Integrable Quantum Circuits from the Star-Triangle Relation

Yuan Miao1,2 and Eric Vernier3

1Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
2Galileo Galilei Institute for Theoretical Physics, INFN, Largo Enrico Fermi, 2, 50125 Firenze, Italy
3Laboratoire de Probabilités, Statistique et Modélisation CNRS - Univ. Paris Cité - Sorbonne Univ. Paris, France

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The star-triangle relation plays an important role in the realm of exactly solvable models, offering exact results for classical two-dimensional statistical mechanical models. In this article, we construct integrable quantum circuits using the star-triangle relation. Our construction relies on families of mutually commuting two-parameter transfer matrices for statistical mechanical models solved by the star-triangle relation, and differs from previously known constructions based on Yang-Baxter integrable vertex models. At special value of the spectral parameter, the transfer matrices are mapped into integrable quantum circuits, for which infinite families of local conserved charges can be derived. We demonstrate the construction by giving two examples of circuits acting on a chain of $Q-$state qudits: $Q$-state Potts circuits, whose integrability has been conjectured recently by Lotkov et al., and $\mathbb{Z}_Q$ circuits, which are novel to our knowledge. In the first example, we present for $Q=3$ a connection to the Zamolodchikov-Fateev 19-vertex model.

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