DIMERS ON TWO AND THREEDIMENSIONAL 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 threedimensional 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 Naxes, 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 fractallike 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. 
Language:  English 
Published: 
1988

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Physics Letter A, Volume 130, number 6, 7, 1988. 
author 
Phares, Alain J. 
author_s 
Phares, Alain J. 
spellingShingle 
Phares, Alain J. DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
authorletter 
Phares, Alain J. 
author_sort_str 
Phares, Alain J. 
author2 
Wunderlich, Francis J. 
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Wunderlich, Francis J. 
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DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
title 
DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
title_short 
DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
title_full 
DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
title_fullStr 
DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
title_full_unstemmed 
DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
collection_title_sort_str 
dimers on two and threedimensional lattices: shift operator matrix solution. 
title_sort 
dimers on two and threedimensional lattices: shift operator matrix solution. 
description 
We use the shift operator matrix (SOM) method developed by McQuistan and Hock to obtain the dimer occupational degeneracies
for a threedimensional 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 Naxes, 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 fractallike 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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1988 
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DIMERS ON TWO AND THREEDIMENSIONAL LATTICES: SHIFT OPERATOR MATRIX SOLUTION. 
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dc.title 
DIMERS ON TWO AND THREEDIMENSIONAL 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 threedimensional 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 Naxes, 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 fractallike 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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Physics Letter A, Volume 130, number 6, 7, 1988. 
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