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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Universality and double critical end points

J. A. Plascak, David Paul Landau

Published Jan 24, 2003• 66 citations
Publisher

Abstract

A double critical end point in the two-dimensional spin-3/2 Blume-Capel model is studied via extensive Monte Carlo simulations. The resultant scaling character of the probability distribution of the mixing scaling operators allows us to locate the double critical end point precisely and also to convincingly show that it indeed belongs to the same universality class as the critical points.

DOI
10.1103/physreve.67.015103

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A double critical end point in the two-dimensional spin-3/2 Blume-Capel model is studied via extensive Monte Carlo simulations. The resultant scaling character of the probability distribution of the mixing scaling operators allows us to locate the double critical end point precisely and also to convincingly show that it indeed belongs to the same universality class as the critical points.
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Monte Carlo simulations are used to investigate the tricritical point properties of a two-dimensional (2D) spin fluid. Measurements of the scaling operator distributions are employed in conjunction with a finite-size scaling analaysis to locate the tricritical point and determine the directions of the relevant scaling fields and their associated tricritical exponents. The scaling operator distributions and exponents are shown to match quantitatively those of the 2D Blume-Capel model, confirming that both models belong to the same universality class. Mean-field calculations of the tricritical point location are also compared with the simulation measurements. \textcopyright{} 1996 The American Physical Society.
We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to L=1600L=1600L=1600. As a check, also simulations of the spin-12\frac{1}{2}21​ Ising model are performed. We find ${f}_{\ensuremath{\sigma}\ensuremath{\sigma}\ensuremath{\epsilon}}=1.051(1)$ and ${f}_{\ensuremath{\epsilon}\ensuremath{\epsilon}\ensuremath{\epsilon}}=1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of NNN-point functions. Finite size corrections vanish with ${L}^{\ensuremath{-}{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}}$, where LLL is the linear size of the lattice and ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\epsilon}}$ is the scaling dimension of the leading Z2{Z}_{2}Z2​-even scalar $\ensuremath{\epsilon}$.
The phase diagram of the two-dimensional Blume-Capel model is investigated by using the technique of phenomenological finite-size scaling. The location of the tricritical point and the values of the critical and tricritical exponents are determined. The location of the tricritical point (Tt{T}_{t}Tt​=0.610\ifmmode\pm\else\textpm\fi{}0.005, Dt{D}_{t}Dt​=1.9655\ifmmode\pm\else\textpm\fi{}0.0010) is well outside the error bars for the value quoted in previous Monte Carlo simulations but in excellent agreement with more recent Monte Carlo renormalization-group results. The values of the critical and tricritical exponents, with the exception of the leading thermal tricritical exponent, are in excellent agreement with previous calculations, conjectured values, and Monte Carlo renormalization-group studies.
We investigate the critical properties of the spin-3/2 Blume-Capel model in two dimensions on a random lattice with quenched connectivity disorder. The disordered system is simulated by applying the cluster hybrid Monte Carlo update algorithm and re-weighting techniques. We calculate the critical temperature as well as the critical point exponents gamma/ n, b / n, a /nu, and nu. We find that, contrary of what happens to the spin-1/2 case, this random system does not belong to the same universality class as the regular two-dimensional ferromagnetic model.
We investigate the critical properties of the spin-3/2 Blume-Capel model in two dimensions on a random lattice with quenched connectivity disorder. The disordered system is simulated by applying the cluster hybrid Monte Carlo update algorithm and re-weighting techniques. We calculate the critical temperature as well as the critical point exponents gamma/ n, b / n, a /nu, and nu. We find that, contrary of what happens to the spin-1/2 case, this random system does not belong to the same universality class as the regular two-dimensional ferromagnetic model.