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Physical review. B, Condensed matter

Surface free energies of three-dimensional Ising models and universality of finite-size scaling amplitudes by direct Monte Carlo sampling

K. K. Mon, M. P. Nightingale

Published May 1, 1985• 16 citations
Publisher

Abstract

Direct Monte Carlo sampling is employed to obtain estimates of excess surface free energies of three-dimensional Ising models at criticality. Results for simple and body-centered-cubic lattices provide strong evidence for the universality of finite-size scaling amplitudes and in particular the related interaction per unit area, mediated via the incipient long-ranged correlations, of two free surfaces a finite distance apart.

DOI
10.1103/physrevb.31.6137

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Direct Monte Carlo sampling is employed to obtain estimates of excess surface free energies of three-dimensional Ising models at criticality. Results for simple and body-centered-cubic lattices provide strong evidence for the universality of finite-size scaling amplitudes and in particular the related interaction per unit area, mediated via the incipient long-ranged correlations, of two free surfaces a finite distance apart.
We present an extensive Monte Carlo study of the three-dimensional Ising model with free surfaces, edges, and corners at bulk criticality. Finite-size scaling properties of the excess free energies of free surfaces, edges, and corners are estimated by use of the recently proposed applications of the parametric integration method. The corner excess free energies are obtained for the first time and are consistent with the logarithmic finite-size scaling predictions. Results of the simplecubic and body-centered-cubic lattices provide support for the proposed universality of the finite-size scaling amplitudes for the excess free energies of free surfaces, edges, and corners.
This paper addresses the limitation for routine applications of the proposed Monte Carlo method of direct calculation of absolute free energy by sampling of finite-size dependences and discusses new extensions. The need for multistage sampling is removed and the method is now simpler to apply in practice. To exhibit this, the critical free-energy density for the simple-cubic nearest-neighbor ferromagnetic Ising model is studied for a wide range of lattice sizes, up to 32\ifmmode\times\else\texttimes\fi{}32\ifmmode\times\else\texttimes\fi{}32. The finite-size scaling is demonstrated and the scaling amplitude calculated with higher accuracy than before. These results indicate the possibility that very accurate free-energy density and related quantities such as surface free energy, can be routinely obtained for lattice models from Monte Carlo simulations, even at criticality.
It is shown that the absolute free energy of lattice spin systems can be calculated directly by a novel application of Monte Carlo sampling of finite-size dependence. Results are obtained for the two- and three-dimensional Ising models at Tc{T}_{c}Tc​ and are consistent with recent finite-size scaling-theory predictions. The free energy is in excellent agreement with Onsager's exact solution for two dimensions and with series expansions for three dimensions. The U0{U}_{0}U0​ is measured for the first time. Results are consistent with the proposed universality.
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).