Abstract
We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), 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 [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of {W}_{0,{mathfrak{q}}^{pm 2}} contains diagonal fields (hr,1, hr,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.
Highlights
The mechanism responsible for the emergence of the rich structure of conformal field theories (CFTs) in the continuum limit of discrete lattice models has attracted growing interest in the last few years
While the questions we investigate and the strategy we use are fully general in the context of two-dimensional lattice models having a conformally invariant continuum limit, we focus in this paper on models based on the Temperley-Lieb algebra
We think of these models as providing some lattice analogue of the Virasoro algebra — or more precisely, since we study systems with closed boundary conditions, the product of the left and the right Virasoro algebras, Vir ⊗ Vir — at central charge c ≤ 1
Summary
The mechanism responsible for the emergence of the rich structure of conformal field theories (CFTs) in the continuum limit of discrete (not necessarily integrable) lattice models has attracted growing interest in the last few years. The second purpose of this paper is to find out what kind of Virasoro modules occur in the XXZ chain when the Virasoro representations are degenerate — that is, (some) fields belong to the extended Kac table We will do this straightforwardly, by exploring the action of the lattice Virasoro generators, and checking directly whether the relevant combinations vanish or not — in technical parlance, whether “null states” or “singular vectors” are zero .
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