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Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

M. Matushko, A. Zotov

Abstract

We propose commuting sets of matrix-valued difference operators in terms of trigonometric ${\rm GL}(N|M)$-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 ${\hat{\mathcal U}}_q({\rm gl}(N|M))$. The second is a graded version of the standard $\mathbb Z_n$-invariant $A_{n-1}$ 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 ${\rm gl}(N|M)$-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.

Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

Abstract

We propose commuting sets of matrix-valued difference operators in terms of trigonometric -valued -matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric -matrices are used for this purpose. The first is the one related to the affine quantized algebra . The second is a graded version of the standard -invariant type -matrix. We show that being properly normalized the latter graded -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 -invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.
Paper Structure (27 sections, 4 theorems, 90 equations)

This paper contains 27 sections, 4 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Supersymmetric R-matrices
  3. Notations
  4. $R$-matrices
  5. Example ${\hat{\mathcal{U}}}_q({\rm gl}(2|0))$:
  6. Example ${\hat{\mathcal{U}}}_q({\rm gl}(1|1))$:
  7. Example ${\rm GL}(2|0)$:
  8. Example ${\rm GL}(1|1)$:
  9. Yang-Baxter equations and other properties of $R$-matrices
  10. Associative Yang-Baxter equation
  11. Quantum Yang-Baxter equation
  12. Twist transformation
  13. Graded spin Ruijsenaars-Macdonald operators
  14. Long-range spin chains
  15. Hamiltonians of the spin chain
  16. ...and 12 more sections

Key Result

Theorem 1

The supersymmetric extension of the $\mathbb Z_n$-invariant $R$-matrix (superR) satisfies the associative Yang-Baxter equation (AYBE).

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4