Ann B. Kallin

Measuring Renyi entanglement entropy in quantum Monte Carlo simulations.

We develop a quantum Monte Carlo procedure, in the valence bond basis, to measure the Renyi entanglement entropy of a many-body ground state as the expectation value of a unitary Swap operator acting on two copies of the system. An improved estimator involving the ratio of Swap operators for different subregions enables convergence of the entropy in a simulation time polynomial in the system size. We demonstrate convergence of the Renyi entropy to exact results for a Heisenberg chain. Finally, we calculate the scaling of the Renyi entropy in the two-dimensional Heisenberg model and confirm that the Néel ground state obeys the expected area law for systems up to linear size L=32.

Anomalies in the Entanglement Properties of the Square Lattice Heisenberg Model

Physics Department, University of California, Davis, CA, 95616(Dated: July 15, 2011)We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg modelby a variety of numerical techniques that include valence bond quantum Monte Carlo (QMC),stochastic series expansion QMC, high temperature series expansions and zero temperature couplingconstant expansions around the Ising limit. We find that the area law is always satisfied, but inaddition to the entanglement entropy per unit boundary length, there are other terms that dependlogarithmically on the subregion size, arising from broken symmetry in the bulk and from theexistence of corners at the boundary. We find that the numerical results are anomalous in severalways. First, the bulk term arising from broken symmetry deviates from an exact calculation that canbe done for a mean-field N´eel state. Second, the corner logs do not agree with the known results fornon-interacting Boson modes. And, third, even the finite temperature mutual information shows ananomalous behavior as T goes to zero, suggesting that T → 0 and L → ∞ limits do not commute.These calculations show that entanglement entropy demonstrates a very rich behavior in d > 1,which deserves further attention.I. INTRODUCTION

Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.

Entanglement at a two-dimensional quantum critical point: a numerical linked-cluster expansion study.

We develop a method to calculate the bipartite entanglement entropy of quantum models, in the thermodynamic limit, using a numerical linked-cluster expansion (NLCE) involving only rectangular clusters. It is based on exact diagonalization of all n×m rectangular clusters at the interface between entangled subsystems A and B. We use it to obtain the Renyi entanglement entropy of the two-dimensional transverse field Ising model, for arbitrary real Renyi index α. Extrapolating these results as a function of the order of the calculation, we obtain universal pieces of the entanglement entropy associated with lines and corners at the quantum critical point. They show NLCE to be one of the few methods capable of accurately calculating universal properties of arbitrary Renyi entropies at higher dimensional critical points.

Finite-temperature critical behavior of mutual information.

We study mutual information for Renyi entropy of arbitrary index n, in interacting quantum systems at finite-temperature critical points, using high-temperature expansion, quantum Monte Carlo simulations and scaling theory. We find that, for n>1, the critical behavior is manifest at two temperatures T(c) and nT(c). For the XXZ model with Ising anisotropy, the coefficient of the area law has a t lnt singularity, whereas the subleading correction from corners has a logarithmic divergence, with a coefficient related to the exact results of Cardy and Peschel. For T<nT(c) there is a constant term associated with broken symmetries that jumps at both T(c) and nT(c), which can be understood in terms of a scaling function analogous to the boundary entropy of Affleck and Ludwig.

Corner contribution to the entanglement entropy of an O(3) quantum critical point in 2 + 1 dimensions

The entanglement entropy for a quantum critical system across a boundary with a corner exhibits a sub-leading logarithmic scaling term with a scale-invariant coefficient. Using a Numerical Linked Cluster Expansion, we calculate this universal quantity for a square-lattice bilayer Heisenberg model at its quantum critical point. We find, for this 2+1 dimensional O(3) universality class, that it is thrice the value calculated previously for the Ising universality class. This relation gives substantial evidence that this coefficient provides a measure of the number of degrees of freedom of the theory, analogous to the central charge in a 1+1 dimensional conformal field theory.

Detecting Classical Phase Transitions with Renyi Mutual Information

Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada(Dated: January 8, 2014)By developing a method to represent the Renyi entropies via a replica-trick on classical statisticalmechanical systems, we introduce a procedure to calculate the Renyi Mutual Information (RMI)in any Monte Carlo simulation. Through simulations on several classical models, we demonstratethat the RMI can detect finite-temperature critical points, and even identify their universality class,without knowledge of an order parameter or other thermodynamic estimators. Remarkably, inaddition to critical points mediated by symmetry breaking, the RMI is able to detect topologicalvortex-unbinding transitions, as we explicitly demonstrate on simulations of the XY model.

Finite-size scaling of mutual information in Monte Carlo simulations: Application to the spin-1/2 XXZ model

We develop a quantum Monte Carlo procedure to compute the Renyi mutual information of an interacting quantum many-body system at non-zero temperature. Performing simulations on a spin-1/2 XXZ model, we observe that for a subregion of fixed size embedded in a system of size L, the mutual information converges at large L to a limiting function which displays non-monotonic temperature behavior corresponding to the onset of correlations. For a region of size L/2 embedded in a system of size L, the mutual information divided by L converges to a limiting function of temperature, with apparently nontrivial corrections near critical points.

Neutron scattering studies of spin-phonon hybridization and superconducting spin gaps in the high temperature superconductor La2-x(Sr;Ba)xCuO4

We present time-of-flight neutron-scattering measurements on single crystals of La2-xBaxCuO4 (LBCO) with 0 ≤ x ≤ 0.095 and La2-xSrxCuO4 (LSCO) with x = 0.08 and 0.11. This range of dopings spans much of the phase diagram relevant to high temperature cuprate superconductivity, ranging from insulating, three dimensional commensurate long range antiferromagnetic order for x ≤ 0.02 to two dimensional (2D) incommensurate antiferromagnetism co-existing with superconductivity for x ≥ 0.05. Previous work on lightly doped LBCO with x = 0.035 showed a clear resonant enhancement of the inelastic scattering coincident with the low energy crossings of the highly dispersive spin excitations and quasi-2D optic phonons. The present work extends these measurements across the phase diagram and shows this enhancement to be a common feature to this family of layered quantum magnets. Furthermore we show that the low temperature, low energy magnetic spectral weight is substantially larger for samples with non-superconducting ground states relative to any of the samples with superconducting ground states. Lastly spin gaps, suppression of low energy magnetic spectral weight, are observed in both superconducting LBCO and LSCO samples, consistent with previous observations for superconducting LSCO