On spin chains and field theories

TL;DR

This paper explores the relationship between integrable spin chains and field theories, showing that global symmetries are not necessary for such connections and providing explicit examples including a generalization of the XXZ chain.

Contribution

It introduces a construction linking integrable spin chains to field theories without requiring global symmetries and identifies conditions for such relations.

Findings

Constructed a field theory from an integrable spin chain without global symmetries.

Derived the spectrum and eigenstates for a 2-state chain.

Connected a generalized spin chain to the Leigh-Strassler deformation of N=4 SYM.

Abstract

We point out that the existence of global symmetries in a field theory is not an essential ingredient in its relation with an integrable model. We describe an obvious construction which, given an integrable spin chain, yields a field theory whose 1-loop scale transformations are generated by the spin chain Hamiltonian. We also identify a necessary condition for a given field theory to be related to an integrable spin chain. As an example, we describe an anisotropic and parity-breaking generalization of the XXZ Heisenberg spin chain and its associated field theory. The system has no nonabelian global symmetries and generally does not admit a supersymmetric extension without the introduction of more propagating bosonic fields. For the case of a 2-state chain we find the spectrum and the eigenstates. For certain values of its coupling constants the field theory associated to this general type of chain is the bosonic sector of the Leigh-Strassler deformation of N=4 SYM theory.