Many-Particle Dephasing after a Quench

After a quench in a quantum many-body system, expectation values tend to relax towards long-time averages. However, temporal fluctuations remain in the long-time limit, and it is crucial to study the suppression of these fluctuations with increasing system size. The particularly important case of nonintegrable models has been addressed so far only by numerics and conjectures based on analytical bounds. In this work, we are able to derive analytical predictions for the temporal fluctuations in a nonintegrable model (the transverse Ising chain with extra terms). Our results are based on identifying a dynamical regime of “many-particle dephasing,” where quasiparticles do not yet relax but fluctuations are nonetheless suppressed exponentially by weak integrability breaking.

Figure 1

Many-particle dephasing in a chain with $N=12$. The quench jumps from $J_{\mathrm{pre}}/\mathrm{\Omega }=0$ to $J/\mathrm{\Omega }=0.8$. Without perturbation (blue) the fluctuations at early and late times are similar. A weak NNN coupling of strength $J_{\mathrm{NNN}}/J=0.01$ leads to a significant additional relaxation for $t\to \infty$. (b) The temporal variance $N{\sigma }_A^2$ for two different observables shows an exponential decay in $N$. [Analytical result from Eq. (6).]

Figure 2

Schematic overview of different dynamical regimes after a quench in a finite-size system that is weakly perturbed away from an integrable, effectively noninteracting model (displayed for a local observable in the perturbed transverse Ising model, but valid more generally). First, revivals occur. Second, a transient steady-state with fluctuations ${\sigma }_A\sim 1/\sqrt{N}$ is observed, as predicted for the integrable case. Finally, after a time that scales as the inverse of the integrability-breaking perturbation, the final steady state is reached. There, fluctuations are reduced exponentially in the system size, due to many-particle dephasing.

Figure 3

Analytical predictions, depending on the postquench parameter $J$. (a) Decay constant $\kappa$ for the decay of fluctuations with system size, and (b) prefactor $\mathcal{C}$ (see main text, in the formal limit $N\to \infty$). The quench assumed here jumps from a coupling $J_{\mathrm{pre}}$ to $J$.

Figure 4

Temporal variance ${\sigma }_A^2$ for $N=8$ and different prequench parameters. The squares show results from numerical exact diagonalization. Integrability is weakly broken by $J_{\mathrm{NNN}}/\mathrm{\Omega }=0.01$ (blue) or a longitudinal magnetic field $J_x/\mathrm{\Omega }=0.01$ (red). The black line shows the analytical predictions. With the pre- or postquench parameters in the ferromagnetic phase and nonzero $J_x$, stronger deviations due to the spontaneous symmetry breaking occur. For smaller fields, the agreement is restored (orange crosses) [$J_x/\mathrm{\Omega }=2\times {10}^{-4}$ in (a) and $J_x/\mathrm{\Omega }=5\times {10}^{-4}$ in (b)].