A Characterization of Topological Insulators: Chern Numbers for a Ground State Multiplet

We propose an alternative formulation of the $Z_2$ topological index for quantum spin Hall systems and band insulators when time reversal invariance is not broken. The index is expressed in terms of the Chern numbers of the bands of the model, and a connection with the number of pairs of robust edge states is thus established. The alternative index is easy to compute in most cases of interest. We also discuss connections with the recently proposed spin Chern number for quantu...

This paper is a survey of the $\mathbb{Z}_2$-valued invariant of topological insulators used in condensed matter physics. The $\mathbb{Z}$-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The $\mathbb{Z}_2$ invariant is more mysterious, we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the cla...

If an extensive partition in two dimensions yields a gapful entanglement spectrum of the reduced density matrix, the Berry curvature based on the corresponding entanglement eigenfunction defines the Chern number. We propose such an entanglement Chern number as a useful, natural, and calculable topological invariant, which is potentially relevant to various topological ground states. We show that it serves as an alternative topological invariant for time-reversal invariant sys...

We propose characterization of the three-dimensional topological insulator by using the Chern number for the entanglement Hamiltonian (entanglement Chern number). Here we take the extensive spin partition of the system, that pulls out the quantum entanglement between up spin and down spin of the many-body ground state. In three dimensions, the topological insulator phase is described by the section entanglement Chern number, which is the entanglement Chern number for a period...

We study a special class of topological phase transitions in two dimensions described by the inversion of bands with relative angular momentum higher than 1. A band inversion of this kind, which is protected by rotation symmetry, separates the trivial insulator from a Chern insulating phase with higher Chern number, and thus generalizes the quantum Hall transition described by a Dirac fermion. Higher angular momentum band inversions are of special interest, as the non-vanishi...

We propose a two-dimensional non-Hermitian Chern insulator with inversion symmetry, which is anisotropic and has staggered gain and loss in both x and y directions. In this system, conventional bulk-boundary correspondence holds. The Chern number is a topological invariant that accurately predicts the topological phase transition and the existence of helical edge states in the topologically nontrivial gapped phase. In the gapless phase, the band touching points are isolated a...

We devise local lattice models whose ground states are model fractional Chern insulators---Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the $\mathcal{Z}_k$ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling ...

We have studied extensively the band crossing patterns of the bulk entanglement spectrum (BES) for various lattice Chern insulators. We find that only partitions with dual symmetry can have either stable nodal-lines or nodal-points in the BES when the system is in the topological phase of a nonzero Chern number. By deforming the Hamiltonian to lift the accidental symmetry, one can see that only nodal points are robust. They thus should bear certain topological characteristics...

As first demonstrated by the characterization of the quantum Hall effect by the Chern number, topology provides a guiding principle to realize robust properties of condensed matter systems immune to the existence of disorder. The bulk-boundary correspondence guarantees the emergence of gapless boundary modes in a topological system whose bulk exhibits nonzero topological invariants. Although some recent studies have suggested a possible extension of the notion of topology to ...

Chern insulators are states of matter characterized by a quantized Hall conductance, gapless edge modes but also a singular response to monopole configurations of an external electromagnetic field. In this paper, we describe the nature of such a singular response and show how it can be used to define a class of operators acting as non-local order parameters. These operators characterize the Chern-insulator states in the following way: for a given state, there exists a corresp...