Free-fermion Entanglement Spectrum through Wannier Interpolation

TL;DR

This paper derives bounds on the spacing of entanglement eigenenergies in free-fermion systems using Wannier function decay, revealing their relation to band gaps and edge states, with implications for numerical methods and Toeplitz matrices.

Contribution

It introduces an interpolation method connecting Wannier decay to entanglement spectrum spacing, providing new bounds and insights into the spectrum's structure.

Findings

Entanglement energies decay rate relates to band gap size.

One-to-one correspondence between entanglement spectrum and edge states.

Provides bounds for eigenvalue distributions of Block Toeplitz matrices.

Abstract

Quantum Entanglement plays an ubiquitous role in theoretical physics, from the characterization of novel phases of matter to understanding the efficacy of numerical algorithms. As such, there have been extensive studies on the entanglement spectrum (ES) of free-fermion systems, particularly in the relation between its spectral flow and topological charge pumping. However, far less has been studied about the \emph{spacing} between adjacent entanglement eigenenergies, which affects the truncation error in numerical computations involving Matrix Product States (MPS) or Projected Entangled-Pair States (PEPS). In this paper, we shall hence derive asymptotic bounds for the ES spacings through an interpolation argument that utilizes known results on Wannier function decay. For translation invariant systems, the Entanglement energies are shown to decay at a rate monotonically related to the complex gap between the filled and occupied bands. This interpolation also demonstrates the one-to-one correspondence between the ES and the edge states. Our results also provide asymptotic bounds for the eigenvalue distribution of certain types of Block Toeplitz matrices common in physics, even for those not arising from entanglement calculations.