High-precision Monte Carlo study of the three-dimensional XY model on GPU

Abstract

We perform large-scale Monte Carlo simulations of the classical XY model on a three-dimensional L × L × L cubic lattice using the graphics processing unit (GPU). By the combination of Metropolis single-spin flip, over-relaxation and parallel-tempering methods, we simulate systems up to L = 160. Performing the finite-size scaling analysis, we obtain estimates of the critical exponents for the threedimensional XY universality class: α = −0.01293( 48) and ν = 0.67098(16). Our estimate for the correlation-length exponent ν, in contrast to previous theoretical estimates, agrees with the most recent experimental estimate νexp = 0.6709(1) at the superfluid transition of 4 He in a microgravity environment.

Solving quantum rotor model with different Monte Carlo techniques

“…From Fig. 3 (c) one can read the z = 2.05 (16) for LM, z = 2.05 (8) for OR, and z = 3.60 (5) for HM, and as for the other two update schemes, shorter autocorrelation times are observed, for example, z = 0.84(2) for WC, and z = 1.62 (30) for FA. So these results reveal that at the QCP of (2 + 1)D quantum rotor model, the critical slowing-down is mostly suppressed in WC and FA schemes.…”

“…The quantum rotor model in Eqs. ( 2),( 12) and ( 15) can be solved with various Monte Carlo simulation schemes, includ-ing the local Metropolis (LM) [27], LM plus over-relaxation (OR) [5,18], Wolff-cluster (WC) [17], hybrid Monte Carlo (HM) and hybrid Monte Carlo supplemented with Fourier acceleration (FA) [28]. In this section, we will elucidate the basic steps in these schemes with the detailed explanation in HM [29][30][31] and FA [32] schemes as they are more used in the high-energy community and less so to condensed matter.…”

“…Over-relaxation method [18] is an improvement of the local update. It was introduced in Monte Carlo evaluation of the partition function for multiquadratic actions [34] and have been used in quantum rotor model [5]. For each site, the unit vector θ r acquires a phase between 0 to 2π.…”

“…We thank Ying-Jer Kao for insightful discussion and introduction to us the over-relaxation update scheme in Ref. [5]. WLJ, GPP and ZYM acknowledge the supports from the Ministry of Science and Technology of China through the National Key Research and Development Program (Grant No.…”

“…The study of the critical behavior in XY model dates back to the early stage of renormalization group [1]. To date, very accurate analytical and numerical calculation at the (2+1)D O(2) Wilson-Fisher quantum critical point exist with high precision of exponents determined [2][3][4][5][6][7][8][9], and the rich physics of such transition related with the superconductor-insulator [10,11], superfluid-insulator [12,13] and easy-plane quantum magnetic [14] transitions have been well acknowledged by the community. Moreover, the presence of Kosterlitz-Thouless (KT) transition at finite temperature also illustrates the nontrivial topological character of the setting and the associated vortex excitations are appearing in various material realizations [15].…”