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arXiv:1910.05916 · Physical review. B./Physical review. B

Monte Carlo study of an improved clock model in three dimensions

Martin Hasenbusch B

Published Dec 26, 2019• 107 citations
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Abstract

We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with ${Z}_{8}$ symmetry. At the transition, with increasing length scale, $O(2)$ symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to $L=512$. The field variable requires less memory and the updates are faster than for a model with $O(2)$ symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional $XY$-universality class. In particular, we get $\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)$, and $\ensuremath{\omega}=0.789(4)$. Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.

DOI
10.1103/physrevb.100.224517

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We study a generalized clock model on the simple cubic lattice. The parameter of the model can be tuned such that the amplitude of the leading correction to scaling vanishes. In the main part of the study, we simulate the model with Z8{Z}_{8}Z8​ symmetry. At the transition, with increasing length scale, O(2)O(2)O(2) symmetry emerges. We perform Monte Carlo simulations using a hybrid of local Metropolis and cluster algorithms of lattices with a linear size up to L=512L=512L=512. The field variable requires less memory and the updates are faster than for a model with O(2)O(2)O(2) symmetry at the microscopic level. Our finite-size scaling analysis yields accurate estimates for the critical exponents of the three-dimensional XYXYXY-universality class. In particular, we get \ensuremathη=0.03810(8),\ensuremathν=0.67169(7)\ensuremath{\eta}=0.03810(8),\ensuremath{\nu}=0.67169(7)\ensuremathη=0.03810(8),\ensuremathν=0.67169(7), and \ensuremathω=0.789(4)\ensuremath{\omega}=0.789(4)\ensuremathω=0.789(4). Furthermore, we obtain estimates for fixed point values of phenomenological couplings and critical temperatures.
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We report on large scale finite-temperature Monte Carlo simulations of the classical 120° or e g orbital-only model on the simple cubic lattice in three dimensions with a focus towards its critical properties. This model displays a continuous phase transition to an orbitally ordered phase. While the correlation length exponent ν≈0.665 is close to the 3D XY value, the exponent η≈0.15 differs substantially from O( N ) values. We also introduce a discrete variant of the e g model, called the e g -clock model, which is found to display the same set of exponents. Further, an emergent U(1) symmetry is found at the critical point T c , which persists for T < T c below a crossover length scaling as Λ ∼ ξ a , with an unusually small a ≈1.3.
We perform large-scale Monte Carlo simulations of the classical XY model on a three-dimensional L×L×LL\times L \times LL×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 three-dimensional XY universality class: α=−0.01293(48)α=-0.01293(48)α=−0.01293(48) and ν=0.67098(16)ν=0.67098(16)ν=0.67098(16). Our estimate for the correlation-length exponent ννν, in contrast to previous theoretical estimates, agrees with the most recent experimental estimate νexp=0.6709(1)ν_{\rm exp}=0.6709(1)νexp​=0.6709(1) at the superfluid transition of 4^44He in a microgravity environment.