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arXiv:0909.2654 · Physical Review Letters

Entanglement Spectrum of Topological Insulators and Superconductors

Lukasz Fidkowski

Published Apr 2, 2010• 394 citations
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Abstract

We study two a priori unrelated constructions: the spectrum of edge modes in a band topological insulator or superconductor with a physical edge, and the ground state entanglement spectrum in an extended system where an edge is simulated by an entanglement bipartition. We prove an exact relation between the ground state entanglement spectrum of such a system and the spectrum edge modes of the corresponding spectrally flattened Hamiltonian. In particular, we show that gapless edge modes result in degeneracies of the entanglement spectrum.

DOI
10.1103/physrevlett.104.130502

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We study two a priori unrelated constructions: the spectrum of edge modes in a band topological insulator or superconductor with a physical edge, and the ground state entanglement spectrum in an extended system where an edge is simulated by an entanglement bipartition. We prove an exact relation between the ground state entanglement spectrum of such a system and the spectrum edge modes of the corresponding spectrally flattened Hamiltonian. In particular, we show that gapless edge modes result in degeneracies of the entanglement spectrum.
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How do we uniquely identify a quantum phase, given its ground state wave-function? This is a key question for many body theory especially when we consider phases like topological insulators, that share the same symmetry but differ at the level of topology. The entanglement spectrum has been proposed as a ground state property that captures characteristic edge excitations. Here we study the entanglement spectrum for topological band insulators. We first show that insulators with topological surface states will necessarily also have protected modes in the entanglement spectrum. Surprisingly, however, the converse is not true. Protected entanglement modes can also appear for insulators without physical surface states, in which case they capture a more elusive property. This is illustrated by considering insulators with only inversion symmetry. Inversion is shown to act in an unusual way, as an antiunitary operator, on the entanglement spectrum, leading to this protection. The entanglement degeneracies indicate a variety of different phases in inversion symmetric insulators, and these phases are argued to be robust to the introduction of interactions.
In this study, we discuss a type of bulk-boundary correspondence, which holds for topological insulators and superconductors when the parity-time ( P T ) and/or parity-particle-hole ( P C ) symmetry are present. In these systems, even when the bulk topology is nontrivial, the edge spectrum is generally gapped, and thus the conventional bulk-boundary correspondence does not hold. We find that, instead of the edge spectrum, the single-particle entanglement spectrum becomes gapless when the bulk topology is nontrivial: i.e., the holds in P T - and/or P C -symmetric topological insulators and superconductors. After showing the correspondence using K -theoretic approach, we provide concrete models for each symmetry class up to three dimensions where nontrivial topology because of P T and/or P C is expected. An implication of our results is that, when the bulk topology under P T and/or P C symmetry is nontrivial, the noninteracting many-body entanglement spectrum is multiply degenerate in one dimension and is gapless in two or higher dimensions. Published by the American Physical Society 2024
We study the entanglement spectrum of a Chern insulator on a cylinder geometry, with the cut separating the two partitions taken parallel to the cylinder edge, at varying distances from the edge. In contrast to similar studies on a torus, there is only one cut, and hence only one virtual edge mode in the entanglement spectrum. The entanglement spectrum has a gap when the cut is close enough to the physical edge of the cylinder such that the edge mode spatially extends over the cut. This effect is suppressed for parameter choices where the edge mode is sharply localized at the edge. In the extreme case of a perfectly localized edge mode, the entanglement spectrum is gapless even if the smaller partition consists of a single edge row. For the single-row cut, we construct the corresponding entanglement Hamiltonian, which is a one-dimensional tight-binding Hamiltonian with complex long-range hopping and interesting properties. We also study and explain the effect of two different schemes of flux insertion through a ring described by such an entanglement Hamiltonian.
We present a theoretical approach to incorporate electronic interactions in the study of two-dimensional topological insulators. By exploiting the correspondence between edge-state physics and entanglement spectrum in gapped topological systems, we deconstruct the system into one-dimensional channels. This framework enables a simple and elegant inclusion of fermionic interactions into the discussion of topological insulators. We apply this approach to the Kane-Mele model with interactions and magnetic impurities.