Global properties of the spectrum of the Haldane-Shastry spin chain

Abstract

We introduce four types of SU(2M + 1) spin chains which can be regarded as the BC N versions of the celebrated Haldane-Shastry chain. These chains depend on two free parameters and, unlike the original Haldane-Shastry chain, their sites need not be equally spaced. We prove that all four chains are solvable by deriving an exact expression for their partition function using Polychronakos's "freezing trick". From this expression we deduce several properties of the spectrum, and advance a number of conjectures that hold for a wide range of values of the spin M and the number of particles. In particular, we conjecture that the level density is Gaussian, and provide a heuristic derivation of general formulas for the mean and the standard deviation of the energy.

Exact partition function of supersymmetric Haldane–Shastry spin chain

“…The above defined ordering is effectively a partial ordering, since it does not induce an ordering between ξ p and ξ p ′ whenp =p ′ . It can be shown that the action of H (3.4) on the state vector ξ p yields [7,9] …”

“…To execute this programme, let us first briefly recapitulate the calculation for the partition function of spinless CS model (2.12) at a → ∞ limit [9]. It should be noted that, the eigenvalues in eqn.…”

“…Consequently, in contrast to the restricted choice of k for the case of bosonic spin chain [9], all possible k ∈ P N will contribute to the partition function (4.9) in the case of supersymmetric as well as fermionic HS spin chain. It is well known that the spectrum of bosonic SU(m) HS spin chain (1.1) containing N number of lattice sites can be obtained from motifs like δ ≡ (0, δ 1 , .…”

“…, δ N −1 , 0), where each δ j is either 0 or 1 [6][7][8]. The form of these motifs and corresponding eigenvalues can be reproduced by using the partition function of bosonic SU(m) HS spin chain [9]. Now we want to explore how the motifs associated with SU(m|n) supersymmetric HS spin chain emerge naturally from the expression of partition function (4.9).…”

“…However, it has been found that the NNS distribution for SU(m) bosonic HS spin chain does not follow either Wigner's law or Poisson's law within a wide range of N [9]. Instead, the cumulative NNS distribution for this bosonic spin chain can be well approximated by a function likẽ 12) where v = u/u max with u max being the largest normalized spacing, α and β are two free parameters taking values within the range 0 < α, β < 1, and the value of γ is fixed by requiring that the average normalized spacing be equal to 1.…”