Exact and Scaling Form of the Bipartite Fidelity of the Infinite XXZ Chain

TL;DR

This paper derives an exact formula for the bipartite fidelity in the infinite XXZ chain and confirms its scaling behavior aligns with conformal field theory predictions, linking quantum entanglement to the correlation length.

Contribution

It provides an exact expression for bipartite fidelity in the infinite XXZ chain and demonstrates its large correlation length scaling matches conformal field theory conjectures.

Findings

Exact formula for bipartite fidelity derived

Scaling behavior consistent with c/8 ln(xi)

Supports conformal field theory predictions

Abstract

We find an exact expression for the bipartite fidelity f=|<vac|vac>'|^2, where |vac> is the vacuum eigenstate of an infinite-size antiferromagnetic XXZ chain and |vac>' is the vacuum eigenstate of an infinite-size XXZ chain which is split in two. We consider the quantity -ln(f) which has been put forward as a measure of quantum entanglement, and show that the large correlation length xi behaviour is consistent with a general conjecture -ln(f) ~ c/8 ln(xi), where c is the central charge of the UV conformal field theory (with c=1 for the XXZ chain). This behaviour is a natural extension of the existing conformal field theory prediction of -ln(f) ~ c/8 ln(L) for a length L bipartite system with 0<< L <<xi.