On spin chains and field theories

TL;DR

This work shows how any nearest-neighbor integrable spin chain can be mapped to a 4D field theory whose 1-loop dilatation operator reproduces the spin chain Hamiltonian, even in the absence of nonabelian global symmetries. It develops a concrete anisotropic, parity-breaking spin chain based on a $U(K)$ R-matrix and derives its associated bosonic field theory, including a detailed $K=2$ case that links to a $q$-deformation of ${ m N}=4$ SYM. The Bethe Ansatz is employed to diagonalize the two-field sector, yielding explicit eigenstates and eigenvalues, and revealing BMN-like operator structures with phase-weighted contributions. The results provide a framework for engineering integrable structures in field theories and connect to Leigh-Strassler-type deformations, with potential implications for AdS/CFT-inspired integrability in less symmetric settings.

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.