Fisher-Hartwig formula has been successful applied to describe the von Neumann and Rényi entropies of a block of spins in the ground state of XX spin chain. It was based on a determinant representation. In this paper, we generalize the free fermion method to obtain an exact formulation for the entropy of any finite subsystem in XX spin chain. Based on this, we derive a determinant representation of the entropy of multiple disjoint intervals in the ground state of
XX model.
Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.
We consider the Rényi entropies in the one-dimensional spin-1/2 Heisenberg XX chain in a magnetic field. The case n = 1 corresponds to the von Neumann ‘entanglement’ entropy. Using a combination of methods based on the generalized Fisher–Hartwig conjecture and a recurrence relation connected to the Painlevé VI differential equation we obtain the asymptotic behaviour, accurate to order , of the Rényi entropies for large block lengths . For n = 1, 2, 3, 10 this constitutes the 3, 6, 10, 48 leading terms respectively. The o(1) contributions are found to exhibit a rich structure of oscillatory behaviour, which we analyse in some detail both for finite n and in the limit .
Block entanglement entropy in the ground state of a quantum spin chain is investigated. The spins have Kitaev-type nearest-neighbor interaction, of strength
Jx or
Jy, 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
2L-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
logL at the degeneracy point
h=0 and only for
Jx=Jy. 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.
We consider the Rényi entanglement entropies in the one-dimensional XX spin-chains with open boundary conditions in the presence of a magnetic field. In the case of a semi-infinite system and a block starting from the boundary, we derive rigorously the asymptotic behavior for large block sizes on the basis of a recent mathematical theorem for the determinant of Toeplitz plus Hankel matrices. We conjecture a generalized Fisher–Hartwig form for the corrections to the asymptotic behavior of this determinant that allows the exact characterization of the corrections to the scaling at order for any n . By combining these results with conformal field theory arguments, we derive exact expressions also in finite chains with open boundary conditions and in the case when the block is detached from the boundary.