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Physical Review

Some Remarks on Perturbation Theory and Critical Phenomena

P. T. Herman, J. R. Dorfman

Published Dec 5, 1968• 12 citations
Publisher

Abstract

Thermodynamic perturbation theory is applied to systems that undergo a phase transition. It is shown that certain conditions are necessary in order that the application of the perturbation does not change the qualitative features of the phase transition, e.g., the critical indices. The method is then applied to the two-dimensional-square Ising lattice with nearest-and next-nearest-neighbor interactions. The shift in the critical temperature is determined to two orders in the peturbation parameter, viz., the ratio of the strength of the next-nearest-neighbor force to the strength of the nearest-neighbor force. A similar calculation is also made for a two-dimensional-lattice potential of Kac and Helfand.

DOI
10.1103/physrev.176.295

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Thermodynamic perturbation theory is applied to systems that undergo a phase transition. It is shown that certain conditions are necessary in order that the application of the perturbation does not change the qualitative features of the phase transition, e.g., the critical indices. The method is then applied to the two-dimensional-square Ising lattice with nearest-and next-nearest-neighbor interactions. The shift in the critical temperature is determined to two orders in the peturbation parameter, viz., the ratio of the strength of the next-nearest-neighbor force to the strength of the nearest-neighbor force. A similar calculation is also made for a two-dimensional-lattice potential of Kac and Helfand.
A system with Hamiltonian H0 is assumed to undergo a phase transition at transition temperature T0. When a small perturbation term λH′ is added to H0, the transition temperature will depend on the coupling parameter λ and the critical indices will in general be changed. The change induced by the application of λH′ is classified into three types, and these types discussed for the cases of anisotropic Ising ferromagnets by means of thermodynamic perturbation theory and scaling law equation for spin correlation functions.
The critical values of the ratio of interaction energies at which the ordering of spins in the ground state is indeterminate (and probably Tc=0{T}_{c}=0Tc​=0) are calculated exactly for a variety of Ising lattices with antiferromagnetic next-nearest-neighbor coupling, and for anisotropic nearest-neighbor coupling. Using these results and the exact two-dimensional solutions, the general dependence of the critical point on interaction ratio is sketched. The complex case of the antiferromagnetic fcc is reconsidered.
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.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.