Exact partition function of supersymmetric Haldane–Shastry spin chain

Abstract

By taking the freezing limit of a spin Calogero-Sutherland model containing 'anyon like' representation of the permutation algebra, we derive the exact partition function of SU(m|n) supersymmetric Haldane-Shastry (HS) spin chain. This partition function allows us to study global properties of the spectrum like level density distribution and nearest neighbour spacing distribution. It is found that, for supersymmetric HS spin chains with large number of lattice sites, continuous part of the energy level density obeys Gaussian distribution with a high degree of accuracy. The mean value and standard deviation of such Gaussian distribution can be calculated exactly. We also conjecture that the partition function of supersymmetric HS spin chain satisfies a duality relation under the exchange of bosonic and fermionic spin degrees of freedom.

Thermodynamics of spin chains of Haldane–Shastry type and one-dimensional vertex models

“…Performing the change of variable t = −λ ϕ(x) and proceeding as in the previous section we readily obtain 22) where the error term has been estimated as in Appendix A. Note that if 0 < |B| < B s we must have 0 < x 0 < κ, and consequently ε (x 0 ) > 0.…”

“…In particular, these two chains are also completely integrable. Apart from their integrable character, spin chains of HS type appear in many areas of current interest in both physics and mathematics, such as quantum chaos [19,20], supersymmetry [21,22], conformal field theory [23][24][25], the AdS-CFT correspondence [26,27], one-dimensional anyons [28] and Yangian quantum groups [15,[29][30][31].…”

Low energy properties of the supersymmetric Haldane–Shastry spin chain

“…[20], the SU(m|n) supersymmetric HS model in (2.6) can be transformed to a spin chain. We consider a representation of the permutation algebra (2.5), which acts on a spin state like |α 1 α 2 .…”

“…Next we define the quantities K i = i j=1 k j which denote the partial sums corresponding to the partition k ∈ P N . The partition function of the SU(m|n) HS spin chain, obtained through the freezing technique, is then given by [20] Z (m|n) 15) where E(l) = l(N − l) and q = e −1/T ; here T is the temperature, and we have set the Boltzmann constant k B = 1. Note that the dimension of the summation variable k (i.e., r) in Eq.…”

“…In Ref. [20], this technique has been used to compute the exact partition function of the SU(m|n) HS spin chain. Subsequently, it was shown that the partition function of the SU(m|n) HS spin chain can be expressed through the Schur polynomials associated with the motif representations, and an exact duality relation has been established between the partition functions of the SU(m|n)…”