Initial-state dependence of the quench dynamics in integrable quantum systems. III. Chaotic states

We study sudden quantum quenches in which the initial states are selected to be either eigenstates of an integrable Hamiltonian that is nonmappable to a noninteracting one or a nonintegrable Hamiltonian, while the Hamiltonian after the quench is always integrable and mappable to a noninteracting one. By studying weighted energy densities and entropies, we show that quenches starting from nonintegrable (chaotic) eigenstates lead to an “ergodic” sampling of the eigenstates of the final Hamiltonian, while those starting from the integrable eigenstates do not (or at least it is not apparent for the system sizes accessible to us). This goes in parallel with the fact that the distribution of conserved quantities in the initial states is thermal in the nonintegrable cases and nonthermal in the integrable ones, and means that, in general, thermalization occurs in integrable systems when the quench starts form an eigenstate of a nonintegrable Hamiltonian (away from the edges of the spectrum), while it fails (or requires larger system sizes than those studied here to become apparent) for quenches starting at integrable points. We test those conclusions by studying the momentum distribution function of hard-core bosons after a quench.

Figure 1

Coarse-grained plot of the weights in the DE [(a)–(c) and (e)–(g)], and in the CE [(d) and (h)], $|C_{\alpha }{|}^2$ and $e^{-E_{\alpha }/T_{\mathrm{CE}}}/Z_{\mathrm{CE}}$, respectively. (a)–(c) Results for quench type I for different values of $V$. (e)–(g) Results for quench type II with different values of $t^{\prime }=V^{\prime }$. (d) and (h) Weights in the CE, which correspond to effective temperatures $T_{\mathrm{CE}}=2.006$ ($E_I=-4.976$) and $T_{\mathrm{CE}}=1.989$ ($E_I=-5.011$), obtained for the quenches with $V=1.28$ and $t^{\prime }=V^{\prime }=1.28$, respectively. All results are for $L=24,N=8$, and the temperatures are $T_{\mathrm{CE}}=2.00\pm 0.01$. The color scale indicates the number of states per unit area in the plot.

Figure 2

Weighted energy density functions $P\left(E\right)$ for both the DE and the CE. (a)–(c) Results for quenches type I. (e)–(f) Results for quenches type II. The dashed (blue) lines depict the energy shell, where (a) $E_I=-4.99,\delta E=0.159$, (b) $E_I=-4.99,\delta E=0.571$, (c) $E_I=-4.98,\delta E=1.08$, (d) $E_I=-5.00,\delta E=0.253$, (e) $E_I=-5.00,\delta E=0.936$, and (f) $E_I=-5.01,\delta E=2.66$. Solid lines show the results of the fits of $P\left(E\right)$ to the most appropriate functional form. Thin lines are fits to a Breit-Wigner function with (a) $\overline{E}=-5.01,\Gamma =0.042$, (b) $\overline{E}=-5.04,\Gamma =0.101$, (c) $\overline{E}=-5.17,\Gamma =0.567$, (d) $\overline{E}=-4.99,\Gamma =0.187$, while thick lines are fits to a Gaussian with (e) $\overline{E}=-5.16,\Delta =0.650$, and (f) $\overline{E}=-5.70,\Delta =2.31$. In all cases $L=24$ and $N=8$.

Figure 3

$P\left(\epsilon \right)$ for (a)–(c) quenches type I and (d)–(f) quenches type II. They were obtained by averaging over the nine even states $|{\psi }_{\alpha }\rangle$ that are closest in energy to $E_I$. Before averaging, we shift ${\epsilon }_S$ of each distribution to the one obtained for the state $|{\psi }_{\alpha }\rangle$ that is closest in energy to $E_I$. The dashed (blue) lines depict the energy shell, where (a) ${\epsilon }_S=-3.72,\delta \epsilon =0.139$, (b) ${\epsilon }_S=-3.66,\delta \epsilon =0.557$, (c) ${\epsilon }_S=-3.59,\delta \epsilon =1.11$, (d) ${\epsilon }_S=-3.76,\delta \epsilon =0.221$, (e) ${\epsilon }_S=-3.83,\delta \epsilon =0.885$, and (f) ${\epsilon }_S=-4.10,\delta \epsilon =3.54$. Solid lines show the results of the fits of $P\left(\epsilon \right)$ to the most appropriate functional form. Thin lines are fits to a Breit-Wigner function with (a) $\overline{\epsilon }=-3.70,\Gamma =0.074$, (b) $\overline{\epsilon }=-3.79,\Gamma =0.126$, (c) $\overline{\epsilon }=-3.72,\Gamma =0.308$, (d) $\overline{\epsilon }=-3.82,\Gamma =0.94$, while thick lines are fits to a Gaussian with (e) $\overline{\epsilon }=-3.95,\Delta =0.578$, and (f) $\overline{\epsilon }=-4.37,\Delta =3.54$. In all cases $L=18$ and $N=6$.

Figure 4

$S/L$ for the DE, CE, GGE, and GE in systems with $L=15$, 18, 21, and 24. Results are presented for (a)–(d) quenches type I and (e)–(h) quenches type II.

Figure 5

$(S_{\mathrm{GE}}-S_{\mathrm{DE}})/L$ as a function of (a) $V$ for quench type I and (c) $t^{\prime }=V^{\prime }$ for quench type II. $(S_{\mathrm{GE}}-S_{\mathrm{GGE}})/L$ as a function of (b) $V$ for quench type I and (d) $t^{\prime }=V^{\prime }$ for quench type II. Four different system sizes are presented for comparison.

Figure 6

Distribution of conserved quantities $\langle {\Psi }_I\left|{\widehat{I}}_j\right|{\Psi }_I\rangle$ in the initial state (same as the GGE) and in the GE. Results are presented for (a)–(d) quenches type I, and (e)–(h) quenches type II. In all cases, $L=24$ and $N=8$.

Figure 7

Relative integrated difference of the conserved quantities in the initial state (in the GGE) and the GE, $\Delta I$, as a function of (a) $V$ for quench type I and (b) $t^{\prime }=V^{\prime }$ for quench type II. Results are presented for four system sizes.

Figure 8

Momentum distribution functions in the DE, GGE, and GE. Results are presented for (a)–(d) quenches type I, and (e)–(h) quenches type II. In all cases, $L=24$ and $N=8$.

Figure 9

Relative integrated difference $\Delta m_{\mathrm{DE}}$ as a function of (a) $V$ for quench type I and (b) $t^{\prime }=V^{\prime }$ for quench type II. Results are presented for four system sizes.