Dualities in one-dimensional quantum lattice models: topological sectors
Laurens Lootens, Clement Delcamp, Frank Verstraete
TL;DR
The paper develops a general framework to relate spectra of dual one-dimensional quantum lattice models using categorical symmetry data. It constructs explicit symmetry and duality operators as matrix product operators (MPOs) from module functors over fusion categories and analyzes the full sector structure via the tube algebra and Drinfeld center, yielding a Morita-invariant spectrum mapping. Through concrete examples with Rep(S3) symmetry, it demonstrates how eight topological sectors arise per model and how dualities can permute or preserve these sectors, linking XXZ-type spin chains to their dual boundary theories. The results provide a practical, computation-friendly approach to studying dualities and topological sectors in lattice models and point to broad applications in symmetric tensor networks, boundary conditions, and higher-dimensional generalizations.
Abstract
It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin-$\frac{1}{2}$ Heisenberg XXZ model. One model has symmetry operators forming the fusion category $\mathsf{Rep}(\mathcal S_3)$ of representations of the group $\mathcal S_3$. We find that the mapping between its topological sectors and those of the XXZ model is associated with the non-trivial braided auto-equivalence of the Drinfel'd center of $\mathsf{Rep}(\mathcal S_3)$.