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Journal of Physics F Metal Physics

Monte Carlo evidence for the absence of a phase transition in the two-dimensional Ising spin glass

Alan J. Bray, M A Moore

Published Dec 1, 1977• 53 citations
Publisher

Abstract

Monte Carlo calculations have been performed for an Ising system on a square lattice with nearest-neighbour interactions distributed according to a Gaussian with zero mean. The susceptibility is computed from the time averages of the fluctuating magnetisation and its square, and exhibits the characteristic maximum as a function of temperature. The height and position of this peak are found to depend on the 'observation time' used for the computation of the time averages. This is interpreted as evidence for the absence of a phase transition.

DOI
10.1088/0305-4608/7/12/004

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Monte Carlo calculations have been performed for an Ising system on a square lattice with nearest-neighbour interactions distributed according to a Gaussian with zero mean. The susceptibility is computed from the time averages of the fluctuating magnetisation and its square, and exhibits the characteristic maximum as a function of temperature. The height and position of this peak are found to depend on the 'observation time' used for the computation of the time averages. This is interpreted as evidence for the absence of a phase transition.
Many problems in statistical physics resist exact solution once the number of interacting particles becomes large, and the two-dimensional Ising model is a classic case where numerical simulation fills that gap. This paper reports a Monte Carlo study of the two-dimensional Ising model using the Metropolis algorithm, undertaken to examine how magnetization, internal energy, specific heat, and magnetic susceptibility change with temperature. Spin configurations were generated on a square lattice under periodic boundary conditions, with each proposed spin flip accepted or rejected according to the Boltzmann weight of the resulting energy change. The simulation reproduces the expected transition from an ordered ferromagnetic state at low temperature to a disordered paramagnetic state at high temperature, with magnetization falling sharply and both specific heat and susceptibility peaking near a critical temperature of Tc ≈ 2.3–2.4 (in units of J/kB), consistent with the exact value obtained analytically for this system. These results confirm that the Metropolis algorithm, despite its simplicity, remains a dependable tool for locating second-order phase transitions and extracting thermodynamic quantities in lattice spin systems, and the same computational framework can be adapted to more elaborate models used in condensed matter and materials research.
Monte Carlo simulations are performed on the square lattice Ising system with both the nearest neighbor and the two-dimensional dipolar interactions in order to investigate the critical property of ferromagnetic phase transitions in this 2-dimensional system. Applying the finite size scaling theory to Monte Carlo data just near the transition point, the critical indices of this dipolar system are calculated and α=0.31±0.04, β =0.20±0.02, γ=1.08±0.20 and ν=0.72±0.10 are obtained. However, in the temperature region considerably far from the transition point, different values of α, β and γ seem to be adequate, which are rather close to those of the rigorous result of the 2-dimensional Ising model.
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.