On the composition of an arbitrary collection of SU(2) spins: an enumerative combinatoric approach

Abstract

The whole enterprise of spin compositions can be recast as simple enumerative combinatoric problems. We show here that enumerative combinatorics (EC)[1] is a natural setting for spin composition, and easily leads to very general analytic formulae -many of which hitherto not present in the literature. Based on it, we propose three general methods for computing spin multiplicities; namely, 1) the multi-restricted composition, 2) the generalized binomial and 3) the generating function methods. Symmetric and anti-symmetric compositions of SU (2) spins are also discussed, using generating functions. Of particular importance is the observation that while the common Clebsch-Gordan decomposition (CGD) -which considers the spins as distinguishable -is related to integer compositions, the symmetric and anti-symmetric compositions (where one considers the spins as indistinguishable) are obtained considering integer partitions. The integers in question here are none other but the occupation numbers of the Holstein-Primakoff bosons.The pervasiveness of q−analogues in our approach is a testament to the fundamental role they play in spin compositions. In the appendix, some new results in the power series representation of Gaussian polynomials (or q−binomial coefficients) -relevant to symmetric and antisymmetric compositions -are presented. a

Magnetic Resonance, Index Compression Maps and the Holstein-Primakoff Bosons: Towards a Polynomially Scaling Exact Diagonalization of Isotropic Multispin Hamiltonians

“…Besides allowing an easy, yet analytic, monitoring of the various multispin states and transitions (because η 0 , like all η c , is bijective) -not to mention the yoke of carrying along long strings of indexes which it frees us from -another key advantage of the mapping η 0 has to do with the fact that it indubitably allows an analytic characterization of multispin Hamiltonians without much effort as we saw in sections V A and V B where we were able to predict how many subspaces we should be expecting and their dimensions. We call the reader's attention to the fact that such results were easily obtained due to the connection the mapping η 0 and HP bosons create between spin composition and enumerative combinatorics 6 .…”

“…, 2j i }. The bosons in discussion here are the so-called Holstein-Primakoff bosons (or magnons), and one may think of the spins as vertexes of a complete graph -where each vertex can accommodate not more than a certain fixed number of these particles 6,10 .…”

“…There is thus a one-to-one correspondence between the basis {|j i , m i } and the integer basis {|ν i } under the HP transformation 6 . In explicit terms,…”

“…The analytical determination of Ω A,n is an enumerative combinatoric problem 6 . In particular, it can be proved that the generating function, G A,Ω (q), for the integers {Ω A,n } is of the form 6 :…”

Stellar Representation of Grassmannians

“…Material related to the one presented above, concerning the multiplicities of the irreducible components of the -fold tensor product of the spin-representation of (2), can be found in [24,25], while an enumerative combinatoric approach that rederives the above result, among many others, is undertaken in [26]. Note that the above problem of determining the irreducible components of the -fold wedge product of a spin-representation is a special case of the general plethysm problem (see p. 289 of [21]), which remains open to this day.…”