Entanglement entropy and the complex plane of replicas

TL;DR

This paper investigates the analytic structure of entanglement entropy in quantum critical models, revealing a complex n-plane behavior with a threshold near n=2 that impacts the calculation of entropy.

Contribution

It uncovers a phase-like transition in the analytic properties of the trace of the n-th power of the reduced density matrix in quantum critical systems.

Findings

No true singularities for n>0

A threshold near n=2 separates two distinct regions

Complex n-plane structure affects entropy computation

Abstract

The entanglement entropy of a subsystem $A$ of a quantum system is expressed, in the replica method, through analytic continuation with respect to n of the trace of the n-th power of the reduced density matrix $\tr\rho_A^n$. We study the analytic properties of this quantity as a function of n in some quantum critical Ising-like models in 1+1 and 2+1 dimensions. Although we find no true singularities for n>0, there is a threshold value of n close to 2 which separates two very different `phases'. The region with larger n is characterized by rapidly convergent Taylor expansions and is very smooth. The region with smaller n has a very rich and varied structure in the complex n plane and is characterized by Taylor coefficients which instead of being monotone decreasing, have a maximum growing with the size of the subsystem. Finite truncations of the Taylor expansion in this region lead to increasingly poor approximations of $\tr\rho_A^n$. The computation of the entanglement entropy from the knowledge of $\tr\rho^n_A$ for positive integer n becomes extremely difficult particularly in spatial dimensions larger than one, where one cannot use conformal field theory as a guidance in the extrapolations to n=1.