Trace Index and Spectral Flow in the Entanglement Spectrum

We investigate the entanglement spectra of topological insulators which manifest edge states on a lattice with spatial boundaries. In the physical energy spectrum, a subset of the edge states that intersect the Fermi level translates to discontinuities in the trace of the single-particle entanglement spectrum, which we call a `trace index'. We find that any free-fermion topological insulator that exhibits spectral flow has a non-vanishing trace index, which provides us with a new description of topological invariants. In addition, we identify the signatures of spectral flow in the single-particle and many-body entanglement spectrum; in the process we present new methods to extract topological invariants and establish a connection between entanglement and quantum Hall physics.