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

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 nondegenerate ground state for $h\ensuremath{\ne}0$ 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 an $L$-dimensional matrix, which enables us to calculate easily the block entanglement for large-$N$ chains numerically. The entanglement entropy grows as $\mathrm{log}L$ at the degeneracy point $h=0$ and only for ${J}_{x}={J}_{y}$. For $h\ensuremath{\ne}0$, the entropy becomes independent of the size, thus obeying the area law. For ${J}_{x}\ensuremath{\ne}{J}_{y}$, the block entropy shows a nonmonotonic behavior for $L<N/2$.