Block entropy for Kitaev-type spin chains in a transverse field

TL;DR

This paper investigates the entanglement entropy in Kitaev-type spin chains under a transverse field, revealing a logarithmic growth at degeneracy points and an area law behavior otherwise, using an exact and efficient computational method.

Contribution

It provides an exact solution for the block entanglement entropy in Kitaev-type spin chains with a transverse field, including a novel efficient method for large systems.

Findings

Entropy grows as log L at h=0 and J_x=J_y.

Entropy obeys area law for nonzero magnetic field.

Non-monotonic entropy behavior for J_x ≠ J_y and L<N/2.

Abstract

Block entanglement entropy in the ground state of a quantum spin chain is investigated. The spins have Kitaev-type nearest-neighbor interaction, of strength J_x or J_y, through either x or y components of the spins on alternating bonds, along with a transverse magnetic field h. An exact solution is obtained through Jordan-Wigner fermionization, and it exhibits a macroscopically degenerate ground state for h=0, and a non-degenerate ground state for nonzero h and for all interaction strengths. For a chain of N spins, we study the block entropy of a partition of L contiguous spins. The block entanglement entropy needs the eigenvalues of the 2^L-dimensional reduced density matrix. We employ an efficient method that reduces this problem to evaluating eigenvalues of a L-dimensional matrix, which enables us to calculate easily the block entanglement for large-N chains numerically. The entanglement entropy grows as log L, at the degeneracy point h=0, and only for J_x=J_y. For nonzero magnetic field, the entropy becomes independent of the size, thus obeying the area law. For unequal J_x and J_y, the block entropy shows a non-monotonic behavior for L<N/2.