Transfer Matrix Analysis of Random Tiling Models using the Bethe Ansatz

Jan de Gier, Bernard Nienhuis

We consider the two-dimensional random tiling model introduced by Cockayne, i.e., the ensemble of all possible coverings of the plane without gaps or overlaps with squares and various hexagons. At the appropriate relative densities the correlations have eightfold rotational symmetry. We reformulate the model in terms of a random tiling ensemble with identical rectangles and isosceles triangles. The partition function of this model can be calculated by diagonalizing a transfer matrix using the Bethe Ansatz (BA). The BA equations can be solved providing exact values of the entropy and elastic constants.

Marc Bellon, S. Boukraa, J.-M. Maillard +1 more

Smita Krishnaswamy, Igor L. Markov, John P. Hayes

R. S. Vieira, A. Lima-Santos

We study in this article the algebraic Bethe Ansatz from a combinatorial perspective.The commutation relations are represented by combinatorial diagrams so that the action of the transfer matrix over the nth excited state can be found through the analysis of some combinatorial trees.This allows us to compute the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations, in a straightforward way.As examples, we consider the six-vertex model with both periodic and non-periodic boundary conditions.In parallel, we also present a comprehensive introduction to the usual algebraic Bethe Ansatz (for both the periodic and the non-periodic cases).

Frank Göhmann, Alexander Seel

The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same R -matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with 9 × 9, rational, gl(1|2) invariant R -matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t – J model, the supersymmetric U model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.

Barbara Lulek, Tadeusz Lulek