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Journal of Physics A Mathematical and Theoretical

From fermionic spin-Calogero–Sutherland models to the Haldane–Shastry chain by freezing

Jules Lamers, Didina Serban

Published May 14, 2024• 7 citations
Publisher

Abstract

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 .

DOI
10.1088/1751-8121/ad4b79

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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 1/21/21/2, explicit highest-weight eigenvectors featuring (the case α=1/2α= 1/2α=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\mathit{fermionic}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)SU(r)SU(r)-version of the Haldane-Shastry chain arise by freezing SU(r−1)SU(r-1)SU(r−1) spin-Calogero-Sutherland eigenvectors at α=1/2α= 1/2α=1/2.
Abstract We study the -analogue of the Haldane–Shastry model, a partially isotropic (xxz-like) long-range spin chain that by construction enjoys quantum-affine (really: quantum-loop) symmetries at finite system size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of D. Uglov, via ‘freezing’ from the affine Hecke algebra. To this end we first obtain explicit expressions for the spin-Macdonald operators of the (trigonometric) spin-Ruijsenaars model. Through freezing these give rise to the higher Hamiltonians of the spin chain, including another Hamiltonian of the opposite ‘chirality’. The sum of the two chiral Hamiltonians has a real spectrum also when ∣q∣=1|\mathsf {q}|=1∣q∣=1 |q|=1 , so in particular when is a root of unity. For generic q\mathsf {q}q q the eigenspaces are known to be labelled by ‘motifs’. We clarify the relation between these patterns and the corresponding degeneracies (multiplicities) in the crystal limit q→∞\textsf {q}\rightarrow \infty q→∞ q→∞ . For each motif we obtain an explicit expression for the exact eigenvector, valid for generic , that has (‘pseudo’ or ‘l-’) highest weight in the sense that, in terms of the operators from the monodromy matrix, it is an eigenvector ofAand Dand annihilated byC. It has a simple component featuring the ‘symmetric square’ of the -Vandermonde polynomial times a Macdonald polynomial—or more precisely its quantum spherical zonal special case. All other components of the eigenvector are obtained from this through the action of the Hecke algebra, followed by ‘evaluation’ of the variables to roots of unity. We prove that our vectors have highest weight upon evaluation. Our description of the exact spectrum is complete. The entire model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are theY-operators from the affine Hecke algebra. From a more mathematical perspective the key step in our diagonalisation is as follows. We show that on a subspace of suitable polynomials the first M‘classical’ (i.e. no difference part)Y-operators inNvariables reduce, upon evaluation as above, toY-operators inMvariables with parameters at the quantum zonal spherical point.
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 su(1|1) symmetric version of the Haldane-Shastry spin chain is diagonalized by means of a linear transformation. The same transformation applied to the original su(2) model yields simple expressions for the Hamiltonian and the generators of the Yangian symmetry of the model in terms of spin wave operators. Several years ago Haldane and Shastry independently discovered a soluble spin chain with exchange interaction falling off as the inverse square of the distance between the spins [1, 2]. This model can be thought of as the static limit [3] of the Sutherland model with internal degrees of freedom [4, 5, 6]. The idea of putting internal degrees of freedom into the Calogero-Sutherland models has proved to be very fruitful since then. A rich family of related models has been found. These long range interacting systems are under active investigation now. A common algebraic structure underlying their integrability is still waiting for its discovery. In this letter we discuss a linear transformation, that turns the su(m|n) supersymmetric extension of the original Haldane-Shastry Hamiltonian into a more compact form,
We study the trigonometric quantum spin-Calogero–Sutherland model, and the Haldane–Shastry spin chain as a special case, using a Bethe-Ansatz analysis. We harness the model’s Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero–Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe Ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero–Sutherland model that generalises the Yangian Gelfand–Tsetlin basis of Takemura–Uglov. The Bethe-Ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe Ansatz works in the presence of fusion, may be of independent interest.
We study the trigonometric quantum spin-Calogero-Sutherland model, and the Haldane-Shastry spin chain as a special case, using a Bethe-ansatz analysis. We harness the model's Yangian symmetry to import the standard tools of integrability for Heisenberg spin chains into the world of integrable long-range models with spins. From the transfer matrix with a diagonal twist we construct Heisenberg-style symmetries (Bethe algebra) that refine the usual hierarchy of commuting Hamiltonians (quantum determinant) of the spin-Calogero-Sutherland model. We compute the first few of these new conserved charges explicitly, and diagonalise them by Bethe ansatz inside each irreducible Yangian representation. This yields a new eigenbasis for the spin-Calogero-Sutherland model that generalises the Yangian Gelfand-Tsetlin basis of Takemura-Uglov. The Bethe-ansatz analysis involves non-generic values of the inhomogeneities. Our review of the inhomogeneous Heisenberg XXX chain, with special attention to how the Bethe ansatz works in the presence of fusion, may be of independent interest.