Classification of interacting Floquet phases with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> symmetry in two dimensions

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 $\ensuremath{\pi}(z)=a(z)/b(z)$, where $a(z)$ and $b(z)$ are polynomials obeying certain conditions and $z$ is a formal parameter. The physical meaning of $\ensuremath{\pi}(z)$ involves the stroboscopic edge dynamics of the corresponding Floquet system: in the case of bosonic systems, $\ensuremath{\pi}(z)=\frac{p}{q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\pi}}(z)$, where $\frac{p}{q}$ is a rational number that characterizes the flow of quantum information at the edge during each driving period and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\pi}}(z)$ 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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\pi}}(z)$ 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 noninteracting fermionic Floquet phases.

Topics & Concepts