The open XXZ chain at $\Delta =-1/2$ and the boundary quantum Knizhnik-Zamolodchikov equations

Abstract

The open XXZ spin chain with the anisotropy parameter $\Delta =-\frac{1}{2}$ and diagonal boundary magnetic fields that depend on a parameter $x$ is studied. For real $x>0$, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in $x$ with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik-Zamolodchikov equations associated with the $R$-matrix and diagonal $K$-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

arXiv comment36 pages, no figures; v2: corrected typos and conjectures 2.1, 5.5

References in corpus (29)

• Spin chains and combinatorics
• Boundary K-matrices for the XYZ, XXZ AND XXX spin chains
• The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions
• The Importance of being Odd
• Difference Equations in Spin Chains with a Boundary
• Magic in the spectra of the XXZ quantum chain with boundaries at Delta=0 and Delta=-1/2
• Spin chains and combinatorics: twisted boundary conditions
• One-boundary Temperley-Lieb algebras in the XXZ and loop models
• Universal behavior of a bipartite fidelity at quantum criticality
• Non-local space-time supersymmetry on the lattice
• Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
• The XXZ spin chain at $\Delta =-1/2$: Bethe roots, symmetric functions and determinants
• Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
• Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2
• Logarithmic corrections to the free energy from sharp corners with angle $2\pi$
• Open spin chains with dynamic lattice supersymmetry
• Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations
• Koornwinder polynomials and the XXZ spin chain
• Finite size emptiness formation probability of the XXZ spin chain at $\Delta =-1/2$
• Boundary transfer matrices and boundary quantum KZ equations
• Finite-size corrections for universal boundary entropy in bond percolation
• Lattice Supersymmetry in the Open XXZ Model: An Algebraic Bethe Ansatz Analysis
• Boundary quantum Knizhnik-Zamolodchikov equations and Bethe vectors
• Integral solutions to boundary quantum Knizhnik-Zamolodchikov equations
• On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions
• On some ground state components of the O(1) loop model
• Boundary emptiness formation probabilities in the six-vertex model at $\Delta =-\frac{1}{2}$
• Bipartite fidelity of critical dense polymers
• Refined Razumov-Stroganov conjectures for open boundaries

Cited by in corpus (1)

• On the logarithmic bipartite fidelity of the open XXZ spin chain at $\Delta =-1/2$