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Physical Review

Ising-Model Reformulation. III. Quadruplet Spin Averages

Frank H. Stillinger

Published Jun 3, 1966• 2 citations
Publisher

Abstract

A previously developed method for diagrammatic expansion of the Ising-model partition function and pair distribution function is applied to calculation of the field-free quadruplet spin averages $〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}{\ensuremath{\mu}}_{3}{\ensuremath{\mu}}_{4}〉$. The central four-vertex function $Q$ generated by these averages is topologically analyzed in standard fashion in terms of an irreducible four-vertex quantity $e$. Several rigorously necessary conditions that must be satisfied by $Q$ are listed. It is furthermore pointed out that the existence of a logarithmic specific-heat anomaly puts an additional constraint on quadruplet averages (which is verified in an Appendix by direct calculation on the Ising-Onsager two-dimensional square lattice). Upon making a simplifying functional assumption for $Q$, this latter quantity may be entirely determined above ${T}_{c}$ by one of the necessary conditions. This leads at ${T}_{c}$ to a ${k}^{\frac{d}{2}}$ spectrum ($d=\mathrm{dimensionality}$) and above ${T}_{c}$ to a logarithmic specific heat, both of which were adduced by Abe's Ising-model version of the ${\mathrm{He}}^{4}$ speculative analysis due to Patashinskii and Pokrovskii (but for different reasons from those in the present analysis). Sinec the Abe-Patashinskii-Pokrovskii spectrum almost certainly exhibits an incorrect exponent, an alternative and more powerful functional assumption for $Q$ is suggested which still yields a soluble theory in principle, but construction of the solution is not attempted here.

DOI
10.1103/physrev.146.209

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The partition function (ZZZ) for the classical Ising model of cooperative phenomena in lattice of arbitrary dimensionality and with interactions of unspecified range is expressed as a vacuum-state expectation value of a product of two operators, each constructed from boson annihilation and creation operators. In the absence of external fields, ZZZ, and similarly the spin-pair correlation function $\ensuremath{\psi}(\mathrm{r})$, are expanded thereupon into a series of Feynman diagrams. In the case of $\ensuremath{\psi}(\mathrm{r})$, a formally exact diagram summation (1) shows how the spherical model may be recovered in low-order approximation, (2) suggests a way of introducing systematic corrections to this approximation, and (3) leads to a generalized criterion for suppression of antiferro-magnetic order in "nonfitting" lattices. A full topological-diagram reduction to restricted sets of "elementary" subdiagrams is carried out.
Series for the reduced susceptibility $\overline{\ensuremath{\chi}}$, the reduced specific heat C‾H{\overline{C}}_{H}CH​, and second moment ${\ensuremath{\mu}}_{2}$ of the static correlation function for the three-dimensional S=12S=\frac{1}{2}S=21​ Ising model on both the simple cubic (sc) and face-centered cubic (fcc) lattices with different coupling strengths in different lattice directions have been analyzed to determine the crossover exponent $\ensuremath{\phi}$ describing the behavior of the critical temperature as a function of the anisotropy parameter RRR in the Hamiltonian $\mathcal{H}=\ensuremath{-}{J}_{\mathrm{xy}}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}\ensuremath{-}{J}_{z}\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j}\ensuremath{\equiv}\ensuremath{-}{J}_{\mathrm{xy}}(\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{\mathrm{xy}}{s}_{i}{s}_{j}+R\ensuremath{\Sigma}{〈\mathrm{ij}〉}^{z}{s}_{i}{s}_{j})$. Here si=+1{s}_{i}=+1si​=+1, the first sum is over all nearest-neighbor pairs in the xy\mathrm{xy}xy plane, and the second sum is over all pairs coupled in the zzz direction. The constant gap exponent we obtain for successive derivatives of $\overline{\ensuremath{\chi}}$ and C‾H{\overline{C}}_{H}CH​ with respect to RRR confirms the exponent predictions of scaling in the parameter RRR for thermodynamic functions, while the results of the ${\ensuremath{\mu}}_{2}$ series confirm the exponent predictions of scaling with respect to RRR for the two-spin correlation function. Our results agree with the predictions for $\ensuremath{\phi}$ of Abe and Suzuki, and also with rigorous relations satisfied by the exponents describing the derivatives of the various functions. Our results do not agree with previously published results, which are based on an analysis of only the susceptibility on only the sc lattice.
This thesis deals with two two-dimensional lattice models in statistical mechanics. The first of these is the zero-field 20-vertex model on a triangular lattice. This model is the analogue to the ice-type, zero-field six-vertex model on a square lattice. We solve this problem exactly for a restricted set of vertex weights. The other model is the q-component Potts model on a square lattice. This model is equivalent to a staggered ice-type model on a square lattice. Using a variational method we estimate the critical exponent ß for several values of q. In part I we outline the fundamental statistical mechanical ideas needed, including the definition of the partition function and its relation to the thermodynamic quantities. Mention is also made of the existence and uniqueness of the thermodynamic limit for these models. The 20-vertex model is discussed in part II. The transfer matrix approach is adopted and a Bethe ansatz is used for the eigenvector. Unlike the square lattice ice-type model, the triangular lattice 20-vertex model is only soluble by this method for a restricted set of vertex weights. In particular, the ten parameters determining the vertex weights must be expressed in terms of four independent parameters. This results in only one type of non-trivial temperature independent solution. This solution exhibits critical behaviour similar to the square lattice KDP model. The general solution has three states: disordered, antiferroelectric and ferroelectric. Which state the model is in is determined by one of the above four parameters. The transition from the disordered to the antiferroelectric state is of infinite-order (i.e the free energy has an essential singularity), and from the disordered to the ferroelectric state is of finite-order. We also compare our results to the free-fermion solution of the same model and note where the two methods overlap. In part III the Potts model is considered. The model is defined and shown to be equivalent to a staggered ice-type model. A variational principle for the eigenvalues of the transfer matrix of the staggered ice model is determined. Using the zero-temperature solution as a ^uide a simple form for a sequence of approximations to the eigenvector, below T , is developed. By using the variational principle we arrive at a set of matrix equations determining the eigenvector at a given level of approximation. By noting some of the symmetries of the transfer matrix we are able to separate some of the dependences of the matrix elements and consequently simplify the original set of equations. We are also able to express the percolation probability ( the analogue of the spontaneous magnetization for general values of q ) of the Potts model as a simple function of these matrix elements. At each level of approximation there is a set of algebraic equations to solve. We have used an electronic computer to solve these equations for the first five levels of approximation, and for several values of q between zero and four. With our results we estimate the critical exponent 3 for the Potts model. These estimates agree reasonably well with previous work and indicate 3=0.145 ±.017 for q=l.
It is shown that an application of the techniques of division and integration of lattice sites reduces Ising models to equivalent vertex models. When the Ising model is on a two-dimensional lattice and no crossing occurs for any two of line segments connecting two lattice sites between which an interaction exists, the equivalent vertex model is shown to be a free-fermion vertex model. An argument is also given to show that the present method provides a method of expressing the partition function of a general Ising model on the square lattice, with noncrossing interactions within a square plaquette, in terms of a determinant of a 4×4 matrix.