Generalized duality study of classical analog models of synthetic dimensions

In physics, bond-algebraic duality maps are direct mappings between operators that preserve their algebraic structure. It has been used to analytically predict phase transitions in the Ising and Potts Models. In this work, we have investigated the underlying duality of a classical spin model inspired from dipolar-coupled Rydberg atom arrays (or polar molecules) in synthetic dimensions. The model, first proposed in Ref.[1], adds an energy offset term J1 to the Potts model with coupling J0 [2]. We analytically derive a closed form of the self-dual line for p<=5 (number of discrete spin states). It has two regions: for small J1/J0, there is only one phase transition line; for J1 comparable to J0, there are two phase transitions. Our derived self-dual line overlaps with the single-phase transition computed numerically using Monte Carlo. Furthermore, we confirm that, under the duality map, the transition lines in the double-phase transition region map onto each other. We illustrate the computational power of the duality map by showing that it accurately predicts the thermodynamic observables of its dual points. By computing spin-spin correlation functions, we provide evidence for a BKT-type transition in the model. The limitations of the bond-algebraic duality are discussed, and future research directions are outlined.

[1] M. Cohen, M. Casebolt, Y. Zhang, K. R. A. Hazzard, and R. Scalettar, Classical analog of quantum models in synthetic dimensions, Phys. Rev. A 109, 013303 (2024).

[2] R. B. Potts, Some generalized order-disorder transformations, Mathematical Proceedings of the Cambridge Philosophical Society 48, 106–109 (1952).