Ising model

About: Ising model is a research topic. Over the lifetime, 25508 publications have been published within this topic receiving 555000 citations.

New numerical method to study phase transitions.

Abstract: A powerful method of detecting first order transitions by numerical simulations of finite systems is presented. The method relies on simulations and the finite size scaling properties of free energy barriers between coexisting states. It is demonstrated that the first order transitions in d = 2, q = 5 and d = q = 3 Potts models are easily seen with modest computing time. The method can also be used to obtain quite accurate estimates of critical exponents by studying the barriers in the vicinity of a critical point. Some new results on exponents and conformal charge in frustrated XY models and a related coupled XY-Ising model in d = 2 are presented. These show that the transitions in these models are in new universality classes and that the conformal charge varies with a parameter.

Density fluctuations and field mixing in the critical fluid

Abstract: The authors develop a finite-size-scaling theory describing the joint density and energy fluctuations in a near-critical fluid. As a result of the mixing of the temperature and chemical potential in the two relevant scaling fields, the energy operator features in the critical density distribution as an antisymmetric correction to the limiting scale-invariant form. Both the limiting form and the correction are predicted to be functions that are characteristic of the Ising universality class and are independently known. The theory is tested with extensive Monte Carlo studies of the two-dimensional Lennard-Jones fluid, within the grand canonical ensemble. The simulations and scaling framework together are shown to provide a powerful way of identifying the location of the liquid-gas critical point, while confirming and clarifying its essentially Ising character. The simulations also show a clearly identifiable signature of the field-mixing responsible for the failure of the law of rectilinear diameter.

25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple cubic lattice

Abstract: 25th-order high-temperature series are computed for a general nearest-neighbor three-dimensional Ising model with arbitrary potential on the simple cubic lattice. In particular, we consider three improved potentials characterized by suppressed leading scaling corrections. Critical exponents are extracted from high-temperature series specialized to improved potentials, obtaining $\ensuremath{\gamma}=1.2373(2),$ $\ensuremath{ u}=0.63012(16),$ $\ensuremath{\alpha}=0.1096(5),$ $\ensuremath{\eta}=0.03639(15),$ $\ensuremath{\beta}=0.32653(10),$ and $\ensuremath{\delta}=4.7893(8).$ Moreover, biased analyses of the 25th-order series of the standard Ising model provide the estimate $\ensuremath{\Delta}=0.52(3)$ for the exponent associated with the leading scaling corrections. By the same technique, we study the small-magnetization expansion of the Helmholtz free energy. The results are then applied to the construction of parametric representations of the critical equation of state, using a systematic approach based on a global stationarity condition. Accurate estimates of several universal amplitude ratios are also presented.

Multicritical phase diagrams of the Blume-Emery-Griffiths model with repulsive biquadratic coupling.

Abstract: Six new phase diagrams, including a novel multicritical topology and two new ordered phases, high-entropy ferrimagnetic and antiquadrupolar, are found in the spin-1 Ising model with only nearest-neighbor interactions, for negative biquadratic couplings. Thus, the global phase diagram of this simple spin system includes nine distinct topologies and three ordered phases. It is indicated that these results, obtained by mean-field theory, are applicable to three-dimensional systems.