Correlations and commuting transfer matrices in integrable unitary circuits

TL;DR

This paper explores integrable unitary circuits with R-matrix gates satisfying the Yang-Baxter equation, revealing their spectral properties, eigenstates, and correlation functions through transfer matrix formalism and Bethe ansatz.

Contribution

It introduces a novel transfer matrix approach for non-Hermitian, integrable unitary circuits and analyzes their eigenstates and correlation functions, including the homogeneous limit and Jordan block structure.

Findings

Transfer matrices are mutually commuting but non-Hermitian.

Bethe ansatz eigenstates can be constructed, with special considerations at the homogeneous limit.

Eigenstates and correlation functions are related to an integrable spin-1 chain with SU(2) symmetry.

Abstract

We consider a unitary circuit where the underlying gates are chosen to be R-matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.