Dimers on Two- and Three-dimensional Lattices: Shift Operator Matrix Solution

We use the shift operator matrix (SOM) method developed by McQuistan and Hock to obtain the dimer occupational degeneracies for a three-dimensional L×M×Nlattice. The solution is given in terms of the shift operator, R, which reduces size Nof the lattice by one unit, and the annihilation operators, U, Vand W, of dimers parallel to the L-, M-, and N-axes, respectively. We show that the factorization observed by McQuistan and Hock for monomers distributed on the planar lattice is possible for dimers, and the degeneracy is obtained as a solution of an eigenvalue problem. In particular, it is interesting to note that, as in the monomer case, R J plays the role of an eigenvalue operator of a matrix which is the Hadamard product of two matrices, one depending on W, and the other on U and V. The matrix depending on W it itself the Kronecker product of order LM of a 2 × 2 matrix. The matrix depending on Uand Vhas an interesting fractal-like structure, which may be used to reduce the complexity of the problem. Our previous work done with dimers having orientational dependent activities on planar lattices, 1 XM×N, was possible by considering only one lattice size at a time, M=2,3,4 or 5. The major advantage of the SOM method is to provide a general expression valid for any M and L.