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arXiv:0808.0600 · Physical Review A

Entanglement of two blocks of spins in the critical Ising model

Paolo Facchi A, Giuseppe Florio, Carmen Invernizzi, Saverio Pascazio

Published Nov 3, 2008• 44 citations
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Abstract

We compute the entropy of entanglement of two blocks of $L$ spins at a distance $d$ in the ground state of an Ising chain in an external transverse magnetic field. We numerically study the von Neumann entropy for different values of the transverse field. At the critical point we obtain analytical results for blocks of size $L=1$ and 2. In the general case, the critical entropy is shown to be additive when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally, based on simple arguments, we derive an expression for the entropy at the critical point as a function of both $L$ and $d$. This formula is in excellent agreement with numerical results.

DOI
10.1103/physreva.78.052302

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Bipartite entanglement in the ground state of a chain of NNN quantum spins can be quantified either by computing pairwise concurrence or by dividing the chain into two complementary subsystems. In the latter case the smaller subsystem is usually a single spin or a block of adjacent spins and the entanglement differentiates between critical and noncritical regimes. Here we extend this approach by considering a more general setting: our smaller subsystem SA{S}_{A}SA​ consists of a comb of LLL spins, spaced ppp sites apart. Our results are thus not restricted to a simple area law, but contain nonlocal information, parametrized by the spacing ppp. For the XXXXXX model we calculate the von Neumann entropy analytically when $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and investigate its dependence on LLL and ppp. We find that an external magnetic field induces an unexpected length scale for entanglement in this case.
We study the scaling of the R\'enyi and entanglement entropy of two disjoint blocks of critical Ising models as function of their sizes and separations. We present analytic results based on conformal field theory that are quantitatively checked in numerical simulations of both the quantum spin chain and the classical two-dimensional Ising model. Theoretical results match the ones obtained from numerical simulations only after taking properly into account the corrections induced by the finite length of the blocks to their leading scaling behavior.
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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{J}_{x}Jx​ or Jy{J}_{y}Jy​, through either xxx or yyy components of the spins on alternating bonds, along with a transverse magnetic field hhh. An exact solution is obtained through Jordan-Wigner fermionization, and it exhibits a macroscopically degenerate ground state for h=0h=0h=0 and a nondegenerate ground state for $h\ensuremath{\ne}0$ for all interaction strengths. For a chain of NNN spins, we study the block entropy of a partition of LLL contiguous spins. The block entanglement entropy needs the eigenvalues of the 2L{2}^{L}2L-dimensional reduced density matrix. We employ an efficient method that reduces this problem to evaluating eigenvalues of an LLL-dimensional matrix, which enables us to calculate easily the block entanglement for large-NNN chains numerically. The entanglement entropy grows as logL\mathrm{log}LlogL at the degeneracy point h=0h=0h=0 and only for Jx=Jy{J}_{x}={J}_{y}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/2L<N/2L<N/2.
Using free-fermionic techniques we study the entanglement entropy of a block of contiguous spins in a large finite quantum Ising chain in a transverse field, with couplings of different types: homogeneous, periodically modulated and random. We carry out a systematic study of finite-size effects at the quantum critical point, and evaluate subleading corrections both for open and for periodic boundary conditions. For a block corresponding to a half of a finite chain, the position of the maximum of the entropy as a function of the control parameter (e.g. the transverse field) can define the effective critical point in the finite sample. On the basis of homogeneous chains, we demonstrate that the scaling behavior of the entropy near the quantum phase transition is in agreement with the universality hypothesis, and calculate the shift of the effective critical point, which has different scaling behaviors for open and for periodic boundary conditions.