We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field
h, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to
2.6%, has no spontaneous magnetization, and gives an effective critical-isotherm exponent
δ→1. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at
βh=iπ/2) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to
0.15% for all
β and
h, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm
M∝h1/δ with
δ=15.2 and amplitude
1.055, compared with the exact
δ=15 and
1.059. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm,
δ=15) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to
4×10−5.
Abstract We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems ( ). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.
There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).
We have performed large-scale numerical simulations for the ± J Ising model on the simple cubic and the square lattices with asymmetric weight of ferromagnetic and antiferromagnetic bonds. The Monte Carlo renormalization group method is used to estimate T c , γ/ν, β/ν and ν at various points along the phase boundary between ferromagnetic and paramagnetic phases including the tricritical point. The obtained values of T c and the resulting phase diagrams of the simple cubic and the square lattices are much more accurate than those estimated so far. In the simple cubic lattice, we find evidence for the possibility that universality of the ferromagnetic critical exponents holds along the critical line except at the tricritical point, while weak universality holds even at the tricritical point. Weak universality seems to hold also in the square lattice.
The interface method is used to derive the order-disorder critical temperature of the two-dimensional Ising model on a rectangular lattice with ferromagnetic interactions in the x direction and antiferromagnetic interactions in the y direction. Employing this approximation in two different ways gives different critical temperatures
Tc for nonzero uniform magnetic fields H. However, both of the phase boundaries
Tc(H) reduce to the exact results available at H=0 and T=0. .AE
From the exact partition functions of the Ising model on L x L square lattices (up to L = 14) in an arbitrary nonzero external magnetic field H, the precise distributions of the Fisher zeros in the complex temperature (a = e(2betaJ)) plane are obtained. Our results indicate that in the limit L-->infinity the Fisher zeros for H not = 0 do cut the positive real axis at the antiferromagnetic critical point ac(H); these values are compared with the results of closed-form approximations for the antiferromagnetic critical line. From the Fisher zeros the thermal scaling exponent yt = 1 is also obtained along the critical line, indicating the logarithmic singularity of the specific heat for the antiferromagnetic Ising model even in a strong uniform magnetic field.