Rational quantum integrable systems ofDNtype with polarized spin reversal operators

Abstract

We study the spin Calogero model of D N type with polarized spin reversal operators, as well as its associated spin chain of Haldane-Shastry type, both in the antiferromagnetic and ferromagnetic cases. We compute the spectrum and the partition function of the former model in closed form, from which we derive an exact formula for the chain's partition function in terms of products of partition functions of Polychronakos-Frahm spin chains of type A. Using a recursion relation for the latter partition functions that we derive in the paper, we are able to numerically evaluate the partition function, and thus the spectrum, of the D N -type spin chain for relatively high values of the number of spins N . We analyze several global properties of the chain's spectrum, such as the asymptotic level density, the distribution of consecutive spacings of the unfolded spectrum, and the average degeneracy. In particular, our results suggest that this chain is invariant under a suitable Yangian group, and that its spectrum coincides with that of a Yangian-invariant vertex model with linear energy function and dispersion relation.paper, we shall extend the above results to spin Calogero models and their corresponding spin chains of Haldane-Shastry type based on the D N root system.In order to present our work in an appropriate context, let us briefly recall the origin and significance of the latter models. The Haldane-Shastry (HS) spin chain, introduced independently by these authors in the late eighties [4,5], is perhaps the best known example of an exactly solvable one-dimensional lattice model with long-range interactions. More precisely, this model describes a circular array of equispaced spins with two-body interactions inversely proportional to the square of the (chord) distance between the spins. The motivation for introducing this chain was the construction of a simple model with an exact ground state given by the U → ∞ limit of Gutzwiller's variational wavefunction for the ground state of the one-dimensional Hubbard model [6][7][8]. Over the years, the HS chain has appeared in many areas of interest both in Physics and Mathematics, such as fractional statistics and one-dimensional anyons [9-12], quantum entanglement [13], characterization of integrability vs. quantum chaos [14][15][16][17], quantum integrability via the asymptotic Bethe ansatz [18][19][20], Yangian quantum groups [12,[21][22][23], and conformal field theory [18,[24][25][26].One of the key properties of the HS chain -already noted by Haldane and Shastry in their original papers-is its intimate connection with the scalar (trigonometric) Sutherland model [27,28]. This connection was subsequently elucidated by Polychronakos in Ref. [2], who showed how to derive the HS chain from the spin Sutherland model [29][30][31] by a technique that he called the "freezing trick". The main idea behind this technique is to note that when the coupling constant in the spin Sutherland model goes to infinity the particles tend to concentrate on the coordinates of the (essential...

Partition functions of Polychronakos like spin chains associated with polarized spin reversal operators

“…Finally it should be noted that, apart from the context of the BC N type of spin Calogero model and PF spin chain, the BC N type of Weyl algebra plays a very important role in context of the BC N type of spin Sutherland model, related HS spin chain and also for the cases of supersymmetric generalizations of these models [37,56]. Therefore, one can use the PSRO (2.24) to construct novel exactly solvable variants of the BC N type of spin Sutherland model and HS spin chain [57]. Furthermore, it is also possible to construct supersymmetric generalization of the PSRO (2.24) and apply such operator to find out exactly solvable variants of the supersymmetric spin Calogero model and PF spin chain associated with the BC N root system [58].…”

Super Rogers–Szegö polynomials associated with BC type of Polychronakos spin chains

“…For the special cases like m 1 = m 2 or m 1 = m 2 ± 1, P (m 1 ,m 2 ) i reduces to the spin reversal operator P i (up to a sign factor) through a similarity transformation. Choosing the reflection operators as such PSRO, new exactly solvable spin Calogero models of BC N and D N type have been constructed and the partition functions of the related PF chains have also been computed by using the freezing trick via the direct approach [49,50]. Furthermore, exactly solvable spin Calogero models of BC N type have been constructed by taking reflection operators as supersymmetric analogues of PSRO (SAPSRO) [51].…”

Appearance of branched motifs in the spectra of BC type Polychronakos spin chains

“…It can be shown that, in some particular cases like m 1 " m 2 or m 1 " m 2 ˘1, P pm 1 ,m 2 q i reduces to the spin reversal operator P i through a similarity transformation. Choosing the reflection operators as such PSRO, new variants of BC N and D N type of PF spin chains have been obtained and the partition functions of these spin chains have also been calculated by using the freezing trick [47,48]. Finally, by choosing the reflection operators as supersymmetric analogues of PSRO (SAPSRO), an even larger class of BC N type of PF spin chains have been obtained [49].…”