Onset of quantum chaos in one-dimensional bosonic and fermionic systems and its relation to thermalization

By means of full exact diagonalization, we study level statistics and the structure of the eigenvectors of one-dimensional gapless bosonic and fermionic systems across the transition from integrability to quantum chaos. These systems are integrable in the presence of only nearest-neighbor terms, whereas the addition of next-nearest-neighbor hopping and interaction may lead to the onset of chaos. We show that the strength of the next-nearest-neighbor terms required to observe clear signatures of nonintegrability is inversely proportional to the system size. Interestingly, the transition to chaos is also seen to depend on particle statistics, with bosons responding first to the integrability breaking terms. In addition, we discuss the use of delocalization measures as main indicators for the crossover from integrability to chaos and the consequent viability of quantum thermalization in isolated systems.

Figure 1

(Color online) Level spacing distribution for hard-core bosons averaged over all $k$’s in Table , for $L=24$, and $t^{\prime }=V^{\prime }$. For comparison purposes, we also present the Poisson and Wigner-Dyson distributions. Bottom right panel: energy difference between first excited state $E_1$ and ground state $E_0$ in the full spectrum times $L$, for $L=18$ (circles), 21 (squares), and 24 (triangles).

Figure 2

(Color online) As in Fig. 1 but for spinless fermions.

Figure 3

(Color online) Average $\alpha$ over all $k$’s; $t^{\prime }=V^{\prime }$. Top panel: difference $\alpha \left(\text{fermions}\right)-\alpha \left(\text{bosons}\right)$. Left bottom panel, bosons; right bottom panel, fermions. Semilogarithmic plot in main panels and linear plot in insets. Circles, $L=18$; squares, $L=21$; triangles, $L=24$.

Figure 4

(Color online) Peak position of the level spacing distribution averaged over all $k$’s; $t^{\prime }=V^{\prime }$. Top panel, difference between the peak position of bosons and fermions, $\text{peak}\left(\text{bosons}\right)-\text{peak}\left(\text{fermions}\right)$. Left bottom panel, bosons; right bottom panel, fermions. Semilogarithmic plot in main panels and linear plot in insets. Circles, $L=18$; squares, $L=21$; triangles, $L=24$. Dashed line indicates the peak position of $P_{WD}\left(s\right)$.

Figure 5

(Color online) Level number variance averaged over all $k$’s. In each panel, solid lines from top to bottom, $t^{\prime }=V^{\prime }=0,0.02,0.04,0.08,0.16,0.32,0.64$. Dashed line, GOE; dotted-dashed line, Poisson.

Figure 6

(Color online) Shannon entropy in the mean-field basis vs effective temperature for bosons, $L=24$, $k=2$, and $t^{\prime }=V^{\prime }$. The dashed line gives the GOE averaged value $S_{\text{GOE}}\sim \text{ln}\left(0.48D\right)$.

Figure 7

(Color online) As in Fig. 6 for fermions.

Figure 8

(Color online) Shannon entropy in the mean-field basis vs energy when the system is close to integrability (top panels) and in the chaotic limit (bottom panels); $k=2$ and $t^{\prime }=V^{\prime }$. Panels on the left, bosons; panels on the right, fermions. Curves from bottom to top: $L=18,21,24$.

Figure 9

(Color online) $T_{all}$ vs energy, where $T_{all}$ stands for the effective temperature computed considering the eigenvalues of all symmetry sectors; $L=24$. Curves from bottom to top: $t^{\prime }=V^{\prime }=0.00$, 0.02, 0.03, 0.04, 0.06, 0.08, 0.12, 0.16, 0.24, 0.32, 0.48, 0.64, 0.96, 1.28.

Figure 10

(Color online) Temperature difference vs $T_{all}$. $T_{diff}=100\left|T_{all}-T_{k=2}\right|/T_{all}$, where $T_{k=2}$ is computed considering only the eigenvalues from the $k=2$ sector. Left panels, bosons; right panels, fermions; $t^{\prime }=V^{\prime }$. Negligible differences are seen between the curves for $L=24$ [light gray (green online)] and $L=21$ [dark gray (red online)] when $T_{all}>2$, while the curve for $L=18$ (black) saturates at a higher level (especially noticeable for fermions).

Figure 11

(Color online) Shannon entropy in the mean-field basis vs energy for bosons; $L=24$, $k=2$, and $t^{\prime }=V^{\prime }$. The dashed line gives the GOE averaged value $S_{\text{GOE}}\sim \text{ln}\left(0.48D\right)$.

Figure 12

(Color online) Same as in Fig. 11 for fermions.

Figure 13

Inverse participation ratio in the mean-field basis vs energy for bosons; $L=24$, $k=2$, and $t^{\prime }=V^{\prime }$. The GOE result ${\text{IPR}}_{\text{GOE}}\sim D/3$ is beyond the chosen scale.

Figure 14

Same as in Fig. 13 for fermions.

Figure 15

(Color online) Shannon entropy in the $k$ basis vs energy for bosons; $L=24$, $k=2$, and $t^{\prime }=V^{\prime }$. The dashed line gives the GOE averaged value $S_{\text{GOE}}\sim \text{ln}\left(0.48D\right)$.

Figure 16

(Color online) Same as in Fig. 15 for fermions.