The action of the Virasoro algebra in quantum spin chains. I. The non-rational case

TL;DR

This paper analyzes how discretized Virasoro generators act in the critical XXZ quantum spin chain, revealing the structure of continuum Virasoro modules and their relation to lattice operators using Bethe ansatz and form factors.

Contribution

It provides a detailed study of the continuum limit of Virasoro modules in the XXZ spin chain and explores the convergence of lattice Virasoro generators to their continuum counterparts.

Findings

Identified indecomposable but non-logarithmic modules in the continuum limit.

Discovered dual structures of Verma and co-Verma modules from lattice modules.

Established that the convergence of Koo-Saleur generators matches Virasoro generators up to the central term.

Abstract

We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra ("Koo-Saleur generators"[arXiv:hep-th/9312156]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [arXiv:2007.11539] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules $\mathcal{W}_{j,1}$ for $j\neq 0$ contains pairs of "conjugate states" with conformal weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ that give rise to dual structures: Verma or co-Verma modules. The limit of $\mathcal{W}_{0,\mathfrak{q}^{\pm2}}$ contains diagonal fields $(h_{r,1},h_{r,1})$ and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $\mathfrak{q}^{\pm 2}$. In order to obtain matrix elements of Koo-Saleur generators at large system size $N$ we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.