Entanglement between distant qubits in cyclic XX chains

TL;DR

This paper provides an exact analysis of entanglement between any two spins in a cyclic XX chain under a transverse magnetic field at various temperatures, revealing how entanglement persists and varies with field strength, temperature, and chain size.

Contribution

It introduces an exact method to evaluate pairwise entanglement in cyclic XX chains at finite temperature, highlighting finite size effects and entanglement behavior near quantum phase transitions.

Findings

Entanglement exists between any two spins at T=0 within a narrow field range.

At low temperatures, entanglement persists at high fields for all spin pairs.

Limit temperatures for entanglement approach a non-zero constant, decreasing with chain length.

Abstract

We evaluate the exact concurrence between any two spins in a cyclic XX chain of n spins placed in a uniform transverse magnetic field, both at zero and finite temperature, by means of the Jordan-Wigner transformation plus a number parity projected statistics. It is shown that while at T=0 there is always entanglement between any two spins in a narrow field interval before the transition to the aligned state, at low but non-zero temperatures the entanglement remains non-zero for arbitrarily high fields, for any pair separation L, although its magnitude decreases exponentially with the field. It is also demonstrated that all associated limit temperatures approach a constant non-zero value in this limit, which decreases as 1/L^2 for L<<n but exhibit special finite size effects for distant qubits (L approx. n/2). Related aspects such as the different behavior of even and odd antiferromagnetic chains, the existence of n ground state transitions and the thermodynamic limit are also discussed.