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arXiv:1709.00693 · Physical Review A

Gutzwiller Monte Carlo approach for a critical dissipative spin model

Wim Casteels, Ryan Wilson, Michiel Wouters

Published Jun 7, 2018• 28 citations
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Abstract

We use the Gutzwiller Monte Carlo approach to simulate the dissipative $XYZ$ model in the vicinity of a dissipative phase transition. This approach captures classical spatial correlations together with the full on-site quantum behavior while neglecting nonlocal quantum effects. By considering finite two-dimensional lattices of various sizes, we identify a ferromagnetic and two paramagnetic phases, in agreement with earlier studies. The greatly reduced numerical complexity of the Gutzwiller Monte Carlo approach facilitates efficient simulation of relatively large lattice sizes. The inclusion of the spatial correlations allows to capture parts of the phase diagram that are completely missed by the widely applied Gutzwiller decoupling of the density matrix.

DOI
10.1103/physreva.97.062107

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We use the Gutzwiller Monte Carlo approach to simulate the dissipative XYZXYZXYZ model in the vicinity of a dissipative phase transition. This approach captures classical spatial correlations together with the full on-site quantum behavior while neglecting nonlocal quantum effects. By considering finite two-dimensional lattices of various sizes, we identify a ferromagnetic and two paramagnetic phases, in agreement with earlier studies. The greatly reduced numerical complexity of the Gutzwiller Monte Carlo approach facilitates efficient simulation of relatively large lattice sizes. The inclusion of the spatial correlations allows to capture parts of the phase diagram that are completely missed by the widely applied Gutzwiller decoupling of the density matrix.
We study the influence of short-range quantum correlations and classical spatial correlations on the phase diagram of the dissipative XYZ model by using a Gutzwiller Monte Carlo method and a cluster Gutzwiller ansatz for the wave function. Considering lattices of finite size we confirm the emergence of a region with ferromagnetic correlations and two paramagnetic regions and show the emergence of a region with a buildup of ferromagnetic correlations completely missed by mean-field theory. The inclusion of short-range quantum correlations causes drastic alterations of the mean-field correlation functions but our results show that the inclusion of long-range quantum correlations or the use of more sophisticated methods is needed to quantitatively match the exact results. A study of the susceptibility tensor shows that reciprocity is broken, a feature not observed in closed quantum systems. Furthermore, increasing the magnetic field suppresses the magnetization; this is also in contrast with closed quantum systems.
We use quantum Monte Carlo simulations to calculate the phase diagram and the correlation functions for the quantum phase transitions in the two-dimensional dissipative quantum XY model with and without fourfold anisotropy. Without anisotropy, the model describes the superconductor-to-insulator transition in two-dimensional dirty superconductors. With anisotropy, the model represents the loop-current order observed in the underdoped cuprates and its fluctuations, as well as the fluctuations near the ordering vector in simple models of two-dimensional itinerant ferromagnets and itinerant antiferromagnets. These calculations test an analytic solution of the model which reexpressed it in terms of topological excitations: the vortices with interactions only in space but none in time, and warps with leading interactions only in time but none in space, as well as subleading interactions which are both space and time dependent. For parameters where the proliferation of warps dominates the phase transition, the critical fluctuations as functions of the deviation of the dissipation parameter $\ensuremath{\alpha}$ on the disordered side from its critical value ${\ensuremath{\alpha}}_{c}$ are scale invariant in imaginary time $\ensuremath{\tau}$ as the correlation length in time ${\ensuremath{\xi}}_{\ensuremath{\tau}}={\ensuremath{\tau}}_{c}{e}^{{[{\ensuremath{\alpha}}_{c}/({\ensuremath{\alpha}}_{c}\ensuremath{-}\ensuremath{\alpha})]}^{1/2}}$ diverges, where ${\ensuremath{\tau}}_{c}$ is a short-time cutoff. On the other hand, the spatial correlations develop with a correlation length ${\ensuremath{\xi}}_{x}\ensuremath{\approx}{\ensuremath{\xi}}_{0}ln({\ensuremath{\xi}}_{\ensuremath{\tau}})$, with ${\ensuremath{\xi}}_{0}$ of the order of a lattice constant. The dynamic correlation exponent zzz is therefore $\ensuremath{\infty}$. The Monte Carlo calculations also directly show warps and vortices. Their densities and correlations across the various transitions in the model are calculated and related to those of the order-parameter correlations in the dissipative quantum XY model.
This study presents a comprehensive Monte Carlo simulation of the two-dimensional (2-D) Ising model using the Metropolis algorithm to investigate critical phenomena and thermodynamic behavior in spin-lattice systems. The model, implemented in MATLAB with periodic boundary conditions, explores equilibrium properties such as magnetization, internal energy, specific heat, and susceptibility across a range of temperatures. By employing various initial spin configurations—ordered and random—the simulations demonstrate the system's ergodicity and convergence to thermal equilibrium. Key results include a sharp decline in magnetization and a pronounced peak in specific heat near the critical temperature, consistent with second-order phase transition behavior. The simulation captures microscopic domain evolution, highlighting the transition from ferromagnetic to paramagnetic phases as thermal fluctuations increase. The study further evaluates algorithmic efficiency, discusses the impact of lattice size on statistical accuracy, and proposes improvements using advanced cluster algorithms and parallel computing frameworks. The findings validate theoretical predictions from Onsager's solution and underscore the versatility of Monte Carlo techniques in modeling collective behavior in magnetic systems. The simulation framework offers a robust foundation for analyzing critical dynamics and extends its relevance to broader applications in material science, computational physics, and complex systems modeling.
Abstract We study the critical properties of finite-dimensional dissipative quantum spin systems with uniform ferromagnetic interactions. Starting from the transverse field Ising model coupled to a bath of harmonic oscillators with Ohmic spectral density, we generalize its classical representation to classical spin systems with O ( n ) symmetry and then take the large- n limit to reduce the system to a spherical model. The exact solution to the resulting spherical model with long-range interactions along the imaginary time axis shows a phase transition with static critical exponents coinciding with those of the conventional short-range spherical model in d + 2 dimensions, where d is the spatial dimensionality of the original quantum system. This implies that the dynamical exponent is z = 2. These conclusions are consistent with the results of Monte Carlo simulations and renormalization group calculations for dissipative transverse field Ising and O ( n ) models in one and two dimensions. The present approach therefore serves as a useful tool for analytically investigating the properties of quantum phase transitions of the dissipative transverse field Ising and other related models. Our method may also offer a platform to study more complex phase transitions in dissipative finite-dimensional quantum spin systems, which have recently received renewed interest in the context of quantum annealing in a noisy environment.
Many-body open quantum systems are exposed to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. As the exact treatment of such models is generally unfeasible, it is favourable to formulate an approximate description by means of the dynamics of the reduced density operator of the system. When studying models with a weak coupling to a memoryless environment, the existence of such solution is granted and the reduced dynamics is governed by the Lindblad quantum master equation. In recent years, open quantum systems have evolved into a major field of studies. Focus of these studies are the characterization of emergent phenomena and dissipative phase transitions, as well as the ongoing debate about whether quantum computing schemes are still hard to simulate classically --- and thus achieve quantum supremacy --- when in presence of some degree of noise-induced decoherence. However, the challenge posed by this task lies in the complexity of the density matrix that increases exponentially with the system size, and the quest for efficient numerical methods is a research field that is still in its infancy. Here, we first develop a real-time full configuration interaction quantum Monte Carlo technique that stems from a class of methods generally known as projector Monte Carlo. The approach enables the stochastic sampling of the Lindblad time evolution of the density matrix thanks to a massively parallel algorithm, thus providing estimates of observables on the non-equilibrium steady state. We present the underlying theory and introduce an initiator technique and importance sampling to reduce statistical error. We demonstrate the efficiency of the approach by applying it to the dissipative two-dimensional XYZ spin-1/2 model on a lattice. As the importance sampling of projector approaches is often combined with variational results, we then introduce a novel method that is based on the variational Monte Carlo methods and on a neural network representation of the density matrix. Thanks to the stochastic reconfiguration scheme, the application of the variational principle is translated into the actual integration of the Lindblad quantum master equation. We test the effectiveness of the method by modeling the steady state of the dissipative two-dimensional XYZ spin-1/2 model through a dissipative phase transition, and also the real-time dynamics of the dissipative Ising model. In addition, we discuss the application of the developed methods and the open questions of the field.
Earlier Monte Carlo calculations on the dissipative two-dimensional XY model are extended in several directions. We study the phase diagram and the correlation functions when dissipation is very small, where it has properties of the classical 3D-XY transition, i.e., one with a dynamical critical exponent z=1z=1z=1. The transition changes from z=1z=1z=1 to the class of criticality with $z\ensuremath{\rightarrow}\ensuremath{\infty}$ driven by topological defects, discovered earlier, beyond a critical dissipation. We also find that the critical correlations have power-law singularities as a function of tuning the ratio of the kinetic energy to the potential energy for fixed large dissipation, as opposed to essential singularities on tuning dissipation keeping the former fixed. A phase with temporal disorder but spatial order of the Kosterlitz-Thouless form is also further investigated. We also present results for the transition when the allowed Caldeira-Leggett form of dissipation and the allowed form of dissipation coupling to the compact rotor variables are both included. The nature of the transition is then determined by the Caldeira-Leggett form.