Even-odd entanglement in boson and spin systems

TL;DR

This paper investigates the entanglement entropy between even and odd halves of bosonic and spin systems, revealing how it varies with system parameters and identifying conditions for extensivity and size-independence.

Contribution

It provides analytical expressions for entanglement entropy in bosonic systems and applies the random phase approximation to spin chains, highlighting effects of magnetic fields and parity-breaking.

Findings

Entanglement entropy is extensive away from criticality in short-range coupled systems.

At strong magnetic fields, entropy is strictly extensive; at weak fields, deviations occur.

Near the factorizing field, entropy becomes size-independent and matches that of a contiguous half.

Abstract

We examine the entanglement entropy of the even half of a translationally invariant finite chain or lattice in its ground state. This entropy measures the entanglement between the even and odd halves (each forming a "comb" of $n/2$ sites) and can be expected to be extensive for short range couplings away from criticality. We first consider bosonic systems with quadratic couplings, where analytic expressions for arbitrary dimensions can be provided. The bosonic treatment is then applied to finite spin chains and arrays by means of the random phase approximation. Results for first neighbor anisotropic XY couplings indicate that while at strong magnetic fields this entropy is strictly extensive, at weak fields important deviations arise, stemming from parity-breaking effects and the presence of a factorizing field (in which vicinity it becomes size-independent and identical to the entropy of a contiguous half). Exact numerical results for small spin s chains are shown to be in agreement with the bosonic RPA prediction.