Properties of the temporal transfer matrix in integrable Floquet circuits

TL;DR

This paper investigates the properties of the temporal transfer matrix in integrable Floquet circuits, revealing its integrability, explicit Bethe ansatz solutions, and local integrals of motion, advancing understanding of non-equilibrium quantum dynamics.

Contribution

It demonstrates the integrability of the temporal transfer matrix in an XXZ spin chain and provides explicit Bethe ansatz expressions for the influence matrix.

Findings

Temporal transfer matrix is integrable and part of a commuting family.

Explicit Bethe wavefunction for the influence matrix.

Identification of local and quasi-local integrals of motion.

Abstract

One possible approach to studying non-equilibrium dynamics is the so-called influence matrix (IM) formalism. The influence matrix can be viewed as a quantum state that encodes complete information about the non-equilibrium dynamics of a boundary degree of freedom. It has been shown that the IM is the unique stationary point of the temporal transfer matrix. This transfer matrix, however, is non-diagonalizable and exhibits a non-trivial Jordan block structure. In this article, we demonstrate that, in the case of an integrable XXZ spin chain, the temporal transfer matrix itself is integrable and can be embedded into a family of commuting operators. We further provide the exact expression for the IM as a particular limit of a Bethe wavefunction, with the corresponding Bethe roots given explicitly. We also focus on the special case of the free-fermionic XX chain. In this setting, we uncover additional local integrals of motion, which enable us to analyze the dimensions and structure of the Jordan blocks, as well as the locality properties of the IM. Moreover, we construct a basis of quasi-local creation operators that generate the IM from the vacuum state.