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arXiv:quant-ph/0506250 · Physical Review A

Single-copy entanglement in critical quantum spin chains

Jens Eisert, M. Cramer

Published Oct 25, 2005• 101 citations
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Abstract

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 $XY$ model. For the $XX$ model, we provide the asymptotic scaling of $\ensuremath{\sim}(1∕6){\mathrm{log}}_{2}(L)$, and contrast it with the block entropy.

DOI
10.1103/physreva.72.042112

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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∼(1∕6)log2(L)\ensuremath{\sim}(1∕6){\mathrm{log}}_{2}(L)\ensuremath∼(1∕6)log2​(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.
The single-copy entanglement of a given many-body quantum system is defined [J. Eisert and M. Cramer, Phys. Rev. A 72, 042112 (2005)10.1103/PhysRevA.72.042112] as the maximal entanglement deterministically distillable from a bipartition of a single specimen of that system. For critical (gapless) spin chains, it was recently shown that this is exactly half the von Neumann entropy [R. Orús, J. I. Latorre, J. Eisert, and M. Cramer, Phys. Rev. A 73, 060303(R) (2006)], itself defined as the entanglement distillable in the asymptotic limit-i.e., given an infinite number of copies of the system. It is an open question as to what the equivalent behavior for gapped systems is. In this Letter, I show that for the paradigmatic spin-S Affleck-Kennedy-Lieb-Tasaki chain (the archetypal gapped chain), the single-copy entanglement is equal to the von Neumann entropy; i.e., all the entanglement present may be distilled from a single specimen.
We study the critical behavior and the ground-state entanglement of a large class of su(1|1) supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short- and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the same universality class as a (1+1)-dimensional conformal field theory (CFT) with central charge equal to the number of connected components of the Fermi sea. We also study the Rényi entanglement entropy of the ground state, deriving its asymptotic behavior as the block size tends to infinity. In particular, we show that this entropy exhibits the logarithmic growth characteristic of (1+1)-dimensional CFTs and one-dimensional (fermionic) critical lattice models, with a central charge consistent with the low-temperature behavior of the free energy. Our results confirm the widely believed conjecture that the critical behavior of fermionic lattice models is completely determined by the topology of their Fermi surface.
We compute the entanglement between separated blocks in certain spin models showing that at criticality this entanglement is a function of the ratio of the separation to the length of the blocks and can be written as a product of a power law and an exponential decay. It thereby interpolates between the entanglement of individual spins and blocks of spins. It captures features of correlation functions at criticality as well as the monogamous nature of entanglement. We exemplify invariant features of this entanglement to microscopic changes within the same universality class. We find this entanglement to be invariant with respect to simultaneous scale transformations of the separation and the length of the blocks. As a corollary, this Rapid Communication estimates the entanglement between separated regions of those quantum fields to which the considered spin models map at criticality.
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.
The power of matrix product states to describe infinite-size translational-invariant critical spin chains is investigated. At criticality, the accuracy with which they describe ground-state properties of a system is limited by the size $\ensuremath{\chi}$ of the matrices that form the approximation. This limitation is quantified in terms of the scaling of the half-chain entanglement entropy. In the case of the quantum Ising model, we find $S\ensuremath{\sim}\frac{1}{6}\text{log}\text{ }\ensuremath{\chi}$ with high precision. This result can be understood as the emergence of an effective finite correlation length ${\ensuremath{\xi}}_{\ensuremath{\chi}}$ ruling all the scaling properties in the system. We produce six extra pieces of evidence for this finite-$\ensuremath{\chi}$ scaling, namely, the scaling of the correlation length, the scaling of magnetization, the shift of the critical point, the scaling of the entanglement entropy for a finite block of spins, the existence of scaling functions, and the agreement with analogous classical results. All our computations are consistent with a scaling relation of the form ${\ensuremath{\xi}}_{\ensuremath{\chi}}\ensuremath{\sim}{\ensuremath{\chi}}^{\ensuremath{\kappa}}$, with $\ensuremath{\kappa}=2$ for the Ising model. In the case of the Heisenberg model, we find similar results with the value $\ensuremath{\kappa}\ensuremath{\sim}1.37$. We also show how finite-$\ensuremath{\chi}$ scaling allows us to extract critical exponents. These results are obtained using the infinite time evolved block decimation algorithm which works in the thermodynamical limit and are verified to agree with density-matrix renormalization-group results and their classical analog obtained with the corner transfer-matrix renormalization group.