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11 Universal Amplitude Ratios in the Ising Model in Three Dimensions

A. Gordillo-Guerrero, Ralph Kenna B, Juan Jesus Ruiz-Lorenzo, F́ısica De Sistemas Complejos (bifi

Published Jan 1, 2011• 7 citations
OpenAlex

Abstract

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.

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Universal amplitude ratios in the Ising model in three dimensions

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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.

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Universal amplitude ratios in the three-dimensional Ising model

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We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).

arXiv:cond-mat/0306272•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 30, 2003•53 citations
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.
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 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 investigate the three-dimensional O(2) model near the critical point by Monte Carlo simulations and calculate the major universal amplitude ratios of the model. The ratio U_0=A+/A- is determined directly from the specific heat data at zero magnetic field. The data do not, however, allow to extract an accurate estimate for alpha. Instead, we establish a strong correlation of U_0 with the value of alpha used in the fit. This numerical alpha-dependence is given by A+/A- = 1 -4.20(5) alpha + O(alpha^2). For the special alpha-values used in other calculations we find full agreement with the corresponding ratio values, e. g. that of the shuttle experiment with liquid helium. On the critical isochore we obtain the ratio xi+/xi-_T=0.293(9), and on the critical line the ratio xi_T^c/xi_L^c=1.957(10) for the amplitudes of the transverse and longitudinal correlation lengths. These two ratios are independent of the used alpha or nu-values.
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 used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).