The spin-1/2 XXZ Heisenberg chain, the quantum algebra U_q[sl(2)], and duality transformations for minimal models

TL;DR

The paper addresses how finite-size spectra of the spin-1/2 XXZ chain with toroidal boundaries can realize the spectra of all $c<1$ minimal models by a generalized projection that includes half-integer angular momentum sectors. It develops an extended projection framework producing $R$- and $L$-model families, derives their Virasoro-character decompositions, and shows how half-integer sectors yield new spinor operators and duality-related boundary conditions. The authors connect observed finite-size degeneracies to intertwining relations in the quantum algebra $U_{q}[\mathrm{sl}(2)]$, and they explicitly construct duality-twisted boundary conditions for the Ising and 3-state Potts chains, providing lattice realizations and numerical checks. The work demonstrates that models with the same central charge and operator content can distribute operators differently among sectors, highlighting nontrivial universality and symmetry structures and suggesting further exploration of duality and boundary phenomena in critical lattice systems.

Abstract

The finite-size scaling spectra of the spin-1/2 XXZ Heisenberg chain with toroidal boundary conditions and an even number of sites provide a projection mechanism yielding the spectra of models with a central charge c<1 including the unitary and non-unitary minimal series. Taking into account the half-integer angular momentum sectors - which correspond to chains with an odd number of sites - in many cases leads to new spinor operators appearing in the projected systems. These new sectors in the XXZ chain correspond to a new type of frustration lines in the projected minimal models. The corresponding new boundary conditions in the Hamiltonian limit are investigated for the Ising model and the 3-state Potts model and are shown to be related to duality transformations which are an additional symmetry at their self-dual critical point. By different ways of projecting systems we find models with the same central charge sharing the same operator content and modular invariant partition function which however differ in the distribution of operators into sectors and hence in the physical meaning of the operators involved. Related to the projection mechanism in the continuum there are remarkable symmetry properties of the finite XXZ chain. The observed degeneracies in the energy and momentum spectra are shown to be the consequence of intertwining relations involving U_q[sl(2)] quantum algebra transformations.