Entanglement and particle fluctuations of one-dimensional chiral topological insulators

TL;DR

This paper proves that in one-dimensional chiral topological insulators, entanglement spectra and particle fluctuations are topologically protected, with specific bounds related to the winding number, using an index theorem directly on the microscopic model.

Contribution

It establishes a direct proof of topological protection of entanglement and fluctuations in 1D chiral topological insulators via an index theorem, without relying on continuum models or bulk-boundary correspondence.

Findings

Protected eigenvalues at 1/2 in entanglement spectrum proportional to winding number

Lower bounds on particle fluctuations and entanglement entropy related to winding number

Results derived directly from microscopic model using index theorem

Abstract

We consider the topological protection of entanglement and particle fluctuations for a general one-dimensional chiral topological insulator with winding number $\mathcal{I}$. We prove, in particular, that when the periodic system is divided spatially into two equal halves, the single-particle entanglement spectrum has $2|\mathcal{I}|$ protected eigenvalues at $1/2$. Therefore the number fluctuations are bounded from below by $\Delta N^2\geq |\mathcal{I}|/2$ and the entanglement entropy by $S\geq 2|\mathcal{I}|\ln 2$. We note that our results are obtained by applying directly an index theorem to the microscopic model and do not rely on an equivalence to a continuum model or a bulk-boundary correspondence for a slow varying boundary.