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∗∼ecνlog3ν, the system barely absorbs energy. Instead, there is an effective local Hamiltonian Dˆ 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.
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∗∼eclog3νν, 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.
Abstract We study the dynamics of a quantum many-body lattice system with a local Hamiltonian subjected to a quasi-periodic driving with finite regularity. For sufficiently large driving frequencies, we prove that the system remains in a prethermal state for times growing polynomially with the frequency, and we show the optimality of this bound by constructing an explicit example that nearly saturates it. Within this prethermal regime, the dynamics is captured by an effective time-independent local Hamiltonian close to the undriven one. The proof relies on a non convergent normal form scheme, combined with original smoothing techniques for finitely differentiable local operators, and Lieb–Robinson bounds.
Abstract Nonequilibrium dynamics in an isolated quantum spin chain with long-range Ising interactions that decay as ( ) with the distance r is studied. It turns out that long-range interactions give rise to a big timescale separation, which causes prethermalization for all . This conclusion is deduced by comparing two important timescales relevant for relaxation dynamics; one is the relaxation time of local permutation operators, which are quasi-conserved quantities in this system, and the other is the timescale of the initial relaxation of local quantities due to the growth of quantum fluctuations. We also explore the entire nonequilibrium dynamics by using the discrete truncated Wigner approximation, which is consistent with the result mentioned above.
We prove that prethermalization is a generic property of gapped local many-body quantum systems, subjected to small perturbations, in any spatial dimension. More precisely, let H_{0} be a Hamiltonian, spatially local in d spatial dimensions, with a gap Δ in the many-body spectrum; let V be a spatially local Hamiltonian consisting of a sum of local terms, each of which is bounded by ε≪Δ. Then, the approximation that quantum dynamics is restricted to the low-energy subspace of H_{0} is accurate, in the correlation functions of local operators, for stretched exponential timescale τ∼exp[(Δ/ε)^{a}] for any a<1/(2d-1). This result does not depend on whether the perturbation closes the gap. It significantly extends previous rigorous results on prethermalization in models where H_{0} was frustration-free. We infer the robustness of quantum simulation in low-energy subspaces, the existence of athermal "scarred" correlation functions in gapped systems subject to generic perturbations, the long lifetime of false vacua in symmetry broken systems, and the robustness of quantum information in non-frustration-free gapped phases with topological order.
The dynamical phase diagram of interacting disordered systems has seen substantial revision over the past few years. Theory must now account for a large prethermal many-body localized regime in which thermalization is extremely slow, but not completely arrested. We derive a quantitative description of these dynamics in short-ranged one-dimensional systems using a model of successive many-body resonances. The model explains the decay timescale of mean autocorrelators, the functional form of the decay-a stretched exponential-and relates the value of the stretch exponent to the broad distribution of resonance timescales. The Jacobi method of matrix diagonalization provides numerical access to this distribution, as well as a conceptual framework for our analysis. The resonance model correctly predicts the stretch exponents for several models in the literature. Successive resonances may also underlie slow thermalization in strongly disordered systems in higher dimensions, or with long-range interactions.