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arXiv:2010.14427 · Physical review. E

Percolation of the two-dimensional XY model in the flow representation

Bao‐Zong Wang, Pengcheng Hou, Chun-Jiong Huang, Youjin Deng

Published Jun 21, 2021• 8 citations
arXivHTMLPDF

Abstract

We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K_{perc} from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K_{perc}, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K_{perc}=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K_{BKT}=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.

DOI
10.1103/physreve.103.062131

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We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K_{perc} from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K_{perc}, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K_{perc}=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K_{BKT}=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.
We study a percolation problem on a substrate formed by two-dimensional XY spin configurations using Monte Carlo methods. For a given spin configuration, we construct percolation clusters by randomly choosing a direction x in the spin vector space, and then placing a percolation bond between nearest-neighbor sites i and j with probability p(ij)=max(0,1-e(-2Ks(i)(x)s(j)(x))), where K>0 governs the percolation process. A line of percolation thresholds K(c)(J) is found in the low-temperature range J≥J(c), where J>0 is the XY coupling strength. Analysis of the correlation function g(p)(r), defined as the probability that two sites separated by a distance r belong to the same percolation cluster, yields algebraic decay for K≥K(c)(J), and the associated critical exponent depends on J and K. Along the threshold line K(c)(J), the scaling dimension for g(p) is, within numerical uncertainties, equal to 1/8. On this basis, we conjecture that the percolation transition along the K(c)(J) line is of the Berezinskii-Kosterlitz-Thouless type.
Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.
We define a percolation problem on the basis of spin configurations of the two-dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the existence of percolation transitions when the conducting angle is varied, and we determine the transition point for several values of the XY coupling. It appears that the critical behavior of this percolation model can be well described by the standard percolation theory. The critical exponents of the percolation transitions, as determined by finite-size scaling, agree with the universality class of the two-dimensional percolation model on a uniform substrate. This holds over the whole temperature range, even in the low-temperature phase where the XY substrate is critical in the sense that it displays algebraic decay of correlations.
The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of $\ensuremath{\rho}$, the power of the interaction strength decay. For large enough $\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}$, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\ensuremath{\rho}}_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ${|r|}^{\ensuremath{-}\ensuremath{\rho}}$, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that ${\ensuremath{\rho}}_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for $\ensuremath{\rho}\ensuremath{\ge}4$, and to present a second-order transition for $\ensuremath{\rho}<4$.
The disorder-induced quantum phase transition between superfluid and nonsuperfluid states of bosonic particles in one dimension is generally expected to be of the Berezinskii-Kosterlitz-Thouless (BKT) type. Here, we show that hard-core lattice bosons with power-law hopping decaying with distance as $1/{r}^{\ensuremath{\alpha}}$ with a finite integral over space---corresponding in spin language to an XYXYXY model with power-law couplings---undergo a non-BKT continuous phase transition instead. We use exact quantum Monte Carlo methods to determine the phase diagram for different values of the exponent $\ensuremath{\alpha}$, focusing on the regime $\ensuremath{\alpha}>2$. We find that the scaling of the superfluid stiffness with the system size is scale invariant at the transition point for any $\ensuremath{\alpha}\ensuremath{\le}3$---a behavior incompatible with the BKT scenario and typical of continuous phase transitions in higher dimension. By a scaling analysis near the transition point, we find that our data are consistent with a correlation length exponent satisfying the Harris bound $\ensuremath{\nu}\ensuremath{\ge}2$ and demonstrate a universal behavior of disordered bosons in one dimension. For $\ensuremath{\alpha}>3$ our data are consistent with a BKT scenario where the liquid is pinned by infinitesimal disorder.
Abstract In a two-dimensional (2D) spin system, the XY model, characterized by planar rotational symmetry, exhibits a unique phenomenon known as the Berezinskii-Kosterlitz-Thouless (BKT) transition. In contrast, the clock model, which introduces discrete rotational symmetry, exhibits the BKT transition at two different temperatures due to this discreteness. In this study, we numerically investigate the BKT transition for XY and six-state clock models over various two-dimensional lattices. We employ two primary methods: the Monte Carlo method, which analyzes the size dependence of the ratio of the correlation functions for two different distances, and a machine-learning approach to classify the different phases --- namely, the low-temperature ordered phase, the intermediate BKT phase, and the high-temperature disordered phase. We identify the BKT transition temperatures for the XY and six-state models on honeycomb, kagome, and diced lattices. Combined with the previously calculated data for the triangular lattice, we then compare these values with the second-order phase transition temperatures of the 2D Ising model, for which exact solutions are known. Our results indicate that the ratio of the BKT transition temperatures for each lattice relative to the Ising model transition temperatures are close, although the values are not universal.