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

TL;DR

This paper derives sum rules for the ground states of the O(1) loop model and XXZ spin chain at Delta=-1/2, linking them to combinatorial numbers through integrability and knot theory techniques.

Contribution

It introduces a novel approach combining spectral parameters, size mapping, and skein relations to express ground state sums in terms of combinatorial numbers.

Findings

Ground state sums are expressed via combinatorial numbers.

Methods involve spectral parameters and integrability techniques.

Knot-theoretic skein relations are used in the analysis.

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.