Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems

We study how the proximity to an integrable point or to localization as one approaches the atomic limit, as well as the mixing of symmetries in the chaotic domain, may affect the onset of thermalization in finite one-dimensional systems. We consider systems of hard-core bosons at half-filling with nearest-neighbor hopping and interaction, and next-nearest-neighbor interaction. The latter breaks integrability and induces a ground-state superfluid to insulator transition. By full exact diagonalization, we study chaos indicators and few-body observables. We show that when different symmetry sectors are mixed, chaos indicators associated with the eigenvectors, contrary to those related to the eigenvalues, capture the onset of chaos. The results for the complexity of the eigenvectors and for the expectation values of few-body observables confirm the validity of the eigenstate thermalization hypothesis in the chaotic regime, and therefore the occurrence of thermalization. We also study the properties of the off-diagonal matrix elements of few-body observables in relation to the transition from integrability to chaos and from chaos to localization.

Figure 1

(Color online) Top panels: Level spacing distributions. Results are shown for the average between sectors $k=0$ and $k=L/2$ (black solid line), the average between all other $k$-sectors [light (red) solid line], the Poisson distribution, and the Wigner-Dyson (W-D) distribution. Bottom panels: inverse participation ratio in momentum space for sectors $k=0$ and $k=L/2$ [black dots] and all other $k$-sectors [light (red) dots]. In all cases the system size is $L=22$.

Figure 2

(Color online) Structural entropy in the momentum basis vs energy per site for all $k$-sectors. Black dots: $L=22$, light (green) dots: $L=20$. The values of $V^{\prime }$ are indicated in the panels. The solid line correspond to $S_{\text{str}}^{\text{GOE}}\approx 0.3646$.

Figure 3

(Color online) Standard deviation of $S_{\text{str}}$ vs $V^{\prime }$ for eigenstates in the middle of the spectrum with energies varying from $-5$ to 5. The inset shows the gap (between the lowest energy states) times $L$ vs $V^{\prime }$. Results are shown for lattices with $L=18$, $L=20$, and $L=22$ sites.

Figure 4

(Color online) Eigenstate expectation values of $\widehat{K}$ and $\widehat{I}$ vs energy per site for the full spectrum, including all momentum sectors. Black dots: $L=22$, light (green) dots: $L=20$.

Figure 5

(Color online) Eigenstate expectation values of $\widehat{n}\left(k\right)$ and $\widehat{N}\left(k\right)$ vs energy per site for the full spectrum, including all momentum sectors. Black dots: $L=22$, light (green) dots: $L=20$.

Figure 6

(Color online) Average relative deviation of eigenstate expectation values with respect to the microcanonical result as a function of $V^{\prime }$. The average is performed over all eigenstates (including all momentum sectors) that lie within the window $\left[E-\Delta E,E+\Delta E\right]$, with $\Delta E=0.4$. Results are shown for lattices with $L=18$, $L=20$, and $L=22$ sites. The effective temperature is $T=5$, which for $L=22$ corresponds to $E=-2.9680$ for $V^{\prime }=0$, $E=-4.3126$ for $V^{\prime }=2$, $E=-7.7462$ for $V^{\prime }=4$, $E=-13.1636$ for $V^{\prime }=6$, $E=-20.4001$ for $V^{\prime }=8$, and $E=-29.2205$ for $V^{\prime }=10$.

Figure 7

(Color online) Average relative deviation of eigenstate expectation values of $\widehat{n}\left(k\right)$ with respect to the microcanonical prediction as a function of the effective temperature $T$ of the eigenstates. Panels (a) and (d) depict results for $L=20$ and $L=22$. In panels (b) and (c) only $L=22$ is shown, since the results for $L=20$ are very similar. The average is performed over all eigenstates (including all momentum sectors) that lie within the window $\left[E-\Delta E,E+\Delta E\right]$, with $\Delta E=0.1$.

Figure 8

(Color online) Normalized extremal fluctuations of eigenstate expectation values as a function of $V^{\prime }$. All eigenstates from all momentum sectors that lie within the window $\left[E-\Delta E,E+\Delta E\right]$, with $\Delta E=0.4$ are taken into account. Results are shown for lattices with $L=18$, $L=20$, and $L=22$ sites. The effective temperature is $T=5$, which for $L=22$ corresponds to $E=-2.9680$ for $V^{\prime }=0$, $E=-4.3126$ for $V^{\prime }=2$, $E=-7.7462$ for $V^{\prime }=4$, $E=-13.1636$ for $V^{\prime }=6$, $E=-20.4001$ for $V^{\prime }=8$, and $E=-29.2205$ for $V^{\prime }=10$.

Figure 9

(Color online) Matrix elements of $\widehat{n}\left(k=0\right)$ (top panels) and $\widehat{N}\left(k=\pi \right)$ [bottom panels]; $L=22$, $k=0$. We select as the central state the one that has the closest energy to the energy in the canonical ensemble corresponding to an effective temperature $T=5.0$. For the cases depicted in the figure this corresponds to $E=-2.9680$ for $V^{\prime }=0$, $E=-3.3748$ for $V^{\prime }=1$, $E=-4.3126$ for $V^{\prime }=2$, $E=-13.1636$ for $V^{\prime }=6$, and $E=-29.2205$ for $V^{\prime }=10$. A total of 500 eigenstates around the central one are considered; both $\alpha$ and $\beta$ run from $-250$ to 250.

Figure 10

(Color online) Matrix elements of $\widehat{K}$ [top panels] and $\widehat{I}$ [bottom panels]; $L=22$, $k=0$. The data correspond to the same eigenstates of the Hamiltonian as in Fig. 9. In the bottom right panel (for $V^{\prime }=10$), the diagonal elements of $\widehat{I}$ are beyond the interval presented in the plot.

Figure 11

(Color online) Shannon entropy in the $k$-basis vs energy per site for all $k$-sectors; $L=22$. Black dots: sectors $k=0$ and $k=L/2$; light (red) dots: all other $k$-sectors.

Figure 12

(Color online) Shannon entropy in the $k$-basis vs energy for all $k$-sectors; $L=20$. Black dots: sectors $k=0$ and $k=L/2$; light (red) dots: all other $k$-sectors.

Figure 13

(Color online) Average relative deviation of eigenstate expectation values with respect to the microcanonical result as a function of $V^{\prime }$; $L=22$. The average is performed over all eigenstates (including all momentum sectors) that lie within the window $\left[E-\Delta E,E+\Delta E\right]$. The ten curves shown in each panel are obtained for $\Delta E=0.05,0.1,0.15,\dots 0.5$; the thick dashed line corresponds to $\Delta E=0.4$, which is the value considered in Fig. 6. The effective temperature is $T=5$.