Time fluctuations in isolated quantum systems of interacting particles

Numerically, we study the time fluctuations of few-body observables after relaxation in isolated dynamical quantum systems of interacting particles. Our results suggest that they decay exponentially with system size in both regimes, integrable and chaotic. The integrable systems considered are solvable with the Bethe ansatz and have a highly nondegenerate spectrum. This is in contrast with integrable Hamiltonians mappable to noninteracting ones. We show that the coefficient of the exponential decay depends on the level of delocalization of the initial state with respect to the energy shell.

Figure 1

Density of states $L=16,{\mathcal{S}}^z=0$. (a) $\Delta =0.5,\lambda =0$; (b) $\Delta =1,\lambda =0$; and (c) $\Delta =0.5,\lambda =1$.

Figure 2

Level spacing distribution for a single subspace: $L=18,{\mathcal{S}}^z=-3$, and eigenstates with even parity. For comparison, the Poisson and Wigner-Dyson distributions are shown with dashed lines. (a)–(e) have $\lambda =0$ and $\Delta =0.0,{10}^{-3},{10}^{-2},0.1$, and 0.5, respectively. (f) $\Delta =0.5$ and $\lambda =1$.

Figure 3

Distribution of the weights of the initial Néel state in the energy representation $L=16,{\mathcal{S}}^z=0$. The Hamiltonians and $E_{\mathrm{ini}}$ are as follows: (a) ${\widehat{H}}_{\Delta =1,\lambda =0}$ and $-3.750$, (b) ${\widehat{H}}_{\Delta =0.5,\lambda =0}$ and $-1.875$, (c) ${\widehat{H}}_{\Delta =1,\lambda =0.4}$ and $-2.350$, (d) ${\widehat{H}}_{\Delta =1,\lambda =1}$ and $-0.250$, and (e) ${\widehat{H}}_{\Delta =0.5,\lambda =1}$ and $-0.125$. The solid line corresponds to the energy shell: Gaussian of width $\delta E_{\mathrm{ini}}=1.936$.

Figure 4

Top panels: distribution of the weights of the initial state in the energy representation for ${\widehat{H}}_{\Delta =1,\lambda =0},L=16$, and ${\mathcal{S}}^z=0$. (a) $|\mathrm{DW}\rangle ,E_{\mathrm{ini}}=3.250$, and $\delta E_{\mathrm{ini}}=0.500$; (b) $|\mathrm{NS}\rangle ,E_{\mathrm{ini}}=-3.750$, and $\delta E_{\mathrm{ini}}=1.936$; (c) $|\mathrm{PP}\rangle ,E_{\mathrm{ini}}=-0.250$, and $\delta E_{\mathrm{ini}}=1.414$; and (d) $|{\xi }_{{\mathcal{S}}^z=0}\rangle ,E_{\mathrm{ini}}=-0.246$, and $\delta E_{\mathrm{ini}}=1.719$.

Figure 5

Logarithmic plot of the standard deviation of the time fluctuations of different observables vs $L$ for (circles) ${\widehat{H}}_{\Delta =1,\lambda =0}$, (squares) ${\widehat{H}}_{\Delta =0.5,\lambda =0}$, (plus) ${\widehat{H}}_{\Delta =1,\lambda =0.4}$, (up triangle) ${\widehat{H}}_{\Delta =1,\lambda =1}$, and (down triangle) ${\widehat{H}}_{\Delta =0.5,\lambda =1}$. $|\mathrm{ini}\rangle$ is the Néel state, [100, 500] is the time interval for the averages, and ${\sigma }_O^N={\sigma }_O/\overline{O}$. The solid lines correspond to logarithmic fits.

Figure 6

Logarithmic plots of the standard deviation of the time fluctuations for different observables vs $L$ for (circles) $|\mathrm{DW}\rangle$, (squares) $|\mathrm{NS}\rangle$, (plus) $|\mathrm{PP}\rangle$, (up triangle) $|{\xi }_{{\mathcal{S}}^z=0}\rangle$, and (down triangle) $|{\xi }_{2^L}\rangle$. The solid lines correspond to logarithmic fits, and ${\sigma }_O^N={\sigma }_O/\overline{O}$. All panels: ${\widehat{H}}_{\Delta =1,\lambda =0}$ and averages performed in the time interval [100, 500], except for the domain wall state which used $[5\times {10}^2,5\times {10}^3]$.

Figure 7

Left panels: $1-R^2$ values for the (square) exponential and (circle) power-law fits of the decay of the temporal fluctuations of ${\widehat{C}}_{L/2,L/2+1}^x$ with system size. The initial states are (a) $|\mathrm{NS}\rangle$ and (b) $|{\xi }_{{\mathcal{S}}^z=0}\rangle$. Right panel: logarithmic plots of the standard deviation of the time fluctuations of $\widehat{\mathrm{KE}}$ vs $L$ for (c) $|\mathrm{DW}\rangle$ with ${\widehat{H}}_{\Delta =1,\lambda =0}$, (d) $|\mathrm{NS}\rangle$ with ${\widehat{H}}_{\Delta =0.5,\lambda =0}$, and (e) $|{\xi }_{{\mathcal{S}}^z=0}\rangle$ with ${\widehat{H}}_{\Delta =0.5,\lambda =0}$. The solid line corresponds to the fitting for the exponential decay, and the dashed line corresponds to the power-law decay. The averages are performed in the time interval [100, 500], except for the domain wall state, which used $[5\times {10}^2,5\times {10}^3]$.

Figure 8

Time dependence of the kinetic energy for $|\mathrm{ini}\rangle =|\mathrm{DW}\rangle$. (a) ${\widehat{H}}_{\Delta =1,\lambda =0}$ and $L=24,18,12$ from top to bottom. (b) ${\widehat{H}}_{\Delta =1,\lambda =0},{\widehat{H}}_{\Delta =0.5,\lambda =0}$, and ${\widehat{H}}_{\Delta =1.5,\lambda =0}$ from top to bottom, $L=22$.

Figure 9

Relaxation of the structure factor in the $z$ direction; $L=22,{\mathcal{S}}^z=0$. Momentum: (black circles) $k=2\pi /11$, (red squares) $3\pi /11$, (green up triangles) $4\pi /11$, and (blue down triangles) $5\pi /11$. Top panels: $|\mathrm{ini}\rangle =|\mathrm{DW}\rangle$, (a) ${\widehat{H}}_{\Delta =1,\lambda =0}$ and (b) ${\widehat{H}}_{\Delta =0.5,\lambda =0}$. Middle panels: $|\mathrm{ini}\rangle =|\mathrm{DW}\rangle$, (c) ${\widehat{H}}_{\Delta =1,\lambda =1}$ and (d) ${\widehat{H}}_{\Delta =0.5,\lambda =1}$. Bottom panels: ${\widehat{H}}_{\Delta =1,\lambda =0}$, (e) $|\mathrm{ini}\rangle =|\mathrm{NS}\rangle$ and (f) $|{\xi }_{{\mathcal{S}}^z=0}\rangle$.