantum Hall effect, or a magnetoelectric polarization. In one dimension, we establish a link between the product of the inversion eigenvalues of all occupied bands at all inversion symmetric points and charge polarization. In two dimensions, we prove a link between the product of the inversion eigenvalues and the parity of the Chern number of the occupied bands. In three dimensions, we find a topological constraint on the product of the inversion eigenvalues thereby showing that some $3$D materials are protected topological metals; we show the link between the inversion eigenvalues and the $3$D Quantum Hall Effect, and analyze the magnetoelectric polarization ($\theta$ vacuum) in the absence of time-reversal symmetry.

Figure 1

Energy spectra for the simple 1D two-band model with open boundary conditions for (a) $\alpha =0\hspace{0.5em}m=-1$ (nontrivial) (b) $\alpha =0\hspace{0.5em}m=1$ (trivial). Entanglement spectra for the two cases are shown in (c) and (d), respectively, for a half filled Fermi sea ground state with periodic boundary conditions.

Figure 2

(a)–(e) Energy spectra for $H_4$ in cases 1–5, respectively, with open boundary conditions. (f)–(j) Entanglement spectra for $H_4$ in cases 1–5, respectively, with periodic boundary conditions.

Figure 3

The adiabatic transport is carried over $\gamma ,\gamma {}^{{}^{\prime }}$.

Figure 4

The two paths $\gamma$ and $\gamma$ used in the monodromy argument in two dimensions.

Figure 5

Entanglement spectrum for the two-band 1D model with (a) random site disorder, (b) random site disorder that is inversion symmetric around the center of the 1D chain. We show the entanglement spectra for many different random disorder configurations.

Figure 6

(a) 1D dimerized chain; (b) 2D dimerized square lattice model. Solid and dotted lines represent different hopping amplitudes.

Figure 7

Entanglement spectra for (a) dimerized chain; (b) dimerized square lattice.

Figure 8

Energy spectrum with an open boundary, entanglement spectrum with a cut parallel to the $y$ direction, and entanglement energies for [left panel, (a),(d),(g)] a trivial insulator; [middle panel, (b),(e),(h)] nontrivial Chern insulator with negative inversion eigenvalue at $(k_x,k_y)=(0,0)$ implying entanglement midgap modes at $k_y=0$ [right panel, (c),(f),(i)]. Chern insulator with negative inversion eigenvalues at $(k_x,k_y)=(0,0),(\pi ,0)$, and $(0,\pi ).$ This implies entanglement midgap modes at $k_y=\pi .$

Figure 9

Energy spectrum with an open boundary, entanglement spectrum with a cut parallel to the $y$ direction, and entanglement energies for [left panel, (a),(d),(g)] a trivial insulator, [middle panel, (b),(e),(h)] nontrivial quantum spin Hall insulator with time-reversal symmetry, [right panel, (c),(f),(i)] quantum spin Hall insulator with mildly broken time-reversal symmetry. Comparing (h) and (i) one can see that all Kramers degeneracies are lifted when time reversal is broken except the ones at $\epsilon =0.$ By just looking at (e) and (f) it is difficult to tell the difference in the two cases, i.e., that spectral flow is broken.

Figure 10

Energy spectrum with open boundary conditions for (a) eight-band model with $T$ symmetry, (b) eight-band model with random inversion preserving perturbation. Entanglement spectrum for (c) eight-band model with $T,$ (d) eight-band model with random inversion preserving perturbation.

Figure 11

(a) Energy spectrum for the small anisotropic eight-band Hamiltonian with periodic boundary conditions. The difference in negative parity eigenvalues at $k_y=0,\pi$ is 3. (b) Entanglement spectrum for the same. At $k_y=0$ there are six modes at $\xi =1/2.$ The lines are only filled in as guides to the eye. The only allowed $k$ values are $k_y=0$ and $k_y=\pi .$ This is why the spectra appear to have kinks.

Figure 12

Entanglement spectra for (a) 3D QHE, (b) 3D weak topological insulator (WTI) with a cut parallel to the $z$-$x$ plane plotted along a line in the Brillouin zone. The states in (b) are all doubly degenerate compared to (a). The coordinates are in the form $\mathbf{k}=(k_z,k_x).$

Figure 13

Left column: Strong topological insulator. (a) Energy spectrum plotted along a line in the Brillouin zone for open boundary conditions along the $z$ direction; (c) entanglement spectrum with a cut parallel to the $x$-$y$ plane with periodic boundary conditions; (e) entanglement energies. Right column: Strong topological insulator with time-reversal breaking (b) energy spectrum, (d) entanglement spectrum, (f) entanglement energies. In (a) and (b) the midgap states are localized on the surfaces. In (b) there is a gap in the surface states due to time-reversal symmetry breaking. In (f) all Kramers degeneracies are lifted except the one at $\epsilon =0.$