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Journal of Physics A Mathematical and General

Statistical mechanics of non-intersecting line systems

Franz S. Rys, W. Helfrich

Published Feb 1, 1982• 22 citations
Publisher

Abstract

Statistical assemblies of non-intersecting closed lines (loops) with a variable loop multiplicity are considered which may have relevance for defect models of certain phase transitions. A loop gas model on a lattice is formulated and shown to be formally equivalent to an m-anisotropic n-vector model in the limit n=0. In two dimensions, the equivalence with the eight-vertex model in a field is discussed. In a few special cases the novel model is equivalent to known ones.

DOI
10.1088/0305-4470/15/2/026

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Statistical assemblies of non-intersecting closed lines (loops) with a variable loop multiplicity are considered which may have relevance for defect models of certain phase transitions. A loop gas model on a lattice is formulated and shown to be formally equivalent to an m-anisotropic n-vector model in the limit n=0. In two dimensions, the equivalence with the eight-vertex model in a field is discussed. In a few special cases the novel model is equivalent to known ones.
Any planar set of intersecting straight lines forms a four-coordinated graph, or 'lattice', provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagome lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
Abstract Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.
The solution of the zero-field "eight-vertex" model is presented. This model includes the square lattice Ising, dimer, ice, FFF, and KDP models as special cases. It is found that in general the free energy has a branch-point singularity at a phase transition, with a continuously variable exponent.
Many statistical mechanics problems can be framed in terms of random curves; we consider a class of three-dimensional loop models that are prototypes for such ensembles. The models show transitions between phases with infinite loops and short-loop phases. We map them to CP(n-1) sigma models, where n is the loop fugacity. Using Monte Carlo simulations, we find continuous transitions for n=1, 2, 3, and first order transitions for n≥5. The results are relevant to line defects in random media, as well as to Anderson localization and (2+1)-dimensional quantum magnets.