Flow equations for disordered Floquet systems

Abstract

In this work, we present a new approach to disordered, periodically driven (Floquet) quantum many-body systems based on flow equations. Specifically, we introduce a continuous unitary flow of Floquet operators in an extended Hilbert space, whose fixed point is both diagonal and time-independent, allowing us to directly obtain the Floquet modes. We first apply this method to a periodically driven Anderson insulator, for which it is exact, and then extend it to driven many-body localized systems within a truncated flow equation ansatz. In particular we compute the emergent Floquet local integrals of motion that characterise a periodically driven many-body localized phase. We demonstrate that the method remains well-controlled in the weakly-interacting regime, and allows us to access larger system sizes than accessible by numerically exact methods, paving the way for studies of two-dimensional driven many-body systems.

Flow-equation approach to quantum systems driven by an amplitude-modulated time-periodic force

“…This is in contrast to Ref. [54] where the static Kamiltonian is fully diagonalized. Second, the Kamiltonian has a special structure which allows us to write it in terms of the shift operators (15).…”

Local Integrals of Motion in Quasiperiodic Many-Body Localized Systems

“…We emphasize that as compared to previous implementations of the flow-equation method [41,42,69,74] we have now introduced an off-diagonal interaction term, encoded in the tensor Γ ijkq (l). While this will eventually decay to zero at long flow times its presence throughout the flow will play an important role, as we are going to discuss in the following.…”

“…The technique was originally proposed in condensed matter physics by Wegner [54] under the name 'flow equations' (later popularized by Kehrein [55] and coworkers [56][57][58][59][60]), independently in high-energy physics by Glazek and Wilson under the name 'similarity transform' [61,62], and also in mathematics under the names 'isospectral flow' and 'double bracket flow' [63][64][65]. Since then the method has also been generalized to time-dependent systems in a variety of forms [56,[66][67][68][69] including driven and dissipative scenarios, however here we will focus on Hamiltonians with no explicit time dependence.…”

Efficient flow equations for dissipative systems

“…By using generator schemes with renormalizing properties and preserving band-diagonality, the error introduced by truncating the system to only the most relevant components can be kept small [50]. The flow equation approach can also be extended and combined with other formalisms such as the Floquet theory [51][52][53].…”