# Entanglement entropy of two disjoint intervals in c=1 theories

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Luca Tagliacozzo, Jul 02, 2015 Available from:- [Show abstract] [Hide abstract]

**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 · 2.06 Impact Factor - [Show abstract] [Hide abstract]

**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 · 2.06 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We compute the time evolution of the mutual information in out of equilibrium quantum systems whose gravity duals are Vaidya spacetimes in three and four dimensions, which describe the formation of a black hole through the collapse of null dust. We find the holographic mutual information to be non monotonic in time and always monogamous in the ranges explored. We also find that there is a region in the configuration space where it vanishes at all times. We show that the null energy condition is a necessary condition for both the strong subadditivity of the holographic entanglement entropy and the monogamy of the holographic mutual information.Journal of High Energy Physics 10/2011; 2012(1). DOI:10.1007/JHEP01(2012)102 · 6.22 Impact Factor