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.
The free energy of the Ising model on the square lattice with alternating interactions in both horizontal and vertical directions is exactly derived. This model is distinct from the checkerboard Ising model. The result includes Onsager’s free energy as a special case, and also includes Lee–Yang’s free energy with an imaginary field, and relates these two solutions via continuous parameters. The result includes a generalization of Lee–Yang’s result to cases with four different couplings. It is also derived that each imaginary magnetic field iπ/2 applied to a lattice site corresponds to a single frustrated square in its dual lattice.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds
n lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
The Ising model in a random magnetic field h is studied on a square lattice. h has the same value along a row, but varies randomly from row to row. h=\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty} with probability p/2 and is zero with probability 1-p. The quenched free energy F and specific heat C are calculated exactly. We find C\ensuremath{\sim}lnp for small p at the Onsager temperature
Tc. It is shown that for p\ensuremath{\ne}0, F(p,T) is not analytic in T at T=
Tc, i.e., it has a Griffiths-like singularity.