Onset of chaos and relaxation in isolated systems of interacting spins: Energy shell approach

We study the onset of chaos and statistical relaxation in two isolated dynamical quantum systems of interacting spins 1/2, one of which is integrable and the other chaotic. Our approach to identifying the emergence of chaos is based on the level of delocalization of the eigenstates with respect to the energy shell, the latter being determined by the interaction strength between particles or quasiparticles. We also discuss how the onset of chaos may be anticipated by a careful analysis of the Hamiltonian matrices, even before diagonalization. We find that despite differences between the two models, their relaxation processes following a quench are very similar and can be described analytically with a theory previously developed for systems with two-body random interactions. Our results imply that global features of statistical relaxation depend on the degree of spread of the eigenstates within the energy shell and may happen to both integrable and nonintegrable systems.

Figure 1

Absolute values of the matrix elements of model 1 (left panel) and model 2 with $\lambda =0.5$ (right panel) for $L=12$ (therefore $D_P\sim 250$) and $\mu =0.5$. The MF basis is ordered in energy. Only even states are considered. Light color indicates large values.

Figure 2

Information about the matrix elements of model 1 (two left columns) and model 2 (two right columns). The matrices are written in the MF basis, which is ordered from lowest to highest energy. The perturbation strength for each column is shown in the top panels. The top panels show the diagonal elements. The bottom panels show the average values of the absolute values of the off-diagonal elements vs the distance $k$ from the diagonal.

Figure 3

Connectivity of each line $n$ for model 1 (left) and model 2 (right).

Figure 4

Ratio of the average coupling strength $v_n$ to the mean level spacing $d_n$ between directly coupled states for each line $n$ of model 1 (left panels) and model 2 (right panels). The horizontal (green) line stands for $v_n/d_n=1$.

Figure 5

Density of states for model 1 (left panels) and model 2 (right panels); the bin size is equal to 0.1. The solid (black) line gives the best Gaussian fit: $\mu =0.1\to \langle E\rangle =0.034,\sigma =1.330$; $\mu =0.4\to \langle E\rangle =0.131,\sigma =1.375$; $\mu =1.5\to \langle E\rangle =0.363,\sigma =1.857$; $\lambda =0.1\to \langle E\rangle =0.157,\sigma =1.400$; $\lambda =0.4\to \langle E\rangle =0.051,\sigma =1.494$; and $\lambda =1.5\to \langle E\rangle =0.037,\sigma =1.920.$

Figure 6

The top panels show the level spacing distribution. The bottom panels show the parameter $\beta$ of the Brody distribution vs the perturbation strength. The left panels show model 1 and the right panels show model 2.

Figure 7

Examples of eigenstates from the center of the spectrum for model 1 (left) and model 2 (right). They become more extended from top to bottom.

Figure 8

Matrix of squared components of the eigenstates for model 1 (left), $\mu =0.5$ (top) and $\mu =1.5$ (bottom), and model 2 (right), $\lambda =0.5$ (top) and $\lambda =1.0$ (bottom). Only even states are shown, $L=12$. Light areas indicate large values.

Figure 9

Strength functions for model 1 (left) and model 2 (right) obtained by averaging over five even unperturbed states in the middle of the spectrum. The average is performed after shifting the center of SFs to zero. Circles give the fitting curves. The middle panels show the Breit-Wigner function with $\varepsilon +\delta =-0.015$, $\Gamma =0.302$ (left) and $\varepsilon +\delta =0.072$, $\Gamma =0.345$ (right). The bottom panels show the Gaussian function with $\langle E\rangle =-0.072$, $\sigma =1.322$ (left) and $\langle E\rangle =-0.022$, $\sigma =0.936$ (right). Solid curves correspond to the Gaussian form of the energy shells with $\sigma =0.090$ for $\mu =0.1$, $\sigma =0.359$ for $\mu =0.4$, $\sigma =1.345$ for $\mu =1.5$, $\sigma =0.103$ for $\lambda =0.1$, $\sigma =0.412$ for $\lambda =0.4$, and $\sigma =1.029$ for $\lambda =1.0$.

Figure 10

Eigenstates for model 1 (left) and model 2 (right) obtained by averaging over five even perturbed states in the middle of the spectrum. The average is performed after shifting the center of the EFs to zero. They are shown with filled curves. Solid curves correspond to the Gaussian form of the energy shells.

Figure 11

Shannon entropy for all eigenstates written in the MF basis for model 1 (left) and model 2 (right).

Figure 12

Shannon entropy for the strength functions written in the basis of the eigenstates for model 1 (left) and model 2 (right).

Figure 13

Overlaps of neighboring eigenstates for model 1 (left panels) and model 2 (right panels). Dark (Black) and light (red) points indicate eigenstates of even or odd parity. Horizontal (green) lines indicate the GOE prediction $\Omega =1/D$.

Figure 14

Shannon entropy vs time for model 1 (left) and model 2 (right). Circles stand for numerical data, dashed lines show the linear dependence (22), and solid curves correspond to Eq. (23). The horizontal (orange) solid lines represent the value of $S_{\mathrm{GOE}}\sim 6.58$.