The Haldane–Shastry spin chain can be mapped to the infinite coupling limit of the SU (2) spin Calogero–Sutherland model. We use the 2 Jack polynomial technology to compute the form factors of the spin operator on the multi-spinon spectrum. The spin structure factor is obtained through a form factor expansion. The expansion is proved to converge in the small momentum limit. Numerics based on two- and four-spinon contributions give an approximate result for the infinite temperature static and dynamic spin structure factor.
The Haldane–Shastry spin chain can be mapped to the infinite coupling limit of the SU (2) spin Calogero–Sutherland model. We use the 2 Jack polynomial technology to compute the form factors of the spin operator on the multi-spinon spectrum. The spin structure factor is obtained through a form factor expansion. The expansion is proved to converge in the small momentum limit. Numerics based on two- and four-spinon contributions give an approximate result for the infinite temperature static and dynamic spin structure factor.
Abstract The Haldane–Shastry spin chain has a myriad of remarkable properties, including Yangian symmetry and, for spin 1 / 2 , explicit highest-weight eigenvectors featuring (the case α = 1 / 2 of) Jack polynomials. This stems from the spin-Calogero–Sutherland model, which reduces to Haldane–Shastry in a special ‘freezing’ limit. In this work we clarify various points that, to the best of our knowledge, were missing in the literature. We have two main results. First, we show that freezing the fermionic spin-1/2 Calogero–Sutherland model naturally accounts for the precise form of the Haldane–Shastry wave functions, including the Vandermonde factor squared. Second, we use the fermionic framework to prove the claim of Bernard–Gaudin–Haldane–Pasquier that the Yangian highest-weight eigenvectors of the SU ( r ) -version of the Haldane–Shastry chain arise by freezing SU ( r − 1 ) spin-Calogero–Sutherland eigenvectors at α = 1 / 2 .
The Haldane-Shastry spin chain has a myriad of remarkable properties, including Yangian symmetry and, for spin 1/2, explicit highest-weight eigenvectors featuring (the case α=1/2 of) Jack polynomials. This stems from the spin-Calogero-Sutherland model, which reduces to Haldane-Shastry in a special `freezing' limit. In this work we clarify various points that, to the best of our knowledge, were missing in the literature. We have two main results. First, we show that freezing the fermionic spin-1/2 Calogero-Sutherland model naturally accounts for the precise form of the Haldane-Shastry wave functions, including the Vandermonde factor squared. Second, we use the fermionic framework to prove the claim of Bernard-Gaudin-Haldane-Pasquier that the Yangian highest-weight eigenvectors of the SU(r)-version of the Haldane-Shastry chain arise by freezing SU(r−1) spin-Calogero-Sutherland eigenvectors at α=1/2.
In this chapter, we discuss the static and dynamic properties of the spin chain which is often referred to as the Haldane–Shastry model [77, 161]. In contrast with more familiar spin chains with the nearest-neighbor exchange interaction, the Haldane–Shastry model has a particular form of the longrange exchange interaction. In spite of its peculiar shape, it has turned out that the Haldane–Shastry model is the most fundamental one-dimensional spin system. We shall start with a discussion of how the model is solved for the ground state in Sections 4.1 to 4.3, and proceed to static correlation functions in Sections 4.4 and 4.5. The excitation spectrum is interpreted by magnons in Section 4.6, and by spinons in Section 4.7. The spinon picture gives the correct degeneracy of the energy levels in Section 4.8, and leads to the derivation of thermodynamics in Section 2.4, and the dynamical correlation function in Section 4.11. Most results have been obtained in an analytic form without any approximation. There is a close connection between the spectrum of the spin chain and that of the Sutherland model. The most remarkable fact is that the spin chain has a correspondence with two different values of the coupling parameter λ in the Sutherland model. Namely, the spectrum is mapped to the case with either λ = 2 or λ = ∞. In the latter case, particles crystallize with equal spacing, and a small oscillation from the equilibrium, as well as exchange interactions, make up the spectrum. In both cases of λ, however, the degeneracy of each level in the spin chain is not reproduced. Hence, the mapping as it stands cannot be used for thermodynamics.
The dynamical structure factor $S(q,\ensuremath{\omega})$ of the K-component ( K=2, 3, 4) spin chain with a 1/r2 interaction is derived exactly at zero temperature for the arbitrary size of the system. The result is interpreted in terms of a free quasiparticle picture which is a generalization of the spinon picture in the SU(2) case. The excited states consist of K quasiparticles each of which is characterized by a set of $K\ensuremath{-}1$ quantum numbers. Divergent singularities of $S(q,\ensuremath{\omega})$ at the spectral edges are derived analytically. The analytic result is checked numerically for finite systems.
We compute the exact four-spinon contribution to the zero-temperature dynamical structure factor of the spin-1/2 Heisenberg isotropic antiferromagnet in zero magnetic field, directly in the thermodynamic limit. We make use of the expressions for matrix elements of local spin operators obtained by Jimbo and Miwa using the quantum affine symmetry of the model, and of their adaptation to the isotropic case by Abada, Bougourzi and Si-Lakhal (correcting some overall factors). The four-spinon contribution to the first frequency moment sum rule at fixed momentum is calculated. This shows, as expected, that most of the remaining correlation weight above the known two-spinon part is carried by four-spinon states. Our results therefore provide an extremely accurate description of the exact structure factor.
The hole propagator of a spin 1/2 Calogero–Sutherland model is derived using Uglov's method, which maps the exact eigenfunctions of the model, called the Yangian Gelfand-Zetlin basis, to a limit of Macdonald polynomials (gl 2 -Jack polynomials). To apply this mapping method to the calculation of 1-particle Green's function, we confirm that the sum of the field annihilation operator ψ ↑ + ψ ↓ on a Yangian Gelfand-Zetlin basis is transformed to the field annihilation operator ψ on gl 2 -Jack polynomials by the mapping. The resultant expression for the hole propagator for a finite-size system is written in terms of renormalized momenta and spin of quasi-holes, and the expression in the thermodynamic limit coincides with the earlier result derived by another method. We also discuss the singularity of the spectral function for a specific coupling parameter where the hole propagator of the spin Calogero–Sutherland model becomes equivalent to the dynamical colour correlation function of an SU (3) Haldane–Shastry model.