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arXiv:cond-mat/0607189 · Journal of Statistical Mechanics Theory and Experiment

The three-dimensionalXYuniversality class: a high precision Monte Carlo estimate of the universal amplitude ratioA+/A−

Martin Hasenbusch B

Published Aug 25, 2006• 12 citations
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Abstract

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.

DOI
10.1088/1742-5468/2006/08/p08019

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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 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 use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A + / A − in the three-dimensional Ising model via the impact angle ϕ of complex temperature zeros. We also measure the correlation-length critical exponent ν from finite-size scaling and the specific-heat exponent α through hyperscaling. Extrapolations to the thermodynamic limit yield ϕ = 59.2(1.0)°, A + / A − = 0.56(3), ν = 0.630 48(32) and α = 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.
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 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 use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A + /A − in the three-dimensional Ising model via the impact angle phgr of complex temperature zeros. We also measure the correlation-length critical exponent ν from finite-size scaling and the specific-heat exponent α through hyperscaling. Extrapolations to the thermodynamic limit yield phgr = 59.2(1.0)°, A + /A − = 0.56(3), ν = 0.630 48(32) and α = 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.
Monte Carlo simulations and series expansions data for the energy, specific heat, magnetization, and susceptibility of the three-state Potts model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We estimate these amplitudes using the correction-to-scaling exponents predicted by conformal field theory. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\ensuremath{\leqslant}4$, the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low- and high-temperature phases. We present numerical and analytical support for the form of the first two correction-to-scaling terms. Our results for the amplitude ratios closely agree with the theoretical predictions and the earlier numerical estimates of the specific heat and the susceptibility amplitude ratios.