By developing a method to represent the Renyi entropies via a replica trick on classical statistical mechanical systems, we introduce a procedure to calculate the Renyi mutual information (RMI) in any Monte Carlo simulation. Through simulations on several classical models, we demonstrate that 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, in addition to critical points mediated by symmetry breaking, the RMI is able to detect topological vortex-unbinding transitions, as we explicitly demonstrate on simulations of the XY model.

Although the leading-order scaling of entanglement entropy is nonuniversal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in noninteracting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space-time dimensions-a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertexinduced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shapedependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.

A practical use of the entanglement entropy in a 1d quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3) ln for an interval of length in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2d conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and 3-state Potts models. Introduction.-In studies of new and exotic phases of quantum matter, the entanglement entropy has established itself as an important resource [1]. It is particularly useful in the many 1d quantum critical systems governed by a conformal field theory (CFT) [2] in the large-distance limit. Here the Rényi entanglement entropy S n of the ground-state is universal [3][4][5][6], and the leading piece is proportional to the central charge c of the CFT characterizing the universality class. Namely, for a periodic system of length L cut into two open segments of respective sizes L A and L B = (L − L A ),

We show how the thermodynamic properties of large many-body localized systems can be studied using quantum Monte Carlo simulations. To this end we devise a heuristic way of constructing local integrals of motion of very high quality, which are added to the Hamiltonian in conjunction with Lagrange multipliers. The ground state simulation of the shifted Hamiltonian corresponds to a high-energy state of the original Hamiltonian in case of exactly known local integrals of motion. We can show that the inevitable mixing between eigenstates as a consequence of non-perfect integrals of motion is weak enough such that the characteristics of many-body localized systems are not averaged out in our approach, unlike the standard ensembles of statistical mechanics. Our method paves the way to study higher dimensions and indicates that a full many-body localized phase in 2d, where (nearly) all eigenstates are localized, is likely to exist.Introduction -Many-body localization (MBL) addresses the fundamental question under which conditions quantum systems can avoid ergodicity and thermalization, thereby generalizing Anderson localization to interacting systems [1][2][3][4][5][6][7].The widely accepted mechanism for thermalization in quantum systems is the eigenstate thermalization hypothesis [8][9][10]: Under very mild assumptions, (almost) every eigenstate of the system is thermal. This implies that the reduced density matrix of a small subsystem, obtained by tracing out the degrees of freedom of the considered (eigen)state outside the subsystem, is indistinguishable from the thermal density matrix with an effective temperature that depends on the energy density of the chosen eigenstate. MBL states, on the contrary, retain knowledge of their initial local conditions in local operators for asymptotically large times. The picture of local integrals of motion (LIOM) [11][12][13][14][15][16] can explain most of the unusual phenomenology of MBL states: an area low holds in space leading to a logarithmic growth of the entanglement entropy in time and space, and the dc conductivity is identically zero. The area law was demonstrated explicitly in Ref. [17], while it was also found that rare regions can lead to deviations. Dynamics is a decisive characteristic to distinguish between thermal and MBL states: experiments on trapped ions [18] and 1d cold atoms [19] demonstrated memory of the initial conditions over long periods of time for sufficiently strong disorder.

We introduce a quantum Monte Carlo algorithm to measure the Rényi entanglement entropies in systems of interacting bosons in the continuum. This approach is based on a path integral ground state method that can be applied to interacting itinerant bosons in any spatial dimension with direct relevance to experimental systems of quantum fluids. We demonstrate how it may be used to compute spatial mode entanglement, particle partitioned entanglement, and the entanglement of particles, providing insights into quantum correlations generated by fluctuations, indistinguishability and interactions. We present proof-of-principle calculations, and benchmark against an exactly soluble model of interacting bosons in one spatial dimension. As this algorithm retains the fundamental polynomial scaling of quantum Monte Carlo when applied to sign-problem-free models, future applications should allow for the study of entanglement entropy in large scale many-body systems of interacting bosons.