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

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 $$ {W}_{0,{\mathfrak{q}}^{\pm 2}} $$ W 0 , q ± 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 $$ {\mathfrak{q}}^{\pm 2} $$ q ± 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.

On correlation functions in models related to the Temperley-Lieb algebra

“…Those are not "root of unity" points of the kind discussed above, however we shall still encounter quotients of the form W j,z /W j ′ ,z ′ . The reason is that, as explained in [8] (see also [58,62]), even for generic q representations W j,z become reducible when z = q 2 j+2k , where k is some positive integer, and contain some irreducible submodule isomorphic to W j+k,q 2 j . For Q = 5 we find accordingly, for L = 4…”

“…In fact, we are not aware of any work dealing systematically with this problem. Specific correlation functions in Temperley-Lieb models have been investigated in selected cases [8,9], but we are not aware of a general treatment. Meanwhile, correlation functions of the XXZ spin chain have been investigated for multiple decades (see the book [10] and the habilitation thesis [11]).…”

Conformal Field Theory from Lattice Fermions

“…In [43] (see also [42]), there is very interesting work on the KS approximants in the setting of the XXZ spin chain in terms of Temperley-Lieb algebras as in [60]. The authors consider a certain "weak-scaling" procedure, which we would paraphrase as weak operator convergence of KS approximants and products thereof.…”

“…Interestingly, the weak convergence is restricted to a certain class of scaling states lattice states, which we believe roughly correspond to the states identified by the GNS representation of the scaling limit {A ∞ , ω ∞ } in OAR. Moreover, [43] provides a wealth of conjectures, backed by extensive numerics, concerning the limits appearing in the "weak-scaling" procedure, and, thus, it would be worthwhile to investigate whether our methods allow for a proof at least further progress, potentially exploiting the connection with the formulation based on Temperley-Lieb algebras for general c = 0 (see Section 4.2.1). In particular, it would be interesting whether our methods allow to lift the "weak-scaling" procedure to a "strong" version (in the sense of operator topologies).…”