Factorization and entanglement in general XYZ spin arrays in non-uniform transverse fields

TL;DR

This paper explores the conditions for separable eigenstates and their ground states in XYZ spin arrays under non-uniform transverse fields, revealing how entanglement can emerge between any two spins regardless of their separation.

Contribution

It provides a comprehensive analysis of separability and entanglement in general XYZ spin arrays, including conditions for ground states and entanglement measures for arbitrary spin and range.

Findings

Separable eigenstates can be ground states under specific conditions.

Entanglement between any two spins can occur regardless of their separation.

Exact calculations of entanglement measures like concurrence and negativity.

Abstract

We determine the conditions for the existence of a pair of degenerate parity breaking separable eigenstates in general arrays of arbitrary spins connected through $XYZ$ couplings of arbitrary range and placed in a transverse field, not necessarily uniform. Sufficient conditions under which they are ground states are also provided. It is then shown that in finite chains, the associated definite parity states, which represent the actual ground state in the immediate vicinity of separability, can exhibit entanglement between any two spins regardless of the coupling range or separation, with the reduced state of any two subsystems equivalent to that of pair of qubits in an entangled mixed state. The corresponding concurrences and negativities are exactly determined. The same properties persist in the mixture of both definite parity states. These effects become specially relevant in systems close to the $XXZ$ limit. The possibility of field induced alternating separable solutions with controllable entanglement side limits is also discussed. Illustrative numerical results for the negativity between the first and the $j^{\rm th}$ spin in an open spin $s$ chain for different values of $s$ and $j$ are as well provided.