Entanglement entropy of two disjoint intervals in c=1 theories

Journal of Statistical Mechanics Theory and Experiment (Impact Factor: 2.4). 03/2011; 6(06). DOI: 10.1088/1742-5468/2011/06/P06012
Source: arXiv


We study the scaling of the Renyi entanglement entropy of two disjoint blocks
of critical lattice models described by conformal field theories with central
charge c=1. We provide the analytic conformal field theory result for the
second order Renyi entropy for a free boson compactified on an orbifold
describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual
line. We have checked this prediction in cluster Monte Carlo simulations of the
classical two dimensional AT model. We have also performed extensive numerical
simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor
network techniques that allowed to obtain the reduced density matrices of
disjoint blocks of the spin-chain and to check the correctness of the
predictions for Renyi and entanglement entropies from conformal field theory.
In order to match these predictions, we have extrapolated the numerical results
by properly taking into account the corrections induced by the finite length of
the blocks to the leading scaling behavior.

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Available from: Luca Tagliacozzo, Oct 13, 2015
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    • "The first of such results [10] [11] established a fundamental relation between the logarithimic growth of the EE and the CFT central charge for connected regions in ground states of unitary models, later re-derived using different techniques [12] [13]. Generalizations have included excited states [14] [15], disconnected regions [16] [17] [18] [19] [20] and non-unitary models [21]. The first result beyond criticality [12] showed that the EE of infinite regions saturates to a value which logarithmically grows with the correlation length. "
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    ABSTRACT: We consider the logarithmic negativity, a measure of bipartite entanglement, in a general 1+1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length $r$ and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance $r$. We show that the former saturates to a finite value, and that the latter tends to zero, as $r\rightarrow\infty$. We show that in both cases, the leading corrections are exponential decays in $r$ (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy, the logarithmic negativity displays a very high level of universality, allowing one to extract the mass spectrum. Further, a study of sub-leading terms shows that, unlike the entanglement entropy, a large-$r$ analysis of the negativity allows for the detection of bound states.
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    • "Here c is the central charge of the conformal field theory, c 1 a non-universal constant, while L and L A are the sizes of the full system and of part A, respectively. Universal information can also be extracted from the entanglement between many disjoint blocks [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62], or from the scaling corrections of the von Neumann entropy [63] [64] [65] [66] [67] [68] [69]. "
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    ABSTRACT: We investigate entanglement properties of the excited states of the spin-1/2 Heisenberg (XXX) chain with isotropic antiferromagnetic interactions, by exploiting the Bethe ansatz solution of the model. We consider eigenstates obtained from both real and complex solutions ("strings") of the Bethe equations. Physically, the former are states of interacting magnons, whereas the latter contain bound states of groups of particles. We first focus on the low-density regime, i.e., with few particles in the chain. Using exact results and semiclassical arguments, we derive an upper bound S_MAX for the entanglement entropy. This exhibits an intermediate behavior between logarithmic and extensive, and it is saturated for highly-entangled states. As a function of the eigenstate energy, the entanglement entropy is organized in bands. Their number depends on the number of blocks of contiguous Bethe-Takahashi quantum numbers. In presence of bound states a significant reduction in the entanglement entropy occurs, reflecting that a group of bound particles behaves effectively as a single particle. Interestingly, the associated entanglement spectrum shows edge-related levels. Upon increasing the particle density, the semiclassical bound S_MAX becomes inaccurate. For highly-entangled states S_A\propto L_c, with L_c the chord length, signaling the crossover to extensive entanglement. Finally, we consider eigenstates containing a single pair of bound particles. No significant entanglement reduction occurs, in contrast with the low-density regime.
    Journal of Statistical Mechanics Theory and Experiment 06/2014; 2014(10). DOI:10.1088/1742-5468/2014/10/P10029 · 2.40 Impact Factor
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    • ") n using the replica trick and classical Monte Carlo simulations. This is based on the approach developed in [15] [21] [22] "
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    ABSTRACT: We investigate the behavior of the moments of the partially transposed reduced density matrix \rho^{T_2}_A in critical quantum spin chains. Given subsystem A as union of two blocks, this is the (matrix) transposed of \rho_A with respect to the degrees of freedom of one of the two. This is also the main ingredient for constructing the logarithmic negativity. We provide a new numerical scheme for calculating efficiently all the moments of \rho_A^{T_2} using classical Monte Carlo simulations. In particular we study several combinations of the moments which are scale invariant at a critical point. Their behavior is fully characterized in both the critical Ising and the anisotropic Heisenberg XXZ chains. For two adjacent blocks we find, in both models, full agreement with recent CFT calculations. For disjoint ones, in the Ising chain finite size corrections are non negligible. We demonstrate that their exponent is the same governing the unusual scaling corrections of the mutual information between the two blocks. Monte Carlo data fully match the theoretical CFT prediction only in the asymptotic limit of infinite intervals. Oppositely, in the Heisenberg chain scaling corrections are smaller and, already at finite (moderately large) block sizes, Monte Carlo data are in excellent agreement with the asymptotic CFT result.
    Journal of Statistical Mechanics Theory and Experiment 02/2013; 2013(05). DOI:10.1088/1742-5468/2013/05/P05013 · 2.40 Impact Factor
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