Antiunitary symmetry breaking and a hierarchy of purification transitions in Floquet non-unitary circuits

TL;DR

This paper investigates how antiunitary symmetry breaking in Floquet non-unitary circuits induces a hierarchy of purification transitions, revealing different phases with distinct purification dynamics and critical behaviors.

Contribution

It introduces a detailed analysis of symmetry breaking and purification transitions in Floquet non-unitary circuits, including the effects of integrability and perturbations on purification times.

Findings

Symmetry breaking transition coincides with purification transition.

Weakly purifying phase exhibits size-dependent purification time.

Adding perturbations can induce size-independent purification in large systems.

Abstract

We consider how a maximally mixed state evolves under $(1+1)D$ Floquet non-unitary circuits with an antiunitary symmetry that squares to identity, that serves as a generalized $\mathcal{PT}$ symmetry. Upon tuning a parameter, the effective Hamiltonian of the Floquet operator demonstrates a symmetry breaking transition. We show that this symmetry breaking transition coincides with different kinds of purification transitions. Gaussian non-unitary circuits are mixed (not purifying) on both sides of the symmetry breaking transition, while interacting but integrable non-unitary circuits are mixed on the symmetric side and ``weakly purifying" on the symmetry breaking side. In the weakly purifying phase, the initial mixed state purifies on a time scale proportional to the system size. We obtain numerically the critical exponents associated with the divergence of the purification time at the purification transition, which depend continuously on the parameters of the model. Upon adding a symmetric perturbation that breaks integrability, the weakly purifying phase becomes strongly purifying, purifying in a time independent of the system size, for sufficiently large system size. Our models have an extra $U(1)$ symmetry that divides the Hilbert space into different magnetization sectors, some of which demonstrate logarithmic scaling of entanglement in the weakly purifying phase.