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arXiv:1004.3112 · Physical Review A

Entanglement entropy in quantum spin chains with broken reflection symmetry

Zoltán Kádár, Zoltán Zimborás

Published Sep 29, 2010• 29 citations
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Abstract

We investigate the entanglement entropy of a block of $L$ sites in quasifree translation-invariant spin chains concentrating on the effect of reflection-symmetry breaking. The Majorana two-point functions corresponding to the Jordan-Wigner transformed fermionic modes are determined in the most general case; from these, it follows that reflection symmetry in the ground state can only be broken if the model is quantum critical. The large $L$ asymptotics of the entropy are calculated analytically for general gauge-invariant models, which have, until now, been done only for the reflection-symmetric sector. Analytical results are also derived for certain nongauge-invariant models (e.g., for the Ising model with Dzyaloshinskii-Moriya interaction). We also study numerically finite chains of length $N$ with a nonreflection-symmetric Hamiltonian and report that the reflection symmetry of the entropy of the first $L$ spins is violated but the reflection-symmetric Calabrese-Cardy formula is recovered asymptotically. Furthermore, for noncritical reflection-symmetry-breaking Hamiltonians, we find an anomaly in the behavior of the saturation entropy as we approach the critical line. The paper also provides a concise but extensive review of the block-entropy asymptotics in translation-invariant quasifree spin chains with an analysis of the nearest-neighbor case and the enumeration of the yet unsolved parts of the quasifree landscape.

DOI
10.1103/physreva.82.032334

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We investigate the entanglement entropy of a block of LLL sites in quasifree translation-invariant spin chains concentrating on the effect of reflection-symmetry breaking. The Majorana two-point functions corresponding to the Jordan-Wigner transformed fermionic modes are determined in the most general case; from these, it follows that reflection symmetry in the ground state can only be broken if the model is quantum critical. The large LLL asymptotics of the entropy are calculated analytically for general gauge-invariant models, which have, until now, been done only for the reflection-symmetric sector. Analytical results are also derived for certain nongauge-invariant models (e.g., for the Ising model with Dzyaloshinskii-Moriya interaction). We also study numerically finite chains of length NNN with a nonreflection-symmetric Hamiltonian and report that the reflection symmetry of the entropy of the first LLL spins is violated but the reflection-symmetric Calabrese-Cardy formula is recovered asymptotically. Furthermore, for noncritical reflection-symmetry-breaking Hamiltonians, we find an anomaly in the behavior of the saturation entropy as we approach the critical line. The paper also provides a concise but extensive review of the block-entropy asymptotics in translation-invariant quasifree spin chains with an analysis of the nearest-neighbor case and the enumeration of the yet unsolved parts of the quasifree landscape.
We obtain a formula for the determinant of a block Toeplitz matrix associated with a quadratic fermionic chain with complex coupling. Such couplings break reflection symmetry and/or charge conjugation symmetry. We then apply this formula to compute the R\'enyi entropy of a partial observation to a subsystem consisting of contiguous sites in the limit of large size. The present work generalizes similar results due to Its, Jin, and Korepin [Fields Institute Communications, Universality and Renormalization, Vol. 50, p. 151, 2007] and Its, Mezzadri, and Mo [Commun. Math. Phys. 284, 117 (2008)]. A striking feature of our formula for the entanglement entropy is the appearance of a term scaling with the logarithm of the size. This logarithmic behavior originates from certain discontinuities in the symbol of the block Toeplitz matrix. Equipped with this formula we analyze the entanglement entropy of a Dzyaloshinski-Moriya spin chain and a Kitaev fermionic chain with long-range pairing.
Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in depth the properties of the eigenstates of the enlarged Hamiltonian. In particular we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.
We consider the single-copy entanglement as a quantity to assess quantum correlations in the ground state in quantum many-body systems. We show for a large class of models that already on the level of single specimens of spin chains, criticality is accompanied with the possibility of distilling a maximally entangled state of arbitrary dimension from a sufficiently large block deterministically, with local operations and classical communication. These analytical results---which refine previous results on the divergence of block entropy as the rate at which maximally entangled pairs can be distilled from many identically prepared chains---are made quantitative for general isotropic translationally invariant spin chains that can be mapped onto a quasifree fermionic system, and for the anisotropic XYXYXY model. For the XXXXXX model, we provide the asymptotic scaling of $\ensuremath{\sim}(1∕6){\mathrm{log}}_{2}(L)$, and contrast it with the block entropy.
We establish a quantitative relationship between the entanglement content of a single quantum chain at a critical point and the corresponding entropy of entanglement. We find that, surprisingly, the leading critical scaling of the single-copy entanglement with respect to any bipartitioning is exactly one-half of the entropy of entanglement, in a general setting of conformal field theory and quasifree systems. Conformal symmetry imposes that the single-copy entanglement scales as ${E}_{1}({\ensuremath{\rho}}_{L})=(c∕6)\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L\ensuremath{-}(c∕6)({\ensuremath{\pi}}^{2}∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L)+O(1∕L)$, where LLL is the number of constituents in a block of an infinite chain and ccc denotes the central charge. This shows that from a single specimen of a critical chain, already half the entanglement can be distilled compared to the rate that is asymptotically available. The result is substantiated by a quantitative analysis for all translationally invariant quantum spin chains corresponding to all isotropic quasifree fermionic models. An example of the XYXYXY spin chain shows that away from criticality the above relation is maintained only near the quantum phase transition.
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.
We examine the entanglement entropy of the even half of a translationally invariant finite chain or lattice in its ground state. This entropy measures the entanglement between the even and odd halves (each forming a ``comb'' of n/2n/2n/2 sites) and can be expected to be extensive for short-range couplings away from criticality. We first consider bosonic systems with quadratic couplings, where analytic expressions for arbitrary dimensions can be provided. The bosonic treatment is then applied to finite spin chains and arrays by means of the random-phase approximation. Results for first-neighbor anisotropic XY\mathit{XY}XY couplings indicate that, while at strong magnetic fields this entropy is strictly extensive, at weak fields important deviations arise, stemming from parity-breaking effects and the presence of a factorizing field (in the vicinity of which it becomes size-independent and identical to the entropy of a contiguous half). Exact numerical results for small spin sss chains are shown to be in agreement with the bosonic random-phase approximation prediction.