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Journal of Physics A Mathematical and General

The Ising model below Tc: calculation of nonuniversal amplitudes using a primitive droplet model

C K Harris

Published Feb 21, 1984• 19 citations
Publisher

Abstract

Using recent exact results for the surface free energies of Ising droplets on a square lattice, and applying a primitive droplet theory due to Langer (1967), the author obtains an expression for the principal nonuniversal amplitude appearing in the imaginary part of the analytic continuation of the free energy across the coexistence line for the square lattice Ising model below T c . The significance of this calculation is the known dependence on this amplitude of the large-order terms in the expansion of the physical free energy in powers of H, thus allowing a direct numerical comparison with existing series data to be made. After carrying out a simple renormalisation of the droplet volume term, the author finds excellent numerical agreement.

DOI
10.1088/0305-4470/17/3/009

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arXiv:2412.07328•Physical review. B./Physical review. B•Apr 21, 2025•3 citations
Using recent exact results for the surface free energies of Ising droplets on a square lattice, and applying a primitive droplet theory due to Langer (1967), the author obtains an expression for the principal nonuniversal amplitude appearing in the imaginary part of the analytic continuation of the free energy across the coexistence line for the square lattice Ising model below T c . The significance of this calculation is the known dependence on this amplitude of the large-order terms in the expansion of the physical free energy in powers of H, thus allowing a direct numerical comparison with existing series data to be made. After carrying out a simple renormalisation of the droplet volume term, the author finds excellent numerical agreement.
Using the appropriately generalised finite lattice method, series expansions of the layer ( chi 1 ) and local ( chi 11 ) susceptibilities of the square lattice Ising model have been obtained. They extend existing series by 3 and 13 terms for chi 1 and chi 11 , respectively. Series analysis yields the exponent estimates gamma 1 =1.375+or-0.005 and gamma 11 =0.00+or-0.01, in agreement with scaling predictions. Repetition of the analysis used in the analogous self-avoiding walk problem confirms the breakdown of the renormalisation group scaling relation gamma 11 =v-1 for the square lattice self-avoiding walk problem found in an earlier study.
We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N6) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant 1 scaling fields, arising first at order |T −Tc | 9/4, though high-low temperature symmetry is still preserved. At terms of order |T −Tc | 17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T − Tc) p (log |T − Tc|) q with p ≥ q2. Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858,2-\sqrt{2} \approx 0.5858,2−2​≈0.5858, comparable to the exponent 4/7≈0.57144/7 \approx 0.57144/7≈0.5714 found from renormalization arguments.
Low-temperature expansion of Ising model has long been a topic of significant interest in condensed matter and statistical physics. In this paper we present new results of the coefficients in the low-temperature series of the Ising partition function on the square lattice, in the cases of a zero field and of an imaginary field $i(\ensuremath{\pi}/2){k}_{B}T$. The coefficients in the low-temperature series of the free energy in the thermodynamic limit are represented using the explicit expression of the density function of the Fisher zeros. The asymptotic behavior of the sequence of the coefficients when the order goes to infinity is determined exactly, for both the series of the free energy and of the partition function. Our analytic and numerical results demonstrate that the convergence radius of the sequence is dependent on the accumulation points of the Fisher zeros, which have the smallest modulus. In the zero field case this accumulation point is the physical critical point, while in the imaginary field case it corresponds to a nonphysical singularity. We further discuss the relation between the series coefficients and the energy state degeneracies, using the combinatorial expression of the coefficients and the subgraph expansion.