Antiferromagnetic Ising model in an imaginary magnetic field

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 $\theta$ 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.

Figure 1

Phase diagram of the mean-field approach of [2] to the antiferromagnetic Ising model in the $F-\theta$ plane.

Figure 2

Free energy ($-F\varphi$) at $\theta =0,N=2000$ for the square-lattice AF Ising model in the $k\mathrm{th}$ cumulant approximation.

Figure 3

Nonsingular part of the free energy ($-F\varphi$) at $\theta =\pi ,N=2000$.

Figure 4

Internal energy $e\left(F\right)$ computed for one, four, and eight cumulants at $\theta =0$ and $N=2000$.

Figure 5

Internal energy density $e\left(F\right)$ at $\theta =\pi ,N=2000$ at several cumulant expansions.

Figure 6

Nonsingular part of the internal energy at $\theta =\pi ,N=2000$.

Figure 7

Specific heat at $\theta =\pi ,N=2000$, plotted against the analytical expression.

Figure 8

Specific heat at $\theta =0$, plotted against the analytical solution. At $\theta =0, F_c=log(1+\sqrt{2})/2\approx 0.4407$.

Figure 9

$\langle m_s^2\rangle$ curves at $\theta =2,k=8$. Solid lines are just a guide to the eye.

Figure 10

Scaling of $d\langle m_s^2\rangle /d\theta$ at $\theta =2,k=8$. Solid lines are a guide to the eye.

Figure 11

The critical line $F_c\left(\theta \right)$, computed as the maximum of $d\langle m_s^2\rangle /d\theta$ at $N=2000$. The maximal $F$ points obtained in [1] are also shown.

Figure 12

Specific heat $c_v$ with $k=8$ and $\theta =2$. Solid lines are just a guide to the eye.