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.

Main Author: Phares, Alain J.
Other Authors: Wunderlich, Francis J.
Format: Villanova Faculty Authorship
Language: English
Published: 1988
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dc_source_str_mv Physics Letters A, Vol. 130, pp. 385-389, 1988.
author Phares, Alain J.
author_facet_str_mv Phares, Alain J.
Wunderlich, Francis J.
author_or_contributor_facet_str_mv Phares, Alain J.
Wunderlich, Francis J.
author_s Phares, Alain J.
spellingShingle Phares, Alain J.
DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
author-letter Phares, Alain J.
author_sort_str Phares, Alain J.
author2 Wunderlich, Francis J.
author2Str Wunderlich, Francis J.
dc_title_str DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
title DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
title_short DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
title_full DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
title_fullStr DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
title_full_unstemmed DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
collection_title_sort_str dimers on two- and three-dimensional lattices: shift operator matrix solution.
title_sort dimers on two- and three-dimensional lattices: shift operator matrix solution.
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description 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.
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dc.title DIMERS ON TWO- AND THREE-DIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION.
dc.creator Phares, Alain J.
Wunderlich, Francis J.
dc.description 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.
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