Fluctuation-Dissipation Theorem in an Isolated System of Quantum Dipolar Bosons after a Quench

We examine the validity of fluctuation-dissipation relations in isolated quantum systems taken out of equilibrium by a sudden quench. We focus on the dynamics of trapped hard-core bosons in one-dimensional lattices with dipolar interactions whose strength is changed during the quench. We find indications that fluctuation-dissipation relations hold if the system is nonintegrable after the quench, as well as if it is integrable after the quench if the initial state is an equilibrium state of a nonintegrable Hamiltonian. On the other hand, we find indications that they fail if the system is integrable both before and after quenching.

Figure 1

Correlation functions $C_{\mathrm{Fluc}}(t)$, $C_{\mathrm{Diss}}(t)$, and $C_{\mathrm{Appr}}(t)$ when the observable is $n_{j=L/2}$ vs time $t$. Results are shown for the three quenches (i)–(iii) (from top to bottom, respectively) explained in the text, and for $L=15$ (left panels) and 18 (right panels). Results for $L=12$ are presented in Ref. 26. The insets show normalized histograms of $C_{\mathrm{Fluc}}(t)$ (filled red bars) and $C_{\mathrm{Diss}}(t)$ (empty blue bars) calculated for 2000 data points between $t=0$ and 100.

Figure 2

Normalized variance of $C_{\mathrm{Fluc}}(t)-C_{\mathrm{Diss}}(t)$ and $C_{\mathrm{Fluc}}(t)-C_{\mathrm{Appr}}(t)$ vs the system size for the three quenches explained in the text [identified by $Q$ (i), $Q$ (ii), and $Q$ (iii)], where the normalization factor is the average variance of the two functions for which the differences are calculated, e.g., $\mathrm{Var}(C_{\mathrm{Fluc}}-C_{\mathrm{Diss}})/(1/2)[\mathrm{Var}(C_{\mathrm{Fluc}})+\mathrm{Var}(C_{\mathrm{Diss}})]$. The observable is $n_{j=L/2}$. The variances are calculated for 2000 points between $t=0$ and 100.

Figure 3

Absolute value of the off-diagonal matrix elements of ${\widehat{n}}_{j=L/2}$ and ${\widehat{n}}_{k=0}$ in the eigenenergy basis, in a narrow energy window around $E=E_{\mathrm{tot}}$ (with a width of 0.1) vs the eigenenergy difference $\omega =E_{\alpha }-E_{\beta }$. Results are shown for $L=15$ (left panels) and $L=18$ (right panels). (a),(b) and (c),(d) correspond to the final Hamiltonian in quenches (ii) and (iii), respectively. The green (light gray) symbols are the matrix elements of ${\widehat{n}}_{k=0}$ and the black ones of ${\widehat{n}}_{j=L/2}$. In (a) and (b), we have increased the size of the symbols for $n_{j=L/2}$ by a factor of 20 relative to those for $n_{k=0}$. To increase the resolution of the distribution of values in the case of $L=18$, where a very large number of data points exists, we plot only 1 out of every 10 points for $n_{k=0}$ in (b) and for both observables in (d). Lines are running averages for $n_{k=0}$ with a subset length of 50 for $L=15$ and 200 for $L=18$. Insets show the histograms of the relative differences between the $n_{k=0}$ data and running averages ($f_{\mathrm{avg}}$) with subset sizes of 1000 for $L=15$ and 10 000 for $L=18$. The relative difference is defined as $(|O_{\alpha \beta }|-f_{\mathrm{avg}})/f_{\mathrm{avg}}$.