Classification of interacting Floquet phases with \nU(1)\n 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.