  In recent experiments, time-dependent periodic fields are used to create
exotic topological phases of matter with potential applications ranging from
quantum transport to quantum computing. These nonequilibrium states, at high
driving frequencies, exhibit the quintessential robustness against local
disorder similar to equilibrium topological phases. However, proving the
existence of such topological phases in a general setting is an open problem.
We propose a universal effective theory that leverages on modern free
probability theory and ideas in random matrices to analytically predict the
existence of the topological phase for finite driving frequencies and across a
range of disorder. We find that, depending on the strength of disorder, such
systems may be topological or trivial and that there is a transition between
the two. In particular, the theory predicts the critical point for the
transition between the two phases and provides the critical exponents. We
corroborate our results by comparing them to exact diagonalizations for
driven-disordered 1D Kitaev chain and 2D Bernevig-Hughes-Zhang models and find
excellent agreement. This Letter may guide the experimental efforts for
exploring topological phases.
