General Lattice Model of Phase Transitions
TL;DR: In this article, a general lattice-statistical model which includes all soluble two-dimensional model of phase transitions is proposed, besides the well-known Ising and "ice" models, other soluble cases are also considered.
Abstract: A general lattice-statistical model which includes all soluble two-dimensional model of phase transitions is proposed. Besides the well-known Ising and "ice" models, other soluble cases are also considered. After discussing some general symmetry properties of this model, we consider in detail a particular class of the soluble cases, the "free-fermion" model. The explicit expressions for all thermodynamic functions with the inclusion of an external electric field are obtained. It is shown that both the specific heat and the polarizability of the free-fermion model exhibit in general a logarithmic singularity. An inverse-square-root singularity results, however, if the free-fermion model also satisfies the ice condition. The results are illustrated with a specific example.
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TL;DR: In this paper, the partition function of the zero-field eight-vertices model on a square M by N lattice is calculated exactly in the limit of M, N large.
1,648 citations
TL;DR: In this paper, a transfer-matrix approach is introduced to calculate the "Whitney polynomial" of a planar lattice, which is a generalization of the "percolation" and "colouring" problems.
Abstract: A transfer-matrix approach is introduced to calculate the ‘Whitney polynomial’ of a planar lattice, which is a generalization of the ‘percolation’ and ‘colouring’ problems. This new approach turns out to be equivalent to calculating eigenvalues and traces of Heisenberg type operators on an auxiliary lattice which are very closely related to problems of ‘ice’ or ‘hydrogen-bond’ type that have been solved analytically by Lieb (1967a to d). Solutions for certain limiting cases are already known. The expected numbers of components and circuits can now be calculated for the plane square lattice ‘percolation’ problem in a special class of cases, namely those for which p H + p V = 1 where p H and p V are, respectively, the probabilities that any given horizontal or vertical bond is present. This class of cases is known, from the work of Sykes & Essam (1964, 1966), to be critical in the sense that a connected path across a large lattice exists with probability effectively unity whenever p H + p V ≥ 1. Relations with other problems involving the enumeration of graphs on lattices, such as the tree, Onager and dimer problems are pointed out. It is found that, for the plane square lattice, the treatment of problems of ‘ice’ type is very considerably simplified by building up the lattice diagonally, rather than horizontally or vertically.
1,092 citations
Cites background from "General Lattice Model of Phase Tran..."
...I t has been pointed out by Fan & Wu (1970) tha t symmetry and other relations lead to a very large number of possibilities of transforming these ‘ice-like ’ problems into one another....
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TL;DR: In this article, the six-vertex model is used to generate vectors and permutation relations for a two-dimensional lattice and quantum mechanics on a chain, and the Bethe Ansatz general Bethe-Anatz method is used for the inverse problem.
Abstract: CONTENTSIntroduction § 1. Classical statistical physics on a two-dimensional lattice and quantum mechanics on a chain § 2. Connection with the inverse problem method § 3. The six-vertex model § 4. Generating vectors and permutation relations § 5. The general Bethe Ansatz § 6. Integral equations Conclusion Appendix 1 Appendix 2 References
965 citations
TL;DR: In this paper, it was shown that the hard hexagon model is a special case of this eight-vertex SOS model, in which the Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a variable parameter η.
Abstract: The eight-vertex model is equivalent to a “solid-on-solid” (SOS) model, in which an integer heightl
i
is associated with each sitei of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a variable parameter η. Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which η=K/5 and the heights are restricted to the range 1⩽l
i⩽4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with η=K/r (r an integer) and 1⩽l
i⩽r−1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponentsβ, α,
$$\bar \alpha $$
are obtained.
925 citations
TL;DR: In this paper, the effects of higher-order contributions to the linearized renormalization group equations in critical phenomena are discussed and an exact scaling law for redefined fields is obtained.
Abstract: The effects of higher-order contributions to the linearized renormalization group equations in critical phenomena are discussed. This analysis leads to three quite different results: (i) An exact scaling law for redefined fields is obtained. These redefined fields are normally analytic functions of the physical fields. Corrections to the standard power laws are derived from this scaling law. (ii) The theory explains why logarithmic terms can exist in the free energy. (iii) The case in which the energy scales like the dimensionality is analyzed to show that quite anomalous results may be obtained in this special situation.
824 citations