Entanglement between distant qubits in cyclic XX chains

摘要

We evaluate the exact concurrence between any two spins in a cyclic XX chain of n spins placed in a uniform transverse magnetic field, at both 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 nonzero temperatures the entanglement remains nonzero for arbitrarily high fields, for any pair separation L, although its magnitude decreases exponentially with increasing field. It is also demonstrated that the associated limit temperatures approach a constant nonzero value in this limit, which decreases as L-2 for L < n, but exhibit special finite-size effects for distant qubits (L approximate to 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 n ->infinity are also discussed.