We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable
u=e−4K, where
K=J/(kBT), for nonzero external magnetic field,
H. Exact results are given for the phase diagram in the
u plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real
h=H/(kBT), these results provide new insights into features of our earlier study of this case. We also consider complex
h=H/(kBT) and
μ=e−2h. Calculations of complex-
u zeros of the partition function on sections of the square lattice are presented. For the case of imaginary
h, i.e.,
μ=eiθ, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the
u plane. We find that in this case, the phase boundary
Bu contains a real line segment extending through part of the physical ferromagnetic interval
0≤u≤1, with a right-hand endpoint
urhe at the temperature for which the Yang-Lee edge singularity occurs at
μ=e±iθ. Conformal field theory arguments are used to relate the singularities at
urhe and the Yang-Lee edge.
We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}\ensuremath{\infty}, where \ensuremath{\mu}=${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\beta}}\mathit{H}}$. We also carry out a similar analysis for -\ensuremath{\infty}\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}0. The results for the interval -1\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1 provide a way of continuously connecting the two known exact solutions of this model, viz., at \ensuremath{\mu}=1 (Onsager, Yang) and \ensuremath{\mu}=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For \ensuremath{\mu}0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at \ensuremath{\mu}=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams. \textcopyright{} 1996 The American Physical Society.
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.
Numerical methods are used to examine the thermodynamic characteristics of the two-dimensional Ising model as a function of the number of spins N. Onsager's solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N. Due to the finite extent of the system the critical temperature can only be determined to some accuracy.
In this paper an approach to solving the two- (2D) and three-dimensional Ising models in an external magnetic field $H\ensuremath{\ne}0$ is developed. The general formalism for the approach to the problem is presented for the example of the 2D Ising model in the external magnetic field. The paper presents a method obtaining the Onsager solution and computations of asymptotic forms of low-temperature free energy for the 2D Ising model in the external magnetic field
(H). The free energy in the limiting case of the magnetic field tending to zero ($H\ensuremath{\rightarrow}0,$
N, $M\ensuremath{\rightarrow}\ensuremath{\infty}$) at arbitrary temperature is also considered $(T\ensuremath{\ne}0).$