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Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain

Philippe Di Francesco, Paul Zinn-Justin, Jean-Bernard Zuber

Published Jan 1, 2006• 9 citations
OpenAlex

Abstract

The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L+1, and knot-theoretic skein relations.

Related papers

Sum rules for the ground states of theO(1) loop model on a cylinder and the XXZ spin chain

Philippe Di Francesco, Paul Zinn-Justin, Jean-Bernard Zuber

The sums of components of the ground states of the O (1) loop model on a cylinder or of the XXZ quantum spin chain at Δ = −1/2 of size L are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L +1, and knot-theoretic skein relations.

arXiv:math-ph/0603009•Journal of Statistical Mechanics Theory and Experiment•Aug 11, 2006•33 citations

The open XXZ chain at Δ = −1/2 and the boundary quantum Knizhnik–Zamolodchikov equations

Christian Hagendorf, Jean Liénardy

Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 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:2008.03220•Journal of Statistical Mechanics Theory and Experiment•Jan 1, 2021•1 citation

The quantum symmetricXXZchain at Δ = -½, alternating-sign matrices and plane partitions

Murray T. Batchelor, Jan de Gier, Bernard Nienhuis

We consider the ground-state wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at Δ = -½, along with the ground-state wavefunction of the corresponding O( n ) loop model at n = 1. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating-sign matrices and plane partitions in certain symmetry classes.

arXiv:cond-mat/0101385•Journal of Physics A Mathematical and General•May 2, 2001•145 citations

Combinatorial point for higher spin loop models

Paul Zinn-Justin

Integrable loop models associated with higher representations (spin k/2) of U_q(sl(2)) are investigated at the point q=-e^{i\pi/(k+2)}. The ground state eigenvalue and eigenvectors are described. Introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries.

•Jan 1, 2007•1 citation

On the evaluation of form factors and correlation functions for the integrable spin-s XXZ chains via the fusion method

Tetsuo Deguchi, Chihiro Matsui

Revising the derivation of the previous papers, for the integrable spin-$s$ XXZ chain we express any form factor in terms of a single sum over scalar products of the spin-1/2 XXZ chain. With the revised method we express the spin-$s$ XXZ correlation function of any given entry at zero temperature in terms of a single sum of multiple integrals.

arXiv:1103.4206•Mar 22, 2011•3 citations

Exact expressions for correlations in the ground state of the dense O(1) loop model

Saibal Mitra, Bernard Nienhuis, Jan de Gier +1 more

Conjectures for analytical expressions for correlations in the dense O (1) loop model on semi-infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries 'open'. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to fully packed loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O (1) loop model and the XXZ spin chain for the different types of boundary conditions.

arXiv:cond-mat/0401245•Journal of Statistical Mechanics Theory and Experiment•Oct 1, 2004•63 citations
The sums of components of the ground states of the O(1) loop model on a cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are expressed in terms of combinatorial numbers. The methods include the introduction of spectral parameters and the use of integrability, a mapping from size L to L+1, and knot-theoretic skein relations.
Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 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.
We consider the ground-state wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at Δ = -½, along with the ground-state wavefunction of the corresponding O( n ) loop model at n = 1. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating-sign matrices and plane partitions in certain symmetry classes.
Revising the derivation of the previous papers, for the integrable spin-sss XXZ chain we express any form factor in terms of a single sum over scalar products of the spin-1/2 XXZ chain. With the revised method we express the spin-sss XXZ correlation function of any given entry at zero temperature in terms of a single sum of multiple integrals.
Conjectures for analytical expressions for correlations in the dense O (1) loop model on semi-infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries 'open'. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to fully packed loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O (1) loop model and the XXZ spin chain for the different types of boundary conditions.