Application of Algebraic Combinatorics to Finite Spin Systems

Néel probability and spin correlations in some nonmagnetic and nondegenerate states of the hexanuclear antiferromagnetic ringFe6:Application of algebraic combinatorics to finite Heisenberg spin systems

“…Therefore, we have determined a few starting steps of our procedure: (i) generate ordered partitions [κ] of n into no more than 2s+ 1 nonzero parts; (ii) find a decomposition of an orbit O[κ] into orbits of the Hamiltonian symmetry group G; (iii) for a chosen −ns ≤ M ≤ ns determine all nonordered partitions [k] satisfying the condition (4); (iv) decompositions of orbits O[k] into orbits of G are analogous to those determined in Step (ii). 31,32 Note that orbits determined by the action of G are collected into types labeled by classes of conjugated subgroups in G (in fact a representative U ⊆ G is used as such a label; this subgroup is a stabilizer of an element in an orbit under question). In this way an Ising state µ can be labeled by the following indices: magnetization M , a partition [k], a stabilizer U , a representative ν of an orbit G(ν) ∋ µ (G ν = U ), and a representative g r of a left coset g r U ⊂ G identifying µ in the orbit G(ν) (g r ν = µ).…”

“…It is rather tedious than difficult to derive a general formula for matrix elements H Uνv,U ′ ν ′ v ′ labeled by orbit stabilizers U and U ′ , orbit representatives ν and ν ′ , and indices v and v ′ distinguishing copies of Γ. 31,32 The formula obtained contains products of two factors: a matrix element U νg r |H|U ′ ν ′ e G and a group-theoretical parameter of a model under discussion. Determination of the first one is a numerical problem, so it is not discussed here.…”

“…Since dim L 1 = 4221 then there are 111(= 4332 − 4221) states with S = 0. 32,42 At first we determine such non-ordered partitions [k] = [k 0 , k 1 , . .…”

“…This also leads to some simplifications in the procedure generating operator matrices. 31,40 A general formula for matrix elements 31,32 significantly reduces a number of cases to be considered. Note that this formula needs introduction of double cosets, a structure not mentioned in the earlier papers.…”

“…Such approach to finite spin systems has been discussed recently in a series of papers. 32,33 In this work we aim to present and discuss results obtained for the "small ferric wheel" Fe 6 . The reason of such a choice is three-fold: (i) this molecule has been well investigated, 6,25,26,27,34 so we can verify our results; (ii) the number of states are relatively small (6 6 = 46 656 and dim L 0 = 4332), then many characteristics can be obtained without tedious calculations; (iii) all numerical algorithms and procedures can be tested before going to larger problems.…”