Prethermalization and Thermalization in Models with Weak Integrability Breaking

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.

Figure 1

$\mathcal{G}(L/2,L/2+1;t)$ for a quench where the system is prepared in the ground state of $H(0,0.8,0)$ and time evolved with $H(0,0.4,0.4)$ for a system with $L=256$ sites. The EOM results (red line) are in excellent agreement with t-DMRG computations [41] (circles). Inset: Prethermalized behavior persists over a large time interval.

Figure 2

$\mathcal{G}(L/2,L/2-1;t)$ for a system with Hamiltonian $H(J_2,0.1,0.4)$ and sizes $L=360,320$ initially prepared in a thermal state (7) with density matrix $\rho (2,0,0,0)$. The expected steady state thermal values are indicated by dotted lines, while the black dashed lines are exponential fits to (8).

Figure 3

Real (inset, imaginary) part of $\mathcal{G}(L/2,L/2+2;t)$ for a system with Hamiltonian $H(J_2,0.1,0.4)$ and sizes $L=360,320$, that was initially prepared in a thermal state (7) with density matrix $\rho (2,0,0,0)$. The expected steady state thermal values are indicated by dotted lines, while the black dashed lines are exponential fits to (8).

Figure 4

Occupation numbers $n_{++}(k,t)$ and $n_{--}(k,t)$ initialized in the thermal state (7) $\rho (2,0,0.5,0)$ and time evolved with $H(0.5,0,0.4)$. The solid lines are the results of the EOM $(L=320)$ for various times. The dotted lines are computed by means of the QBE ($L=320$). The black solid line is the thermal value found by means of second-order perturbation theory in $U$.