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.
Abstract The partition function of the square lattice Ising model on the rectangle with open boundary conditions in both directions is calculated exactly for arbitrary system size L × M and temperature. We start with the dimer method of Kasteleyn, McCoy and Wu, construct a highly symmetric block transfer matrix and derive a factorization of the involved determinant, effectively decomposing the free energy of the system into two parts, F ( L , M ) = F s t r i p ( L , M ) + F s t r i p r e s ( L , M ) , where the residual part F s t r i p r e s ( L , M ) contains the nontrivial finite- L contributions for fixed M . It is given by the determinant of a M / 2 × M / 2 matrix and can be mapped onto an effective spin model with M Ising spins and long-range interactions. While F s t r i p r e s ( L , M )
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.