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arXiv:1312.6416 · Nuclear Physics B

A geometric Monte Carlo algorithm for the antiferromagnetic Ising model with “topological” term at θ=π

V. Azcoiti, Gennaro Cortese, E. Follana, Matteo Giordano

Published Apr 12, 2014• 1 citation
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Abstract

In this work we study the two and three-dimensional antiferromagnetic Ising model with an imaginary magnetic field iθ at θ=π. In order to perform numerical simulations of the system we introduce a new geometric algorithm not affected by the sign problem. Our results for the 2D model are in agreement with the analytical solution. We also present new results for the 3D model which are qualitatively in agreement with mean-field predictions.

DOI
10.1016/j.nuclphysb.2014.04.004

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In this work we study the two and three-dimensional antiferromagnetic Ising model with an imaginary magnetic field iθ at θ=π. In order to perform numerical simulations of the system we introduce a new geometric algorithm not affected by the sign problem. Our results for the 2D model are in agreement with the analytical solution. We also present new results for the 3D model which are qualitatively in agreement with mean-field predictions.
2D2D2D and 3D3D3D Antiferromagnetic Ising Model with topological term at θ=πθ=πθ=π
We study the two and three-dimensional Antiferromagnetic Ising Model with an imaginary magnetic field iθiθiθ at θ=πθ= πθ=π. We use a new geometric algorithm which does not present a sign problem. This allows us to perform efficient numerical simulations of this system.
We study the two-dimensional antiferromagnetic Ising model with a purely imaginary magnetic field, which can be thought of as a toy model for the usual θ physics. Our motivation is to have a benchmark calculation in a system which suffers from a strong sign problem, so that our results can be used to test Monte Carlo methods developed to tackle such problems. We analyze here this model by means of analytical techniques, computing exactly the first eight cumulants of the expansion of the effective Hamiltonian in powers of the inverse temperature, and calculating physical observables for a large number of degrees of freedom with the help of standard multiprecision algorithms. We report accurate results for the free energy density, internal energy, standard and staggered magnetization, and the position and nature of the critical line, which confirm the mean-field qualitative picture, and which should be quantitatively reliable, at least in the high-temperature regime, including the entire critical line.
The mean-field approximation of the critical behavior of the antiferromagnetic Ising model in a magnetic field shows that the critical field increases first at low temperature. However, the exact solution of a decorated square lattice proves that the critical field rapidly decreases with increasing temperature for the two-dimensional lattice. Three-dimensional lattices of this model have been studied as well using the high temperature expansion, low temperature expansion, and Monte Carlo simulation. The results are different. In this article we have investigated the critical field of the body centered cubic lattice of this model using the linked-cluster expansion technique. An eighth-order perturbation series for the free energy has been calculated. The magnetization and staggered magnetic susceptibility series have been calculated from the free energy. These series have been analyzed using Padé approximants to obtain the phase diagram in the field-temperature plane. A comparison is made with Monte Carlo simulation calculations.
We study the three-dimensional Ising model at the critical point in the fixed-magnetization ensemble, by means of the recently developed geometric cluster Monte Carlo algorithm. We define a magnetic-field-like quantity in terms of microscopic spin-up and spin-down probabilities in a given configuration of neighbors. In the thermodynamic limit, the relation between this field and the magnetization reduces to the canonical relation M(h). However, for finite systems, the relation is different. We establish a close connection between this relation and the probability distribution of the magnetization of a finite-size system in the canonical ensemble.