Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
Mariya Georgievna Matushko, Andrei Vladimirovich Zotov
We propose commuting sets of matrix-valued difference operators in terms of trigonometric GL(N|M)-valued R-matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric R-matrices are used for this purpose. The first is the one related to the affine quantized algebra Uˆq(gl(N|M)). The second is a graded version of the standard Zn-invariant An−1 type R-matrix. We show that being properly normalized the latter graded R-matrix satisfies the associative Yang-Baxter equation. Next, we discuss construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extends the gl(N|M)-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.
We propose commuting sets of matrix-valued difference operators in terms of trigonometric GL(N|M)-valued R-matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric R-matrices are used for this purpose. The first is the one related to the affine quantized algebra Uˆq(gl(N|M)). The second is a graded version of the standard Zn-invariant An−1 type R-matrix. We show that being properly normalized the latter graded R-matrix satisfies the associative Yang-Baxter equation. Next, we discuss construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extends the gl(N|M)-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.
Abstract We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars–Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin G L M R -matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions for equilibrium position in the underlying classical spinless Ruijsenaars–Schneider model. Trigonometric degenerations are studied as well. For example, in M = 2 case our construction provides q-deformation for anisotropic XXZ Haldane–Shastry model. The standard Haldane–Shastry model and its Uglov’s q-deformation based on U q ( g l ˆ M ) XXZ R-matrix are included into consideration by separate verification.
In this paper we discuss R-matrix-valued Lax pairs for slN Calogero–Moser model and their relation to integrable quantum long-range spin chains of the Haldane–Shastry–Inozemtsev type. First, we construct the R-matrix-valued Lax pairs for the third flow of the classical Calogero–Moser model. Then we notice that the scalar parts (in the auxiliary space) of the M-matrices corresponding to the second and third flows have form of special spin exchange operators. The freezing trick restricts them to quantum Hamiltonians of long-range spin chains. We show that for a special choice of the R-matrix these Hamiltonians reproduce those for the Inozemtsev chain. In the general case related to the Baxter's elliptic R-matrix we obtain a natural anisotropic extension of the Inozemtsev chain. Commutativity of the Hamiltonians is verified numerically. Trigonometric limits lead to the Haldane–Shastry chains and their anisotropic generalizations.
Matrix representations of BCn associative Yang--Baxter equations and related long-range spin chains
We construct a matrix representation of the algebra defined by the associative Yang--Baxter equations of BCn type. In the elliptic case, two sets of generators are represented in terms of the Baxter--Belavin quantum R-matrix in the fundamental representation of GLN, while the remaining generators are represented by the elliptic K-matrix found by Komori and Hikami. We revisit its derivation using the Felder--Pasquier procedure and correct the unitarity relation for odd N. We then use these matrix representations to construct examples of long-range quantum spin chains from R-matrix-valued Lax pairs for the classical Calogero--Inozemtsev system associated with the BCn root system. We also construct R-matrix-valued Lax pairs for Calogero--Moser models in an external field, leading to Haldane--Shastry and Polychronakos--Frahm chains with modified spin positions.
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 |q|=1 , so in particular when is a root of unity. For generic 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→∞ 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.
We consider the construction of quantum-integrable spin chains with q -deformed long-range interactions by ‘freezing’ integrable quantum many-body systems with spins. The input is a (quantum) spin-Ruijsenaars system along with an equilibrium configuration of the underlying spinless classical Ruijsenaars–Schneider system. We focus on the elliptic case. For a distinguished choice of equilibrium, the resulting long-range spin chain has a real spectrum and admits a short-range limit, providing an integrable interpolation from nearest-neighbor to long-range interacting spins. We first define an action of the modular group on the spinless elliptic Ruijsenaars–Schneider system to show that, for a fixed elliptic parameter, it has a whole modular family of classical equilibrium configurations. These typically have constant but nonzero momenta. Then we use the setting of deformation quantization to provide a uniform framework for freezing elliptic spin-Ruijsenaars systems at any classical equilibrium while preserving quantum integrability. As we showed in previous work, the results include the Heisenberg, Inozemtsev and Haldane–Shastry chains along with their xxz -like q -deformations (face type), or the antiperiodic Haldane–Shastry chain of Fukui–Kawakami, its elliptic generalization of Sechin–Zotov, and their completely anisotropic q -deformations due to Matushko–Zotov (vertex type). Finally, we show how freezing fits in the setting of ‘hybrid’ integrable systems.
The Inozemtsev chain is an exactly solvable interpolation between the short-range Heisenberg and long-range Haldane–Shastry (HS) chains. In order to unlock its potential to study spin interactions with tunable interaction range using the powerful tools of integrability, the model’s mathematical properties require better understanding. As a major step in this direction, we present a new generalisation of the Inozemtsev chain with spin symmetry reduced to \textit{U}(1) 𝑈 ( 1 ) , interpolating between a Heisenberg XXZ chain and the XXZ-type HS chain, and integrable throughout. Underlying it is a new quantum many-body system that extends the elliptic Ruijsenaars system by including spins, contains the trigonometric spin-Ruijsenaars–Macdonald system as a special case, and yields our spin chain by ‘freezing’. Our models have potential applications from condensed-matter to high-energy theory, and provide a crucial step towards a general theory for long-range integrability.