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arXiv:1506.02994 · Physical Review Letters

Prethermalization and Thermalization in Models with Weak Integrability Breaking

Bruno Bertini, Fabian H. L. Eßler, Stefan Groha, Neil J. Robinson

Published Oct 29, 2015• 249 citations
arXivHTMLPDF

Abstract

We study the effects of integrability-breaking perturbations on the nonequilibrium evolution of many-particle quantum systems. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalizations of quantum Boltzmann equations. We benchmark our method against time-dependent density matrix renormalization group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalization plateaux for local observables on all accessible time scales. Increasing the strength of the integrability-breaking term induces a "drift" away from the prethermalization plateaux towards thermal behavior. We identify a time scale characterizing this crossover.

DOI
10.1103/physrevlett.115.180601

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We study the effects of integrability-breaking perturbations on the nonequilibrium evolution of many-particle quantum systems. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalizations of quantum Boltzmann equations. We benchmark our method against time-dependent density matrix renormalization group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalization plateaux for local observables on all accessible time scales. Increasing the strength of the integrability-breaking term induces a "drift" away from the prethermalization plateaux towards thermal behavior. We identify a time scale characterizing this crossover.
We employ equation-of-motion techniques to study the nonequilibrium dynamics in a lattice model of weakly interacting spinless fermions. Our model provides a simple setting for analyzing the effects of weak integrability-breaking perturbations on the time evolution after a quantum quench. We establish the accuracy of the method by comparing results at short and intermediate times to time-dependent density matrix renormalization group computations. For sufficiently weak integrability-breaking interactions we always observe prethermalization plateaus, where local observables relax to nonthermal values at intermediate time scales. At later times a crossover towards thermal behavior sets in. We determine the associated time scale, which depends on the initial state, the band structure of the noninteracting theory, and the strength of the integrability-breaking perturbation. Our method allows us to analyze in some detail the spreading of correlations and in particular the structure of the associated light cones in our model. We find that the interior and exterior of the light cone are separated by an intermediate region, the temporal width of which appears to scale with a universal power law t1/3{t}^{1/3}t1/3.
We investigate the crossover of the entanglement entropy toward its thermal value in nearly integrable systems. We employ equations-of-motion techniques to study the entanglement dynamics in a lattice model of weakly interacting spinless fermions after a quantum quench. For weak enough interactions we observe a two-step relaxation of the entanglement entropies of finite subsystems. Initially, the entropies follow a nearly integrable evolution, approaching the value predicted by the generalized Gibbs ensemble (GGE) of the unperturbed model. Then, they start a slow drift toward the thermal stationary value described by a standard Gibbs ensemble (GE). While the initial relaxation to the GGE is independent of the interaction, the slow drift from GGE to GE values happens on timescales proportional to the inverse interaction squared. For asymptotically large times and subsystem sizes the dynamics of the entropies can be predicted using a modified quasiparticle picture that keeps track of the evolution of the fermionic occupations caused by the integrability breaking. This picture gives a quantitative description of the results as long as the integrability-breaking timescale is much larger than the one associated with the (quasi)saturation to the GGE. In the opposite limit, the quasiparticle picture still provides the correct late-time behavior, but it underestimates the initial slope of the entanglement entropy.
In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator eiβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.
A quantum integrable system slightly perturbed away from integrability is typically expected to thermalize on timescales of order $\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}2}$, where $\ensuremath{\lambda}$ is the perturbation strength. We here study classes of perturbations that violate this scaling, and exhibit much longer thermalization times $\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}2\ensuremath{\ell}}$ where $\ensuremath{\ell}>1$ is an integer. Systems with these ``weak integrability breaking'' perturbations have an extensive number of quasiconserved quantities that commute with the perturbed Hamiltonian up to corrections of order ${\ensuremath{\lambda}}^{\ensuremath{\ell}}$. We demonstrate a systematic construction to obtain families of such weak perturbations of a generic integrable model for arbitrary $\ensuremath{\ell}$. We then apply the construction to various models, including the Heisenberg, XXZ, and XYZ chains, the Hubbard model, models of spinless free fermions, and the quantum Ising chain. Our analytical framework explains the previously observed evidence of weak integrability breaking in the Heisenberg and XXZ chains under certain perturbations.
We consider quantum quenches in an integrable quantum chain with tuneable-integrability-breaking interactions. In the case where these interactions are weak, we demonstrate that at intermediate times after the quench local observables relax to a prethermalized regime, which can be described by a density matrix that can be viewed as a deformation of a generalized Gibbs ensemble. We present explicit expressions for the approximately conserved charges characterizing this ensemble. We do not find evidence for a crossover from the prethermalized to a thermalized regime on the time scales accessible to us. Increasing the integrability-breaking interactions leads to a behavior that is compatible with eventual thermalization.
Nonequilibrium time evolution in isolated many-body quantum systems generally results in thermalization. However, the relaxation process can be very slow, and quasistationary nonthermal plateaux are often observed at intermediate times. The paradigmatic example is a quantum quench in an integrable model with weak integrability breaking; for a long time, the state cannot escape the constraints imposed by the approximate integrability. We unveil a new mechanism of prethermalization, based on the presence of a symmetry of the prequench Hamiltonian, which is spontaneously broken at zero temperature and is explicitly broken by the postquench Hamiltonian. The typical time scale of the phenomenon is proportional to the thermal correlation length of the initial state, which diverges as the temperature is lowered. We show that the prethermal quasistationary state can be approximated by a mixed state that violates cluster decomposition property. We consider two examples: the transverse-field Ising chain, where the full-time evolution is computed analytically, and the (nonintegrable) anisotropic next-nearest-neighbor Ising model, which is investigated numerically.