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Physical Review E

Finite-size scaling analysis of pseudocritical region in two-dimensional continuous-spin systems

Yusuke Tomita

Published Sep 10, 2014• 25 citations
Publisher

Abstract

At low temperatures, the two-dimensional continuous-spin systems exhibit large correlation lengths. Some of them show the Berezinskii-Kosterlitz-Thouless-like transitions, and some others show pseudocritical behaviors for which correlation lengths are extremely large but finite. To distinguish pseudo and genuine critical behaviors, it is important to understand the nature of spin-spin correlations and topological defects at low temperatures in continuous-spin systems. In this paper, I develop a finite-size scaling analysis which is suitable for distinguishing the critical behavior and its applications to the two-dimensional XY, Heisenberg, and RP(2) models.

DOI
10.1103/physreve.90.032109

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At low temperatures, the two-dimensional continuous-spin systems exhibit large correlation lengths. Some of them show the Berezinskii-Kosterlitz-Thouless-like transitions, and some others show pseudocritical behaviors for which correlation lengths are extremely large but finite. To distinguish pseudo and genuine critical behaviors, it is important to understand the nature of spin-spin correlations and topological defects at low temperatures in continuous-spin systems. In this paper, I develop a finite-size scaling analysis which is suitable for distinguishing the critical behavior and its applications to the two-dimensional XY, Heisenberg, and RP(2) models.
The classical Heisenberg model is one of the most fundamental models in statistical and condensed matter physics. Extensive theoretical and numerical studies suggest that, in two dimensions, this model does not exhibit a finite-temperature phase transition but instead manifests asymptotic freedom. However, some research has also proposed the possibility of a Berezinskii-Kosterlitz-Thouless (BKT) phase transition over the years. In this study, we revisit the classical two-dimensional (2D) Heisenberg model through large-scale simulations with linear system sizes up to L=16384L=16384L=16384. Our Monte Carlo data, without any extrapolation, clearly reveal an exponential divergence of the correlation length $\ensuremath{\xi}$ as a function of the inverse temperature $\ensuremath{\beta}$, a hallmark of asymptotic freedom. Moreover, extrapolating $\ensuremath{\xi}$ to the thermodynamic limit in the low-temperature regime achieves close agreement with the three-loop perturbative calculations. We further propose a finite-size scaling (FSS) ansatz for $\ensuremath{\xi}$, demonstrating that the pseudocritical point ${\ensuremath{\beta}}_{L}$ diverges logarithmically with LLL. The thermodynamic and finite-size scaling behaviors of the magnetic susceptibility $\ensuremath{\chi}$ are also investigated and corroborate the predictions of asymptotic freedom. Our paper provides solid evidence for asymptotic freedom in the 2D Heisenberg model and advances understanding of finite-size scaling in such systems.
A relation between the correlation entropy and the correlation functions for the general spin-1/2 systems is obtained. It is shown that the correlation entropy catches some characters of correlation behavior and can be used to quantify the quantum and finite-temperature phase transitions, including the infinite order or topological ones. As an example, the Kosterlitz–Thouless transition in the quantum two-dimensional XY model is investigated. The critical temperature and the critical exponents are determined from the finite-size scaling analysis of the correlation entropy.
The thermodynamics of the two-dimensional (2D) XY model is formulated by a transfer-matrix method and analyzed by a density-matrix renormalization group. The finite-size scaling and the beta function of the model are studied by the Roomany-Wyld renormalization-group theory. It is found that the 2D XY model has an essential finite-size effect and the Berezinskii-Kosterlitz-Thouless transition with the critical temperature TBKT=0.892{T}_{\mathrm{BKT}}=0.892TBKT​=0.892 appears in a finite system of 2000--3000 spins as a massless to massive transition with the effective critical temperature ${T}_{c}=1.07\ifmmode\pm\else\textpm\fi{}0.01.$
Research in recent years has shown that combining finite-size scaling theory with the transfer matrix technique yields a powerful tool for the investigation of critical behavior. In particular, the method has been used to study two-dimensional statistical mechanical and one-dimensional quantum mechanical systems. We review finite-size scaling theory from the general point of view of renormalization group theory for both continuous and first-order transitions (both for systems with discrete and continuous symmetries). We review applications where a comparison with exact results can be made. These include the Ising, Baxter, and q-state Potts models and the Ising model with a defect line. Various other applications such as quantum systems, the self-avoiding random walk, percolation, and Kosterlitz-Thouless transitions are briefly reviewed. The Kosterlitz-Thouless transitions and the critical fan in the antiferromagnetic 3-state Potts model are discussed at somewhat greater length.
We test an improved finite-size scaling method for reliably extracting the critical temperature T BKT of a Berezinskii–Kosterlitz–Thouless (BKT) transition. Using known single-parameter logarithmic corrections to the spin stiffness ρ s at T BKT in combination with the Kosterlitz–Nelson relation between the transition temperature and the stiffness, ρ s ( T BKT ) = 2 T BKT /π, we define a size-dependent transition temperature T BKT ( L 1 , L 2 ) based on a pair of system sizes L 1 , L 2 , e.g., L 2 = 2 L 1 . We use Monte Carlo data for the standard two-dimensional classical XY model to demonstrate that this quantity is well behaved and can be reliably extrapolated to the thermodynamic limit using the next expected logarithmic correction beyond the ones included in defining T BKT ( L 1 , L 2 ). For the Monte Carlo calculations we use GPU (graphical processing unit) computing to obtain high-precision data for L up to 512. We find that the sub-leading logarithmic corrections have significant effects on the extrapolation. Our result T BKT = 0.8935(1) is several error bars above the previously best estimates of the transition temperature, T BKT ≈ 0.8929. If only the leading log-correction is used, the result is, however, consistent with the lower value, suggesting that previous works have underestimated T BKT because of the neglect of sub-leading logarithms. Our method is easy to implement in practice and should be applicable to generic BKT transitions.
A numerical study of the neutral two-dimensional (2D) lattice Coulomb gas is performed to examine dynamic scaling, finite size effects, and the dynamic critical exponent zzz of the Kosterlitz-Thouless-Berezinskii transition. By studying large system sizes (L=100L=100L=100), we show z=2.0±0.2z=2.0 \pm0.2z=2.0±0.2 using Fisher-Fisher-Huse (FFH) scaling. We also present evidence that the vortex correlation length is finite below the transition temperature, in contrast to conventional wisdom. Finally, we conclude that previous findings for a variety of experimental systems that z≃5.6z\simeq 5.6z≃5.6 using FFH scaling indicates that finite size effects are ubiquitous in 2D superconductors and Josephson Junction Arrays.