arXiv subjects

Eduardo Dominguez

Publications and source records attributed to Eduardo Dominguez.

6 recordsLinked to original sources

arXiv2020

The Quantum Cluster Variational Method and the Phase Diagram of the quantum ferromagnetic $J_1$-$J_2$ model

We exploit the Quantum Cluster Variational Method (QCVM) to study the $J_1$-$J_2$ model for quantum Ising spins. We first describe the QCVM and discuss how it is related to other Mean Field approximations. The phase diagram of the model is studied at the level of the Kikuchi approximation in square lattices as a function of the ratio between $g = J_2/J_1$ , the temperature and the longitudinal and transverse external fields. Our results show that quantum fluctuations may change the order of the transition and induce a gap between the ferromagnetic and the stripe phases. Moreover, when both longitudinal and transverse fields are present, thermal fluctuations and quantum effects contribute to the appearance of a nematic phase.

cond-mat.dis-nn↗

arXiv2017

Quantum Cluster Variational Method and Message Passing Algorithms Revisited

We present a general framework to study quantum disordered systems in the context of the Kikuchi's Cluster Variational Method (CVM). The method relies in the solution of message passing-like equations for single instances or in the iterative solution of complex population dynamic algorithms for an average case scenario. We first show how a standard application of the Kikuchi's Cluster Variational Method can be easily translated to message passing equations for specific instances of the disordered system. We then present an "ad-hoc" extension of these equations to a population dynamic algorithm representing an average case scenario. At the Bethe level, these equations are equivalent to the dynamic population equations that can be derived from a proper Cavity Ansatz. However, at the plaquette approximation, the interpretation is more subtle and we discuss it taking also into account previous results in classical disordered models. Moreover, we develop a formalism to properly deal with the average case scenario using a Replica-Symmetric ansatz within this CVM for quantum disordered systems. Finally, we present and discuss numerical solutions of the different approximations for the Quantum Transverse Ising model and the Quantum Random Field Ising model in two dimensional lattices.

cond-mat.dis-nn↗

arXiv2016

A simple analytical description of the non-stationary dynamics in Ising spin systems

The analytical description of the dynamics in models with discrete variables (e.g. Ising spins) is a notoriously difficult problem, that can be tackled only under some approximation. Recently a novel variational approach to solve the stationary dynamical regime has been introduced by Pelizzola [Eur. Phys. J. B, 86 (2013) 120], where simple closed equations are derived under mean-field approximations based on the cluster variational method. Here we propose to use the same approximation based on the cluster variational method also for the non-stationary regime, which has not been considered up to now within this framework. We check the validity of this approximation in describing the non-stationary dynamical regime of several Ising models defined on Erdos-Rényi random graphs: we study ferromagnetic models with symmetric and partially asymmetric couplings, models with random fields and also spin glass models. A comparison with the actual Glauber dynamics, solved numerically, shows that one of the two studied approximations (the so-called 'diamond' approximation) provides very accurate results in all the systems studied. Only for the spin glass models we find some small discrepancies in the very low temperature phase, probably due to the existence of a large number of metastable states. Given the simplicity of the equations to be solved, we believe the diamond approximation should be considered as the 'minimal standard' in the description of the non-stationary regime of Ising-like models: any new method pretending to provide a better approximate description to the dynamics of Ising-like models should perform at least as good as the diamond approximation.

cond-mat.dis-nn↗

arXiv2016

Gauge-free cluster variational method by maximal messages and moment matching

We present a new implementation of the Cluster Variational Method (CVM) as a message passing algorithm. The kind of message passing algorithms used for CVM, usually named Generalized Belief Propagation, are a generalization of the Belief Propagation algorithm in the same way that CVM is a generalization of the Bethe approximation for estimating the partition function. However, the connection between fixed points of GBP and the extremal points of the CVM free-energy is usually not a one-to-one correspondence, because of the existence of a gauge transformation involving the GBP messages. Our contribution is twofold. Firstly we propose a new way of defining messages (fields) in a generic CVM approximation, such that messages arrive on a given region from all its ancestors, and not only from its direct parents, as in the standard Parent-to-Child GBP. We call this approach maximal messages. Secondly we focus on the case of binary variables, re-interpreting the messages as fields enforcing the consistency between the moments of the local (marginal) probability distributions. We provide a precise rule to enforce all consistencies, avoiding any redundancy, that would otherwise lead to a gauge transformation on the messages. This moment matching method is gauge free, i.e. it guarantees that the resulting GBP is not gauge invariant. We apply our maximal messages and moment matching GBP to obtain an analytical expression for the critical temperature of the Ising model in general dimensions at the level of plaquette-CVM. The values obtained outperform Bethe estimates, and are comparable with loop corrected Belief Propagation equations. The method allows for a straightforward generalization to disordered systems.

cond-mat.dis-nn↗

arXiv2016

A Cavity Master Equation for the continuous time dynamics of discrete spins models

We present a new method to close the Master Equation representing the continuous time dynamics of Ising interacting spins. The method makes use of the the theory of Random Point Processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean field ferromagnet and the dynamics of the one dimensional Ising system. We then present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the Random Field Ising model, and the Viana-Bray spin-glass model.

cond-mat.dis-nn↗

arXiv2015

Random Field Ising Model in two dimensions: Bethe approximation, Cluster Variational Method and message passing algorithms

We study two free energy approximations (Bethe and plaquette-CVM) for the Random Field Ising Model in two dimensions. We compare results obtained by these two methods in single instances of the model on the square grid, showing the difficulties arising in defining a robust critical line. We also attempt average case calculations using a replica-symmetric ansatz, and compare the results with single instances. Both, Bethe and plaquette-CVM approximations present a similar panorama in the phase space, predicting long range order at low temperatures and fields. We show that plaquette-CVM is more precise, in the sense that predicts a lower critical line (the truth being no line at all). Furthermore, we give some insight on the non-trivial structure of the fixed points of different message passing algorithms.

cond-mat.dis-nn↗