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arXiv:0810.2716 · Journal of Statistical Mechanics Theory and Experiment

A Monte Carlo study of the three-dimensionalXYuniversality class: universal amplitude ratios

Martin Hasenbusch

Published Dec 12, 2008• 23 citations
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Abstract

We simulate lattice models in the three-dimensional XY universality class in the low and the high temperature phase. In addition to the XY model, we study the three-dimensional two-component ϕ 4 model on the simple cubic lattice. The parameter of the ϕ 4 model is chosen such that leading corrections to scaling are small. This allows us to compute a number of universal amplitude ratios with unprecedented precision: R ϒ = 0.411(2), R B = 2.83(1), R ξ + = 0.3562(10) and R ξ − = 0.850(5). These results can be compared with those obtained from other theoretical methods, such as field theoretic methods or the high temperature series expansion and also with experimental results for the λ-transition of 4 He.

DOI
10.1088/1742-5468/2008/12/p12006

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We simulate lattice models in the three-dimensional XY universality class in the low and the high temperature phase. In addition to the XY model, we study the three-dimensional two-component ϕ 4 model on the simple cubic lattice. The parameter of the ϕ 4 model is chosen such that leading corrections to scaling are small. This allows us to compute a number of universal amplitude ratios with unprecedented precision: R ϒ = 0.411(2), R B = 2.83(1), R ξ + = 0.3562(10) and R ξ − = 0.850(5). These results can be compared with those obtained from other theoretical methods, such as field theoretic methods or the high temperature series expansion and also with experimental results for the λ-transition of 4 He.
We simulate the improved three-dimensional two-component ϕ 4 model on the simple cubic lattice in the low and the high temperature phase for reduced temperatures down to | T − T c |/ T c ≈ 0.0017 on lattices of size up to 350 3 . Our new results for the internal energy and the specific heat are combined with the accurate estimates of β c and data for the internal energy and the specific heat at β c recently obtained in Campostrini et al (2006 Preprint cond-mat/0605083 ). We find R α = (1 − A + / A − )/α = 4.01(5), where α is the critical exponent of the specific heat and A ± is the amplitude of the specific heat in the high and the low temperature phase, respectively.
We compute a number of universal amplitude ratios in the three-dimensional Ising universality class. To this end, we perform Monte Carlo simulations of the improved Blume-Capel model on the simple cubic lattice. For example, we obtain ${A}_{+}/{A}_{\ensuremath{-}}=0.536(2)$ and ${C}_{+}/{C}_{\ensuremath{-}}=4.713(7)$, where ${A}_{\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{\ifmmode\pm\else\textpm\fi{}}$ are the amplitudes of the specific heat and the magnetic susceptibility, respectively. The subscripts +++ and $\ensuremath{-}$ indicate the high- and the low-temperature phase, respectively. We compare our results with those obtained from previous Monte Carlo simulations, high- and low-temperature series expansions, field theoretic methods, and experiments.
We improve the theoretical estimates of the critical exponents for the three-dimensional XY\mathrm{XY}XY universality class. We find $\ensuremath{\alpha}=\ensuremath{-}0.0146(8),$ $\ensuremath{\gamma}=1.3177(5),$ $\ensuremath{\nu}=0.67155(27),$ $\ensuremath{\eta}=0.0380(4),$ $\ensuremath{\beta}=0.3485(2),$ and $\ensuremath{\delta}=4.780(2).$ We observe a discrepancy with the most recent experimental estimate of $\ensuremath{\alpha};$ this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.
We present a high precision Monte Carlo study of various universal amplitude ratios of the three-dimensional Ising spin model. Using state of the art simulation techniques we study the model close to criticality in both phases. Great care is taken to control systematic errors due to finite size effects and correction to scaling terms. We obtain , and . Our results are compatible with those obtained by field theoretic methods applied to the theory and high- and low-temperature series expansions of the Ising model. The mismatch with a previous Monte Carlo study by Ruge et al remains to be understood.
We improve the theoretical estimates of the critical exponents for the three-dimensional XY universality class. We fine alpha=-0.0146(8), gamma =1.3177(5), nu =0.671 55(27), eta =0.0380(4), beta =0.3485(2), and delta =4.780(2). We observe a discrepancy with the most recent experimental estimate of alpha; this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.
We present a Monte Carlo study of the two-component 4 model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.