Combinatorial Structures in Spin Models: A Method of Operator Matrices Generation

Abstract

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.

Application of Algebraic Combinatorics to Finite Spin Systems

“…and com puter metho ds (sto ri ng of conÙgura ti ons, acti on of a Ham i l to nia n, searchi ng conÙgura ti ons or, equi valently , ). Po ssibl e sol uti ons of the la tter pro bl em are presented elsewhere [15,16].…”

“…A more deta i l ed presenta ti on of parti ti on-rel ated questi ons can b e found i n som e earl i er pap ers [8], wo rks on num eri cal a lgori thm s [15,16], and, the la st but no t l east, Kerb er' s monogra ph [18].…”

“…Since 2 com mutes wi th the i sotro pi c Heisenberg Ham i lto ni an, then the to ta l spi n can b e used as an addi ti ona l l abel of sta tes. The considera ti ons presented here shoul d b e fol l owed by caref ul i m pl ementati on of equati ons and metho ds i n num eri cal pro cedures [15,16].…”

Combinatorial Structures in Spin Models: Algorithms for Generation of Operator Matrices

“…In the previous paper [1] a method of solving an eigen problem of a Hamiltonian acting in a finite space of states were presented. This method is based on the earlier article [2], where an expression for operator matrix elements was formulated.…”

“…of N into m non-negative parts generated in the first step are input data in the second step. The main ideas of solutions proposed here were presented in the previous work [1], whereas this paper is devoted to more detailed discussion.…”