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arXiv:math-ph/0603009 · Journal of Statistical Mechanics Theory and Experiment

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

Published Aug 11, 2006• 33 citations
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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 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.

DOI
10.1088/1742-5468/2006/08/p08011

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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.
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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.
Abstract We define a new family of overlaps C N , m for the XXZ Hamiltonian on a periodic chain of length N . These are equal to the linear sums of the groundstate components, in the canonical basis, wherein m consecutive spins are fixed to the state ↑. We define the boundary emptiness formation probabilities as the ratios C N , m / C N ,0 of these overlaps. In the associated six-vertex model, they correspond to correlation functions on a semi-infinite cylinder of perimeter N . At the combinatorial point Δ = − 1 2 , we obtain closed-form expressions in terms of simple products of ratios of integers.
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.