Integrable Quantum Circuits from the Star-Triangle Relation
Yuan Miao, Éric Vernier
Abstract
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi><mml:mo>&#x2212;</mml:mo></mml:math>state qudits: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi></mml:math>-state Potts circuits, whose integrability has been conjectured recently by Lotkov et al., and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mi>Q</mml:mi></mml:msub></mml:math> circuits, which are novel to our knowledge. In the first example, we present for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Q</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> a connection to the Zamolodchikov-Fateev 19-vertex model.