General entanglement scaling laws from time evolution

We establish a general scaling law for the entanglement of a large class of ground states and dynamically evolving states of quantum spin chains: we show that the geometric entropy of a distinguished block saturates, and hence follows an entanglement-boundary law. These results apply to any ground state of a gapped model resulting from dynamics generated by a local hamiltonian, as well as, dually, to states that are generated via a sudden quench of an interaction as recently studied in the case of dynamics of quantum phase transitions. We achieve these results by exploiting ideas from quantum information theory and making use of the powerful tools provided by Lieb-Robinson bounds. We also show that there exist noncritical fermionic systems and equivalent spin chains with rapidly decaying interactions whose geometric entropy scales logarithmically with block length. Implications for the classical simulatability are outlined.Comment: 4 pages, 1 figure (see also related work by S. Bravyi, M. Hastings, and F. Verstraete, quant-ph/0603121); replaced with final versio

Entanglement entropy of two disjoint intervals in c=1 theories

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.Comment: 37 pages, 23 figure

Exact boundary conditions in numerical relativity using multiple grids: scalar field tests

Cauchy-Characteristic Matching (CCM), the combination of a central 3+1 Cauchy code with an exterior characteristic code connected across a time-like interface, is a promising technique for the generation and extraction of gravitational waves. While it provides a tool for the exact specification of boundary conditions for the Cauchy evolution, it also allows to follow gravitational radiation all the way to infinity, where it is unambiguously defined. We present a new fourth order accurate finite difference CCM scheme for a first order reduction of the wave equation around a Schwarzschild black hole in axisymmetry. The matching at the interface between the Cauchy and the characteristic regions is done by transfering appropriate characteristic/null variables. Numerical experiments indicate that the algorithm is fourth order convergent. As an application we reproduce the expected late-time tail decay for the scalar field.Comment: 14 pages, 5 figures. Included changes suggested by referee

Quantum Quench from a Thermal Initial State

We consider a quantum quench in a system of free bosons, starting from a thermal initial state. As in the case where the system is initially in the ground state, any finite subsystem eventually reaches a stationary thermal state with a momentum-dependent effective temperature. We find that this can, in some cases, even be lower than the initial temperature. We also study lattice effects and discuss more general types of quenches.Comment: 6 pages, 2 figures; short published version, added references, minor change

Field-theory results for three-dimensional transitions with complex symmetries

We discuss several examples of three-dimensional critical phenomena that can be described by Landau-Ginzburg-Wilson $\phi^4$ theories. We present an overview of field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the $\epsilon$ and of the fixed-dimension d=3 expansions. In particular, we discuss the stability of the O(N)-symmetric fixed point in a generic N-component theory, the critical behaviors of randomly dilute Ising-like systems and frustrated spin systems with noncollinear order, the multicritical behavior arising from the competition of two distinct types of ordering with symmetry O($n_1$) and O($n_2$) respectively.Comment: 9 pages, Talk at the Conference TH2002, Paris, July 200

Scaling of Entanglement Entropy in the Random Singlet Phase

We present numerical evidences for the logarithmic scaling of the entanglement entropy in critical random spin chains. Very large scale exact diagonalizations performed at the critical XX point up to L=2000 spins 1/2 lead to a perfect agreement with recent real-space renormalization-group predictions of Refael and Moore [Phys. Rev. Lett. {\bf 93}, 260602 (2004)] for the logarithmic scaling of the entanglement entropy in the Random Singlet Phase with an effective central charge ${\tilde{c}}=c\times \ln 2$. Moreover we provide the first visual proof of the existence the Random Singlet Phase thanks to the quantum entanglement concept.Comment: 4 pages, 3 figure

Exact relationship between the entanglement entropies of XY and quantum Ising chains

We consider two prototypical quantum models, the spin-1/2 XY chain and the quantum Ising chain and study their entanglement entropy, S(l,L), of blocks of l spins in homogeneous or inhomogeneous systems of length L. By using two different approaches, free-fermion techniques and perturbational expansion, an exact relationship between the entropies is revealed. Using this relation we translate known results between the two models and obtain, among others, the additive constant of the entropy of the critical homogeneous quantum Ising chain and the effective central charge of the random XY chain.Comment: 6 page

Universal parity effects in the entanglement entropy of XX chains with open boundary conditions

We consider the Renyi entanglement entropies in the one-dimensional XX spin-chains with open boundary conditions in the presence of a magnetic field. In the case of a semi-infinite system and a block starting from the boundary, we derive rigorously the asymptotic behavior for large block sizes on the basis of a recent mathematical theorem for the determinant of Toeplitz plus Hankel matrices. We conjecture a generalized Fisher-Hartwig form for the corrections to the asymptotic behavior of this determinant that allows the exact characterization of the corrections to the scaling at order o(1/l) for any n. By combining these results with conformal field theory arguments, we derive exact expressions also in finite chains with open boundary conditions and in the case when the block is detached from the boundary.Comment: 24 pages, 9 figure

Exact and Scaling Form of the Bipartite Fidelity of the Infinite XXZ Chain

We find an exact expression for the bipartite fidelity f=|'|^2, where |vac> is the vacuum eigenstate of an infinite-size antiferromagnetic XXZ chain and |vac>' is the vacuum eigenstate of an infinite-size XXZ chain which is split in two. We consider the quantity -ln(f) which has been put forward as a measure of quantum entanglement, and show that the large correlation length xi behaviour is consistent with a general conjecture -ln(f) ~ c/8 ln(xi), where c is the central charge of the UV conformal field theory (with c=1 for the XXZ chain). This behaviour is a natural extension of the existing conformal field theory prediction of -ln(f) ~ c/8 ln(L) for a length L bipartite system with 0<< L <<xi.Comment: 6 page