Exact dynamics in dual-unitary quantum circuits

TL;DR

This paper introduces a class of solvable initial states in dual-unitary quantum circuits, allowing analytical study of their dynamics, including entanglement growth and correlation functions, revealing differences based on ergodicity and initial state purity.

Contribution

The paper defines and classifies solvable matrix product states in dual-unitary circuits, enabling analytical analysis of their dynamics and entanglement spreading, extending previous models.

Findings

Subsystems reach infinite temperature after time proportional to their size

Correlation functions exhibit different behaviors depending on ergodicity

Entanglement entropy evolution can be expressed in closed form

Abstract

We consider the class of dual-unitary quantum circuits in $1+1$ dimensions and introduce a notion of ``solvable'' matrix product states (MPSs), defined by a specific condition which allows us to tackle their time evolution analytically. We provide a classification of the latter, showing that they include certain MPSs of arbitrary bond dimension, and study analytically different aspects of their dynamics. For these initial states, we show that while any subsystem of size $\ell$ reaches infinite temperature after a time $t\propto \ell$, irrespective of the presence of conserved quantities, the light-cone of two-point correlation functions displays qualitatively different features depending on the ergodicity of the quantum circuit, defined by the behavior of infinite-temperature dynamical correlation functions. Furthermore, we study the entanglement spreading from such solvable initial states, providing a closed formula for the time evolution of the entanglement entropy of a connected block. This generalizes recent results obtained in the context of the self-dual kicked Ising model. By comparison, we also consider a family of non-solvable initial mixed states depending on one real parameter $\beta$, which, as $\beta$ is varied from zero to infinity, interpolate between the infinite temperature density matrix and arbitrary initial pure product states. We study analytically their dynamics for small values of $\beta$, and highlight the differences from the case of solvable MPSs.