A Rigorous Theory of Many-Body Prethermalization for Periodically Driven and Closed Quantum Systems

Abstract

Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency ν. We prove that up to a quasi-exponential time τ * ∼ e c ν log 3 ν , the system barely absorbs energy. Instead, there is an effective local Hamiltonian D that governs the time evolution up to τ * , and hence this effective Hamiltonian is a conserved quantity up to τ * . Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi-Hubbard model where the interaction U is much larger than the hopping J. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time τ * that is (almost) exponential in U/J.

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

“…This differs from the proposal in Ref. [26] that there may be two independent quasiconserved quantities (excluding the energy itself) in the strong coupling regime. We suspect that this difference comes from our separation of operators into independent ones using the orthogonality in the Frobenius inner product.…”

“…Nevertheless, the SW transformation is well-defined for any finite n and can be used to obtain rigorous bounds on the dynamics in the spirit of Refs. [23,24,26]. Thus, one can show that, for small enough , H >n F < O(n 2n+2 n+1 ), see Ref.…”

“…Reference [26] used a renormalization scheme to construct an effective Hamiltonian which commutes with H 0 up to some order in small parameter, which can then be used to describe the prethermalization dynamics. Here, we use an approach with similar spirit but based on the local Schrieffer-Wolff (SW) transformation [36,37] to construct a quasiconserved operator perturbatively.…”

“…We can thus intuitively understand the prethermalization via this perturbative SW construction [23,24,26,31]. The solvable limit H 0 defines different sectors labeled by different integers, which can be viewed as counting the number (up to some off-set) of some emergent "particles" (see also Appendix A).…”

“…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].…”