COMBINATORIAL STRUCTURES IN SPIN MODELS: A METHOD OF OPERATOR MATRICES GENERATION
Bucikiewicz Sylwia, Florek Wojciech
Adam Mickiewicz University, Institute of Physics
Umultowska 85, 61-614 Poznań, Poland
e-mail: bucz@spin.amu.edu.pl, florek@spin.amu.edu.pl
DOI: 10.12921/cmst.2001.07.01.27-39
OAI: oai:lib.psnc.pl:513
Abstract:
Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems of very large dimensions. Efficient, and as exact as possible, solutions of such eigenproblems are very difficult. A method leading to block diagonalization of Hamiltonian matrix is proposed in this paper. For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers etc.) are introduced and briefly discussed. Generation of these structures can be performed by means of algorithms being modifications of standard ones. Main ideas are presented in this paper, whereas the actual form of algorithms will be discussed elsewhere.
References:
[1] J. C. Bonner, M. E. Fisher, Phys. Rev., 135, A640 (1964).
[2] D. Gatteschi, A. Caneschi, L. Pardi, R. Sessoli, Science, 265, 1054 (1994); A. Lascialfari,
D. Gatteschi, A. Cornia, U. Balucani, M. G. Pini, A. Rettori, Phys. Rev. B. 57, 1115 (1998);
A. Caneschi, D. Gatteschi, C. Sangregorio, R. Sessoli, L. Sorace, A. Cornia, M. A. Novak, C.
Paulsen, W. Wernsdorfer, J. Mag. Mag. Mater., 200, 182 (1999).
[3] A. Kerber, Algebraic Combinatorics via Finite Group Actions, BI Wissenshaftsverlag, Mannheim-Wien-Zürich, 1991.
[4] W. Florek, T. Lulek, J. Phys. A: Math. Gen., 26, 2153 (1993).
[5] S. L. Altmann, Induced Representations in Crystals and Molecules, Academic Press, London, 1977.
[6] G. Kamieniarz, R. Matysiak, W. Florek, S. Wałcerz, J. Mag. Mag. Mater., 203, 271 (1999).
[7] W. Florek, Acta Phys. Polon. A, 96, 699 (1999).
[8] B. Lulek, T. Lulek, J. Phys. A: Math. Gen., 17, 3077 (1984).
[9] E. N. Reingold, J. Nivergelt, N. Deo, Combinatorial Algorithms. Theory and Practice, Prentice Hall, Engelwood Cliffs NJ, 1977.
[10] W. Lipski, Combinatorics for Programmers. Polish Sci. Publ. (PWN), Warsaw, 1982 [in Polish].
[11] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art
of Scientific Computing, Cambridge Univ. Press, Cambridge, 1992 (Sec. 2.0).
[12] W. Florek, Comp. Meth. Sci. Tech., this volume.
[13] The GNU MP library has been developed by T. Granlund and now is a standard part in main Linux distributions (it is used to compile SSH2). It is also available at its home page
http://www.swox.com/gmp maintained by T. Granlund.
Finite spin models, applicable to investigations of mesoscopic rings, give rise to eigenproblems of very large dimensions. Efficient, and as exact as possible, solutions of such eigenproblems are very difficult. A method leading to block diagonalization of Hamiltonian matrix is proposed in this paper. For a given symmetry group of a Heisenberg Hamiltonian commuting with the total spin projection (i.e. with the total magnetization being a good quantum number) appropriate combinatorial and group-theoretical structures (partitions, orbits, stabilizers etc.) are introduced and briefly discussed. Generation of these structures can be performed by means of algorithms being modifications of standard ones. Main ideas are presented in this paper, whereas the actual form of algorithms will be discussed elsewhere.
[1] J. C. Bonner, M. E. Fisher, Phys. Rev., 135, A640 (1964).
[2] D. Gatteschi, A. Caneschi, L. Pardi, R. Sessoli, Science, 265, 1054 (1994); A. Lascialfari,
D. Gatteschi, A. Cornia, U. Balucani, M. G. Pini, A. Rettori, Phys. Rev. B. 57, 1115 (1998);
A. Caneschi, D. Gatteschi, C. Sangregorio, R. Sessoli, L. Sorace, A. Cornia, M. A. Novak, C.
Paulsen, W. Wernsdorfer, J. Mag. Mag. Mater., 200, 182 (1999).
[3] A. Kerber, Algebraic Combinatorics via Finite Group Actions, BI Wissenshaftsverlag, Mannheim-Wien-Zürich, 1991.
[4] W. Florek, T. Lulek, J. Phys. A: Math. Gen., 26, 2153 (1993).
[5] S. L. Altmann, Induced Representations in Crystals and Molecules, Academic Press, London, 1977.
[6] G. Kamieniarz, R. Matysiak, W. Florek, S. Wałcerz, J. Mag. Mag. Mater., 203, 271 (1999).
[7] W. Florek, Acta Phys. Polon. A, 96, 699 (1999).
[8] B. Lulek, T. Lulek, J. Phys. A: Math. Gen., 17, 3077 (1984).
[9] E. N. Reingold, J. Nivergelt, N. Deo, Combinatorial Algorithms. Theory and Practice, Prentice Hall, Engelwood Cliffs NJ, 1977.
[10] W. Lipski, Combinatorics for Programmers. Polish Sci. Publ. (PWN), Warsaw, 1982 [in Polish].
[11] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C. The Art
of Scientific Computing, Cambridge Univ. Press, Cambridge, 1992 (Sec. 2.0).
[12] W. Florek, Comp. Meth. Sci. Tech., this volume.
[13] The GNU MP library has been developed by T. Granlund and now is a standard part in main Linux distributions (it is used to compile SSH2). It is also available at its home page
http://www.swox.com/gmp maintained by T. Granlund.
