Scaling of the von Neumann entropy across a finite temperature phase transition

TL;DR

This paper analytically studies how the von Neumann entropy scales across a finite temperature phase transition in a system of hard core bosons, revealing a crossover from logarithmic to linear behavior at the critical temperature.

Contribution

It provides an analytical description of the von Neumann entropy's behavior across a phase transition in a bosonic system, highlighting the classical and quantum contributions.

Findings

VNE scales logarithmically at zero temperature.

VNE becomes linear above the critical temperature.

Intermediate temperatures show a sum of classical and quantum entropy contributions.

Abstract

The spectrum of the reduced density matrix and the temperature dependence of the von Neumann entropy (VNE) are analytically obtained for a system of hard core bosons on a complete graph which exhibits a phase transition to a Bose-Einstein condensate at $T=T_c$. It is demonstrated that the VNE undergoes a crossover from purely logarithmic at T=0 to purely linear in block size $n$ behaviour for $T\geq T_{c}$. For intermediate temperatures, VNE is a sum of two contributions which are identified as the classical (Gibbs) and the quantum (due to entanglement) parts of the von Neumann entropy.