Accuracy of topological entanglement entropy on finite cylinders

TL;DR

This paper investigates finite size effects on topological entanglement entropy (TEE) in $Z_2$ topological phases, revealing that von Neumann entropy provides more accurate TEE estimates than higher-order Renyi entropies in finite systems.

Contribution

The study systematically compares finite size corrections of TEE derived from von Neumann and Renyi entropies, highlighting the superior accuracy of von Neumann entropy in finite cylinders.

Findings

Renyi entropies with n≥2 have larger finite size corrections than von Neumann entropy.

For cylinders with circumference about ten times the correlation length, von Neumann TEE error is around 10^{-3}.

Renyi entropies can have errors exceeding 40% under similar conditions.

Abstract

Topological phases are unique states of matter which support non-local excitations which behave as particles with fractional statistics. A universal characterization of gapped topological phases is provided by the topological entanglement entropy (TEE). We study the finite size corrections to the TEE by focusing on systems with $Z_2$ topological ordered state using density-matrix renormalization group (DMRG) and perturbative series expansions. We find that extrapolations of the TEE based on the Renyi entropies with Renyi index $n\geq 2$ suffer from much larger finite size corrections than do extrapolations based on the von Neumann entropy. In particular, when the circumference of the cylinder is about ten times the correlation length, the TEE obtained using von Neumann entropy has an error of order $10^{-3}$, while for Renyi entropies it can even exceed 40%. We discuss the relevance of these findings to previous and future searches for topological ordered phases, including quantum spin liquids.