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- 12 Heisenberg XXZ model. One model has symmetry operators forming the fusion category Rep(S3) of representations of the group S3 . We find that the mapping between its topological sectors and those of the XXZ model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of Rep(S3) . Published by the American Physical Society 2024
For planar lattices, there exists a dual relation between extended (blocked) Ising models in which interactions are expressed by face elements (blocks) instead of bonds. The total interaction for a block contains all even spin products among spins on the face (g-polygon); for g = 2 or 3, it contains only two-spin product, for g = 4 it contains crossed bond interactions and four-spin product, and so on. Formally, there exists a self-dual lattice for every g. The triangular lattice is self-dual for g = 3 and the self-dual lattice for g ≥4 has non integer coordinate number (a rational number) which suggests a universality for the coordination number.
Considers the pure Z(N) spin systems (including the standard Ising and Potts models) as well as generalised gauge systems (plaquettes or more complex simplex) in d-dimensional hypercubic lattices. These models are self-dual, and the authors show how this duality can be thought of as a series-parallel transformation. The simplicity of the equations enables conjectures to be made on the approximate critical frontier of the diluted version of the above systems, including some particular asymptotic behaviours which are exact. As an illustration the d=2 diluted Z(4) spin system is discussed in some detail: for those regions where exact results are available the agreement is satisfactory.
Motivated by the search for physical two-dimensional systems with the unusual phase structure of the vector Potts (clock) model, we study two
ZN-symmetric periodic Gaussian models (PGM) on a triangular lattice. The models differ from each other only in their vortex structure, and should be in the same universality class, along with the
N-state clock model. One of the models is self-dual. We also argue that the critical inverse temperature of the Gaussian approximation to the
XY model on a square lattice, ${\ensuremath{\beta}}_{\ensuremath{\square}}^{*}$, should be related to ${\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}$, the critical inverse temperature of a similar model on a triangular lattice, by ${\ensuremath{\beta}}_{\ensuremath{\Delta}}^{*}\sqrt{3}={\ensuremath{\beta}}_{\ensuremath{\square}}^{*}$. Using this result and the self-dual property of one of our
ZN PGM's, we find that the equation determining
Nc, the value of
N above which there must be at least two phase transitions, is identical for the self-dual
ZN PGM's on a square and triangular lattice.