Classification of interacting Floquet phases with $U(1)$ symmetry in two dimensions

TL;DR

This paper classifies two-dimensional interacting Floquet phases with U(1) symmetry, linking them to rational functions that describe edge dynamics and charge flow, extending understanding of quantized currents in driven quantum systems.

Contribution

It provides a complete classification of 2D interacting Floquet phases with U(1) symmetry using rational functions, connecting edge dynamics to charge flow and current quantization.

Findings

Classification of Floquet phases via rational functions

Edge dynamics characterized by the rational function π(z)

Relation between π(z) and quantized U(1) current

Abstract

We derive a complete classification of Floquet phases of interacting bosons and fermions with $U(1)$ symmetry in two spatial dimensions. According to our classification, there is a one-to-one correspondence between these Floquet phases and rational functions $\pi(z) = a(z)/b(z)$ where $a(z)$ and $b(z)$ are polynomials obeying certain conditions and $z$ is a formal parameter. The physical meaning of $\pi(z)$ involves the stroboscopic edge dynamics of the corresponding Floquet system: in the case of bosonic systems, $\pi(z) = \frac{p}{q} \cdot \tilde{\pi}(z)$ where $\frac{p}{q}$ is a rational number which characterizes the flow of quantum information at the edge during each driving period, and $\tilde{\pi}(z)$ is a rational function which characterizes the flow of $U(1)$ charge at the edge. A similar decomposition exists in the fermionic case. We also show that $\tilde{\pi}(z)$ is directly related to the time-averaged $U(1)$ current that flows in a particular geometry. This $U(1)$ current is a generalization of the quantized current and quantized magnetization density found in previous studies of non-interacting fermionic Floquet phases.