Prethermalization and Thermalization in Isolated Quantum Systems

Prethermalization has been extensively studied in systems close to integrability. We propose a more general, yet conceptually simpler, setup for this phenomenon. We consider a—possibly nonintegrable—reference dynamics, weakly perturbed so that the perturbation breaks at least one conservation law of the reference dynamics. We argue then that the evolution of the system proceeds via intermediate (generalized) equilibrium states of the reference dynamics. The motion on the manifold of equilibrium states is governed by an autonomous equation, flowing towards global equilibrium in a time of order $g^{-2}$, where $g$ is the perturbation strength. We also describe the leading correction to the time-dependent reference equilibrium state, which is, in general, of order $g$. The theory is well confirmed in numerical calculations of model Hamiltonians, for which we use a numerical linked cluster expansion and full exact diagonalization.

Popular Summary

Prethermalization has emerged as a ubiquitous phenomenon in 1D ultracold quantum gases: Some systems far from equilibrium quickly reach a long-lived state that is not quite in thermal equilibrium before slowly settling into true thermal equilibrium. This two-step process remains challenging to understand in isolated many-body quantum systems in the presence of strong interactions. We put forward a very general and conceptually simple theoretical framework to understand prethermalization in those systems.

We argue that the crucial ingredient for prethermalization to occur is the presence of a perturbation that breaks at least one conservation law of a “nearby” unperturbed system. If one takes the perturbed system far from equilibrium, the ensuing fast relaxation dynamics that occurs on short timescales is close to that of the unperturbed system, and the slow approach to thermal equilibrium that occurs on long timescales can be well described by equilibrium states of the unperturbed system. These equilibrium states are characterized by the slowly changing conserved quantities that are weakly broken. We carefully test our theoretical predictions using numerical experiments in quantum lattice models.

Our theoretical framework and numerical experiments are a step forward in bridging the gap between our understanding of quantum microscopic theory and experimental observations. Specifically, we show that exponential relaxation can occur under purely unitary dynamics, and we provide explicit predictions for the relaxation rates that match the ones observed in the numerical experiments.

Figure 1

Dynamics of the particle filling after quantum quenches ${\widehat{H}}_I\to {\widehat{H}}_1$. (a) Dynamics of the particle filling $n(\tau )$ and (b) dynamics of the “distance” to equilibrium ${\delta }_l^{\mathrm{DE}}[n(\tau )]$; see Eq. (52). NLCE results are shown for $l=17$ [NLCE-17, see legend in (a)] and $l=16$ (NLCE-16, dotted lines). Straight lines in (b) depict fits to the results for $l=17$ in the interval $\tau \in [1,16]$ and to exponential functions $\propto \exp [-{\mathrm{\Gamma }}_{17}^{\mathrm{NLCE}}(g_1)\tau ]$. (c) Thermalization rates ${\mathrm{\Gamma }}^{\mathrm{NLCE}}(g_1)$ (filled symbols) obtained from fits as the ones in (b), for $l=17$ in the interval $\tau \in [1,16]$ (NLCE-17), and for $l=16$ in the interval $\tau \in [1,6]$ (NLCE-16), reported for $g_1\in [0.03,0.12]$. Error bars show 95% confidence bounds for the fits. The straight line is the result of a fit to ${\mathrm{\Gamma }}_{17}^{\mathrm{NLCE}}(g_1)\propto g_1^{\beta }$ for $g_1\in [0.03,0.06]$. The open symbols show the rates ${\mathrm{\Gamma }}^{\text{Fermi}}(g_1)$ obtained evaluating Fermi’s golden rule [see Eqs. (53) and (54)] using full exact diagonalization in chains with $L=18$ (Fermi-18) and $L=17$ (Fermi-17), and periodic boundary conditions. The error bars show the standard deviation from averages over different choices of $\mathrm{\Delta }E$ and $\tau$ (see Appendix pp4).

Figure 2

Dynamics of the one-body nearest neighbor correlation $k(\tau )$ when $g_{\alpha }=0$ after the quench. Results are shown for the 18th (NLCE-18) and 19th (NLCE-19) orders of the NLCE. We also report, as horizontal lines, the results for the 19th order of the diagonal ensemble (${\mathrm{DE}}_{19}$) and for the 19th order of the grand canonical ensemble (${\mathrm{GE}}_{19}$).

Figure 3

Dynamics of the one-body nearest neighbor correlation $k(\tau )$ for (a) $g_1=0.06$, (b) $g_1=0.12$, (c) $g_2=0.06$, and (d) $g_2=0.12$. Results are shown for the 16th (NLCE-16) and 17th (NLCE-17) orders of the NLCE, both for the dynamics [see legends in (a)] and for the dynamics in the projected basis of ${\widehat{H}}_0$ [see legends in (c)]. The horizontal lines show the results for the grand canonical ensemble corresponding to the original dynamics (GE) and to the projected dynamics (${\mathrm{GE}}_0$), both evaluated at the 17th order of the NLCE.

Figure 4

Difference in the equilibrium (grand canonical ensemble) results between the actual dynamics (GE) and the projected dynamics (${\mathrm{GE}}_0$) for the one-body nearest neighbor correlation ($k_{\mathrm{GE}}-k_{{\mathrm{GE}}_0})$ evaluated at the 17th order of the NLCE. (a) $g_1{\widehat{V}}_1$ perturbation (symbols), with linear and linear plus quadratic fits (lines). (b) $g_2{\widehat{V}}_2$ perturbation (symbols), with quadratic and quadratic plus cubic fits (lines).

Figure 5

Momentum distribution function $m_k(\tau )$ [see Eq. (50)] at four times after quenches in which ${\widehat{H}}_I\to {\widehat{H}}_1$ with $g_1=0.03$ [(a)–(d)] and $g_1=0.12$ [(e)–(h)]. $m_k(\tau )$ is evaluated at $\tau =0$ [(a),(e)], $\tau =2$ [(b),(f)], $\tau =10$ [(c),(g)], and $\tau =100$ [(d),(h)]. Results are reported for the 15th (NLCE-15) and 16th (NLCE-16) orders of the NLCE. We show $m_k(\tau )$ for the dynamics [see the legends in (a)], and for the projected dynamics in the basis of ${\widehat{H}}_0$ [see the legends in (b)]. The predictions of the diagonal ensemble (DE) and the grand canonical ensemble (GE) for the dynamics, and of the grand canonical ensemble for the projected dynamics (${\mathrm{GE}}_0$), all evaluated at the 16th order of the NLCE, are shown in (d) and (h) [see the legends in (d)]. The inset in (h) shows the predictions of the latter three ensembles about $k=\pi$.

Figure 6

Dynamics of the momentum distribution function $m_k(\tau )$ [Eq. (50)] after quenches ${\widehat{H}}_I\to {\widehat{H}}_1$. (a) Dynamics of the “distance” to equilibrium ${\delta }^{\mathrm{DE}}[m(\tau )]$; see Eq. (57). NLCE results are shown for $l=16$ [NLCE-16] and $l=15$ (NLCE-15, dotted lines). Straight lines in (a) depict fits to the results for $l=16$ in the interval $\tau \in [2,6]$ and to exponential functions $\propto \exp [-{\mathrm{\Gamma }}_{16}^{\mathrm{NLCE}}(g_1)\tau ]$. (b) Thermalization rates ${\mathrm{\Gamma }}^{\mathrm{NLCE}}(g_1)$ of $m_k(\tau )$ (filled symbols) obtained from fits as the ones in (a), for $l=16$ in the interval $\tau \in [2,6]$ (NLCE-16), and for $l=15$ in the interval $\tau \in [2,5]$ (NLCE-16), reported for $g_1\in [0.03,0.12]$. Error bars show 95% confidence bounds for the fits. The straight line is the result of a fit to ${\mathrm{\Gamma }}_{16}^{\mathrm{NLCE}}(g_1)\propto g_1^{\beta }$ for $g_1\in [0.03,0.06]$. The open circles show the rates ${\mathrm{\Gamma }}^{\text{Fermi}}(g_1)$ [also reported in Fig. 1] obtained by evaluating Fermi’s golden rule [see Eqs. (53) and (54)] using full exact diagonalization in chains with $L=18$ (Fermi-18) and periodic boundary conditions. The error bars show the standard deviation from averages over different choices of $\mathrm{\Delta }E$ and $\tau$ (see Appendix pp4).

Figure 7

Dynamics after quantum quenches ${\widehat{H}}_I\to {\widehat{H}}_2$, for $t_I^{\prime }=V_I^{\prime }=0$ in ${\widehat{H}}_I$ and $t^{\prime }=V^{\prime }=0$ in ${\widehat{H}}_2$ (integrable reference dynamics). Main panel: Thermalization rates ${\mathrm{\Gamma }}^{\mathrm{NLCE}}(g_2)$ for the particle filling dynamics (filled symbols) obtained from fits as the ones shown in the upper inset. The straight line is the result of a fit to ${\mathrm{\Gamma }}_{17}^{\mathrm{NLCE}}(g_2)\propto g_2^{\beta }$. The open symbols show the rates ${\mathrm{\Gamma }}^{\text{Fermi}}(g_2)$, see Eqs. (53) and (54), obtained using full exact diagonalization (see Appendix pp4). Upper inset: ${\delta }_l^{\mathrm{DE}}[n(\tau )]$, see Eq. (52), versus $\tau$. The straight lines depict fits to exponential functions $\propto \exp [-{\mathrm{\Gamma }}_{17}^{\mathrm{NLCE}}(g_2)\tau ]$. The legends in the main panel and in the upper inset follow Fig. 1, and the fits are done in the same intervals as in Fig. 1. Lower inset: Dynamics, and projected dynamics in the basis of ${\widehat{H}}_0$, of $k(\tau )$ for $g_2=0.12$. The horizontal line shows the result for the grand canonical ensemble corresponding to the original dynamics. The legends are identical to those in Fig. 3.

Figure 8

Projected dynamics of $m_k(\tau )$ in the basis of ${\widehat{H}}_0$ at three times after the quench ${\widehat{H}}_I\to {\widehat{H}}_1$, with $g_1=0.12$. We compare results for the projected dynamics within the diagonal ensemble (Projected DE), also reported in Fig. 5, with those obtained within the grand canonical ensemble (Projected GE). The temperature and the chemical potential of the grand canonical ensemble are set so that the energy density $e_0$ and the particle filling $n$ in this ensemble match those in the diagonal ensemble. The results reported are for the 16th order of the NLCE.

Figure 9

Time evolution of the filling $n(\tau )$, for $g_1=0.12$, obtained in the last four orders $l$ of the NLCE (NLCE-$l$ in the legends), and in the three largest chains with $L$ sites and periodic boundary conditions solved using full exact diagonalization (ED-$L$ in the legends). The inset shows an enlargement of the NLCE results between $\tau =7.5$ and 15.

Figure 10

Relative differences, defined in Eqs. (c1)–(c3), for $g_1=0.12$. The exact diagonalization results are obtained in chains with periodic boundary conditions.

Figure 11

Time evolution of the one-body nearest neighbor correlation $k(\tau )$ obtained in the last three orders $l$ of the NLCE (NLCE-$l$ in the legends), and in the three largest chains with $L$ sites and periodic boundary conditions diagonalized (ED-$L$ in the legends). Results for the diagonal ensemble are shown for the last order of the NLCE (${\mathrm{DE}}_{\mathrm{NLCE}\text{−}l}$) and the largest chain diagonalized (${\mathrm{DE}}_{\mathrm{ED}\text{−}L}$), while results for the grand canonical ensemble are shown for the last order of the NLCE (${\mathrm{GE}}_{\mathrm{NLCE}\text{−}l}$), which are converged to machine precision. (a) Quenches ${\widehat{H}}_I\to {\widehat{H}}_0$, for $l=17$, 18, and 19 (NLCE-17, NLCE-18, and NLCE-19), and for $L=18$, 19, and 20 (ED-18, ED-19, and ED-20), along with the diagonal ensembles for $l=19$ (${\mathrm{DE}}_{\mathrm{NLCE}\text{−}19}$) and $L=20$ (${\mathrm{DE}}_{\mathrm{ED}\text{−}20}$), and the grand canonical ensemble for $l=19$ (${\mathrm{GE}}_{\mathrm{NLCE}\text{−}19}$). (b) Quenches ${\widehat{H}}_I\to {\widehat{H}}_1$, in which $g_1=0.12$, for $l=15$, 16, 17 (NLCE-15, NLCE-16, NLCE-17), and for $L=17$, 18, 19 (ED-17, ED-18, ED-19), along with the diagonal ensembles for $l=17$ (${\mathrm{DE}}_{\mathrm{NLCE}\text{−}17}$) and $L=18$ (${\mathrm{DE}}_{\mathrm{ED}\text{−}18}$), and the grand canonical ensemble for $l=17$ (${\mathrm{GE}}_{\mathrm{NLCE}\text{−}17}$). Higher orders in the NLCE, and larger chains in exact diagonalization, are calculated in (a) than in (b) thanks to particle-number conservation in the former.

Figure 12

$f_{\mathrm{\Delta }E}(\tau )$, see Eq. (d1), evaluated for a wide range of energy windows $\mathrm{\Delta }E$ for quenches ${\widehat{H}}_I\to {\widehat{H}}_1$ [see Eq. (36)] with $g_1=0.03$, 0.06, and 0.12. $f_{\mathrm{\Delta }E}(\tau )$ is computed using full exact diagonalization of chains with $L=18$ (see legends) and $L=17$ (dotted lines) and periodic boundary conditions. The vertical lines delimit the range $\mathrm{\Delta }E/L\in [0.03,0.09]$ in which $f_{\mathrm{\Delta }E}(\tau )$ is approximately constant.

Figure 13

${\mathrm{\Gamma }}_{\mathrm{\Delta }E,\tau }(g_1)$ [see Eq. (d2)] evaluated for quenches ${\widehat{H}}_I\to {\widehat{H}}_1$ [see Eq. (36)] with $g_1=0.03$, 0.06, and 0.12. The results reported are from full exact diagonalization of chains with $L=18$ and periodic boundary conditions. The average thermalization rate (horizontal line) is obtained by averaging ${\mathrm{\Gamma }}_{\mathrm{\Delta }E,\tau }(g_1)$ over $\tau =1,1.5,\dots ,5$ (nine values) and over $\mathrm{\Delta }E/L=0.03,0.032,\dots ,0.09$ (31 values), for a total of 279 values of ${\mathrm{\Gamma }}_{\mathrm{\Delta }E,\tau }(g_1)$ entering each average reported.