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

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

Topological phases and topological entropy of two-dimensional systems with finite correlation length

We elucidate the topological features of the entanglement entropy of a region in two dimensional quantum systems in a topological phase with a finite correlation length $\xi$. Firstly, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground state wave function of a quantum eight-vertex model in its topological phase, and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a non-trivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.Comment: 17 pages, 4 figures, published versio

Slow dynamics in critical ferromagnetic vector models relaxing from a magnetized initial state

Within the universality class of ferromagnetic vector models with O(n) symmetry and purely dissipative dynamics, we study the non-equilibrium critical relaxation from a magnetized initial state. Transverse correlation and response functions are exactly computed for Gaussian fluctuations and in the limit of infinite number n of components of the order parameter. We find that the fluctuation-dissipation ratios (FDRs) for longitudinal and transverse modes differ already at the Gaussian level. In these two exactly solvable cases we completely describe the crossover from the short-time to the long-time behavior, corresponding to a disordered and a magnetized initial condition, respectively. The effects of non-Gaussian fluctuations on longitudinal and transverse quantities are calculated in the first order in the epsilon-expansion and reliable three-dimensional estimates of the two FDRs are obtained.Comment: 41 pages, 9 figure

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

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

Nonequilibrium critical dynamics of the two-dimensional Ising model quenched from a correlated initial state

The universality class, even the order of the transition, of the two-dimensional Ising model depends on the range and the symmetry of the interactions (Onsager model, Baxter-Wu model, Turban model, etc.), but the critical temperature is generally the same due to self-duality. Here we consider a sudden change in the form of the interaction and study the nonequilibrium critical dynamical properties of the nearest-neighbor model. The relaxation of the magnetization and the decay of the autocorrelation function are found to display a power law behavior with characteristic exponents that depend on the universality class of the initial state.Comment: 6 pages, 5 figures, submitted to Phys. Rev.

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

Aging at Criticality in Model C Dynamics

We study the off-equilibrium two-point critical response and correlation functions for the relaxational dynamics with a coupling to a conserved density (Model C) of the O(N) vector model. They are determined in an \epsilon=4-d expansion for vanishing momentum. We briefly discuss their scaling behaviors and the associated scaling forms are determined up to first order in epsilon. The corresponding fluctuation-dissipation ratio has a non trivial large time limit in the aging regime and, up to one-loop order, it is the same as that of the Model A for the physically relevant case N=1. The comparison with predictions of local scale invariance is also discussed.Comment: 13 pages, 1 figur

Increasing of entanglement entropy from pure to random quantum critical chains

It is known that the entropy of a block of spins of size $L$ embedded in an infinite pure critical spin chain diverges as the logarithm of $L$ with a prefactor fixed by the central charge of the corresponding conformal field theory. For a class of strongly random spin chains, it has been shown that the correspondent block entropy still remains universal and diverges logarithmically with an "effective" central charge. By computing the entanglement entropy for a family of models which includes the $N$-states random Potts chain and the $Z_N$ clock model, we give some definitive answer to some recent conjectures about the behaviour of the effective central charge. In particular, we show that the ratio between the entanglement entropy in the pure and in the disordered system is model dependent and we provide a series of critical models where the entanglement entropy grows from the pure to the random case.Comment: 4 pages, 2 eps figures, added reference