Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems

Abstract

We establish some general dynamical properties of quantum many-body systems that are subject to a high-frequency periodic driving. We prove that such systems have a quasi-conserved extensive quantity H * , which plays the role of an effective static Hamiltonian. The dynamics of the system (e.g., evolution of any local observable) is well-approximated by the evolution with the Hamiltonian H * up to time τ * , which is exponentially large in the driving frequency. We further show that the energy absorption rate is exponentially small in the driving frequency. In cases where H * is ergodic, the driven system prethermalizes to a thermal state described by H * at intermediate times t τ * , eventually heating up to an infinite-temperature state after times t ∼ τ * . Our results indicate that rapidly driven many-body systems generically exhibit prethermalization and very slow heating. We briefly discuss implications for experiments which realize topological states by periodic driving.

Prethermal time crystals in a one-dimensional periodically driven Floquet system

“…A perturbative study of this 3D related model without disorder leads to a critical time crystal 23 . For general periodically driven systems, when subject to a high-frequency periodic driving, the energy absorption rate is exponentially small in the driving frequency for any initial state 24,25 , and it is shown that Floquet time crystals can be stabilized in the so-called prethermal phases without the need for strong disorder, before eventually heating into a thermal phase 26,27 . Another possibility is that the observed DTC falls into the short time regime before reaching the prethermal state, as suggested in Ref.…”

Interband Heating Processes in a Periodically Driven Optical Lattice

“…The validity of the high-frequency approximation, neglecting resonant excitations within the lowenergy Hubbard description, has been studied for various scenarios in references [24,25,27,28,[30][31][32][33][34][35]. For systems with local energy bound, which includes the fermionic Hubbard model, it has been shown that the heating rates decrease exponentially with the driving frequency [36][37][38][39]. In this paper, we will address the validity of the lowfrequency approximation, where the resonant coupling to excited orbital states is neglected.…”

Explicit construction of quasiconserved local operator of translationally invariant nonintegrable quantum spin chain in prethermalization

“…Note that unlike Refs. [23][24][25][26], where the heating rate is proven to be O(exp( A )), we only obtain O(exp( A √ )). This can be traced back to the estimation of the convergence radius in Appendices B and C to be ρ n ∼ 1/n 2 , hence the squared residual norm μ n ∼ O(n 4n 2n ).…”

“…In particular, various works showed that systems with weak integrability breaking exhibit this phenomenon [20][21][22]. In addition, prethermalization has been shown rigorously to exist in periodically driven many-body systems under strong driving frequencies using the FloquetMagnus expansion [23,24] and renormalization technique [25,26]. The latter also applies to time-independent many-body systems, and in particular can be used to prove rigorously the presence of exponentially long relaxation times of "particles" such as doublons in the Hubbard model in the strong coupling limit [27][28][29].…”