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arXiv:hep-lat/9806012 · Physical review. B, Condensed matter

Critical exponents of the three-dimensional Ising universality class from finite-size scaling with standard and improved actions

Martin Hasenbusch B, K. Pinn, S. Vinti

Published May 1, 1999• 139 citations
arXivHTMLPDF

Abstract

We compute an improved action for the Ising universality class in three dimensions that has suppressed leading corrections to scaling. It is obtained by tuning models with two coupling constants. We studied three different models: the $\ifmmode\pm\else\textpm\fi{}1$ Ising model with nearest-neighbor and body diagonal interaction, the spin-1 model with states $0,\ifmmode\pm\else\textpm\fi{}1$, and nearest-neighbor interaction, and ${\ensuremath{\varphi}}^{4}$ theory on the lattice (Landau-Ginzburg model). The remarkable finite-size scaling properties of the suitably tuned spin-1 model are compared in detail with those of the standard Ising model. Great care is taken to estimate the systematic errors from residual corrections to scaling. Our best estimates for the critical exponents are $\ensuremath{\nu}=0.6298(5)$ and $\ensuremath{\eta}=0.0366(8)$, where the given error estimates take into account the statistical and systematic uncertainties.

DOI
10.1103/physrevb.59.11471

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We compute an improved action for the Ising universality class in three dimensions that has suppressed leading corrections to scaling. It is obtained by tuning models with two coupling constants. We studied three different models: the \ifmmode±\else\textpm\fi1\ifmmode\pm\else\textpm\fi{}1\ifmmode±\else\textpm\fi1 Ising model with nearest-neighbor and body diagonal interaction, the spin-1 model with states 0,\ifmmode±\else\textpm\fi10,\ifmmode\pm\else\textpm\fi{}10,\ifmmode±\else\textpm\fi1, and nearest-neighbor interaction, and \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 theory on the lattice (Landau-Ginzburg model). The remarkable finite-size scaling properties of the suitably tuned spin-1 model are compared in detail with those of the standard Ising model. Great care is taken to estimate the systematic errors from residual corrections to scaling. Our best estimates for the critical exponents are \ensuremathν=0.6298(5)\ensuremath{\nu}=0.6298(5)\ensuremathν=0.6298(5) and \ensuremathη=0.0366(8)\ensuremath{\eta}=0.0366(8)\ensuremathη=0.0366(8), where the given error estimates take into account the statistical and systematic uncertainties.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin- 1 2 Ising model with nearest-neighbor interactions, a spin- 1 2 model with nearest-neighbor and third-neighbor interactions, and a spin-1 model with nearest-neighbor interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behavior reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbor interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), yh =2.4815 (15) and y i = -0.82 (6). The universal ratio Q = #m 2 # 2 /#m 4...
We investigate several three-dimensional lattice models believed to be in the Ising universality class by means of Monte Carlo methods and finite-size scaling. These models include spin-1 / 2 models with nearest-neighbor interactions on the simple-cubic and on the diamond lattice. For the simple cubic lattice, we also include models with third-neighbor interactions of varying strength, and some "equivalent-neighbor" models. Also included are a spin-1 model and a hard-core lattice gas. Separate analyses of the numerical data confirm the Ising-like critical behavior of these systems. On this basis, we analyze all these data simultaneously such that the universal parameters occur only once. This leads to an improved accuracy. The thermal, magnetic, and irrelevant exponents are determined as y(t)=1.5868(3), y(h)=2.4816(1), and y(i)=-0.821(5), respectively. The Binder ratio is estimated as Q=<m(2)>(2)/<m(4)>=0.62 341(4).
We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter DDD on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter DDD, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for D=0.641D=0.641D=0.641 and 0.655 on lattices of a linear size up to L=360L=360L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.
We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).
We study a generalized Blume-Capel model on the simple cubic lattice. In addition to the nearest-neighbor coupling there is a next-to-next-to-nearest-neighbor coupling. In order to quantify spatial anisotropy, we determine the correlation length in the high-temperature phase of the model for three different spatial directions. It turns out that the spatial anisotropy depends very little on the dilution or crystal-field parameter DDD of the model and is essentially determined by the ratio of the nearest-neighbor and the next-to-next-to-nearest-neighbor coupling. This ratio is tuned such that the leading contribution to the spatial anisotropy is eliminated. Next we perform a finite-size scaling (FSS) study to tune DDD such that also the leading correction to scaling is eliminated. Based on this FSS study, we determine the critical exponents $\ensuremath{\nu}=0.629\phantom{\rule{0.16em}{0ex}}98(5)$ and $\ensuremath{\eta}=0.0362\phantom{\rule{0.16em}{0ex}}84(40)$, which are in nice agreement with the more accurate results obtained by using the conformal bootstrap method. Furthermore, we provide accurate results for fixed-point values of dimensionless quantities such as the Binder cumulant and for the critical couplings. These results provide the groundwork for broader studies of universal properties of the three-dimensional Ising universality class.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).