arXiv:cond-mat/0501737 · Physical Review E

Langevin simulation of the chirally decomposed sine-Gordon model

L. Moriconi, Marco Moriconi

Published Jul 21, 2005• 1 citation

Abstract

A large class of quantum and statistical field theoretical models, encompassing relevant condensed matter and non-Abelian gauge systems, are defined in terms of complex actions. As the ordinary Monte Carlo methods are useless in dealing with these models, alternative computational strategies have been proposed along the years. The Langevin technique, in particular, is known to be frequently plagued with difficulties such as strong numerical instabilities or subtle ergodic behavior. Regarding the chirally decomposed version of the sine-Gordon model as a prototypical case for the failure of the Langevin approach, we devise a truncation prescription in the stochastic differential equations which yields numerical stability and is assumed not to spoil the Berezinskii-Kosterlitz-Thouless transition. This conjecture is supported by a finite size scaling analysis, whereby a massive phase ending at a line of critical points is clearly observed for the truncated stochastic model.

DOI: 10.1103/physreve.72.016125

Related papers

Berezinskii-Kosterlitz-Thouless phase transition from lattice sine-Gordon model

Joel Giedt, James Flamino

\nWe obtain nonperturbative results on the sine-Gordon model using the lattice field technique. In particular, we employ the Fourier accelerated hybrid Monte Carlo algorithm for our studies. We find the critical temperature of the theory based autocorrelation time, as well as the finite size scaling of the “thickness” observable used in an earlier lattice study by Hasenbusch et al.\n

Mar 26, 20181 citation

Simulating the Berezinskii-Kosterlitz-Thouless transition with the complex Langevin algorithm

Philipp Heinen, Thomas Gasenzer

Numerical simulations of the full quantum properties of interacting many-body systems by means of field-theoretic Monte Carlo techniques are often limited due to a sign problem. Here we simulate properties of a dilute two-dimensional Bose gas in the vicinity of the Berezinskii-Kosterlitz-Thouless (BKT) transition by means of the complex Langevin (CL) algorithm, thereby extending our previous CL study of the three-dimensional Bose gas to the lower-dimensional case. The purpose of the paper is twofold. On the one hand, it adds to benchmarking of the CL method and thus contributes to further exploring the range of applicability of the method. With the respective results, the universality of the equation of state is recovered, as well as the long-wave-length power-law dependence of the single-particle momentum spectrum below the BKT transition. Analysis of the rotational part of the current density corroborates vortex unbinding in crossing the transition. Beyond these measures of consistency we compute quantum corrections to the critical density and chemical potential in the weakly coupled regime. Our results show a shift of these quantities to lower values as compared to those obtained from classical field theory. It points in the opposite direction as compared to the shift of the critical density found by means of the path-integral Monte Carlo method at larger values of the coupling. Our simulations widen the perspective for precision comparisons with experiment.

arXiv:2304.05699Physical Review ANov 13, 20232 citations

Correlation functions of the 2D sine-Gordon model

K. Nomura

A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.

arXiv:cond-mat/9504074Journal of Physics A Mathematical and GeneralOct 7, 1995104 citations

Critical behavior of the quantum double-sine-Gordon model

S. De Martino, Silvio De Siena, Pasquale Sodano

We investigate the critical behavior of systems modeled by the quantum double-sine-Gordon theory. We find the existence of a Kosterlitz-Thouless transition in a region of the phase space. For different ranges of the physical parameters, the model is shown to be equivalent to a free massive Dirac field.

Physical review. B, Condensed matterSep 1, 19857 citations

Sine-Gordon model at finite temperature: The method of random surfaces

M. Tóth, J. H. Pixley, D. Szász-Schagrin +2 more

The authors study here the sine-Gordon quantum field theory, relevant for many 1D gapped systems and experimental settings. Despite its integrability, computing correlation functions, especially at finite temperatures, remains challenging. Existing methods like perturbative expansions, Monte Carlo simulations, and Hamiltonian truncation have limitations. The authors develop here a numerical approach using the method of random surfaces, which is efficient, scalable, and free from the sign problem. They validate their results against established techniques and propose future applications for multipoint functions and quench dynamics.

arXiv:2408.08828Physical review. B./Physical review. BApr 7, 20252 citations

Complex Langevin approach to interacting Bose gases

Philipp Heinen, Thomas Gasenzer

Quantitative numerical analyses of interacting dilute Bose-Einstein condensates are most often based on semiclassical approximations. Since the complex-valued field-theoretic action of the Bose gas does not offer itself to the direct application of standard Monte Carlo techniques, simulations beyond their scope by now almost exclusively rely on quantum-mechanical techniques. Here we explore an alternative approach based on a Langevin-type sampling in an extended state space, known as the complex Langevin (CL) algorithm. While the use of the CL technique has a long-standing history in high-energy physics, in particular in the simulation of QCD at finite baryon density, applications to ultracold atoms are still in their infancy. Here we examine the applicability of the CL approach for a one- and two-component, three-dimensional nonrelativistic Bose gas in thermal equilibrium, below and above the Bose-Einstein phase transition. By comparison with analytic descriptions at the Gaussian level, including Bogoliubov and Hartree-Fock theory, we find that the method allows computing reliably and efficiently observables in the regime of experimentally accessible parameters. Close to the transition, quantum corrections lead to a shift of the critical temperature which we reproduce within the numerical range known in the literature. With this work, we aim to provide a first test of CL as a potential out-of-the-box tool for the simulation of experimentally realistic situations, including trapping geometries and multicomponent (or multiple species) models.

arXiv:2204.10661Physical Review ADec 14, 202210 citations