Abstract
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.
Highlights
One of the major challenges for high-energy and solid-state theorists is the numerical simulation of systems with a severe sign problem (SSP)
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
Summary
One of the major challenges for high-energy and solid-state theorists is the numerical simulation of systems with a severe sign problem (SSP). We study this system by an exact cumulant expansion to eighth order, followed by the analytic computation of the partition function and other physical quantities for a large number of degrees of freedom with the help of a standard multiprecision algorithm. This amounts essentially to the computation of the effective Hamiltonian up to order T −8, and is expected to work well in the high-temperature regime, and we provide strong evidence that this is the case. Expansion can be found in Appendix A, and several tables with numerical results can be found in Appendix B
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