Characterizing quantum chaoticity of kicked spin chains

Quantum many-body systems are commonly considered as quantum chaotic if their spectral statistics, such as the level spacing distribution, agree with those of random matrix theory (RMT). Using the example of the kicked Ising chain we demonstrate that even if both level spacing distribution and eigenvector statistics agree well with random matrix predictions, the entanglement entropy deviates from the expected RMT behavior, i.e., the Page curve. To explain this observation we propose a quantity that is based on the effective Hamiltonian of the kicked system. Specifically, we analyze the distribution of the strengths of the effective spin interactions and compare them with analytical results that we obtain for circular ensembles. Thereby we group the effective spin interactions corresponding to the number $k$ of spins which contribute to the interaction. By this the deviations of the entanglement entropy can be attributed to significantly different behavior of the $k$-spin interactions compared with RMT.

Figure 1

Level spacing distribution, eigenvector statistics, and von Neumann entropy for the kicked spin chain with $L=12$. Left column show results for parameter set A, right column for parameter set B, see Table 1. Red dashed lines show COE results for level spacing distribution (3) and eigenvector statistics (6). Red circles show predictions for random states (8).

Figure 2

Distribution of the coefficients (16) for COE matrices of size $N=2^{12}$ using 10 realizations. Coefficients for $k=2,4,7,12$ spin interactions are shown in dark to light colors. For each interaction order we use 500 coefficients of each realization. The red dashed line shows the prediction (18) for the COE.

Figure 3

Distribution of the coefficients (16) for kicked spin chains of length $L=12$. Panel (a) shows result for parameter set A, panel (b) for parameter set B, see Table 1. Coefficients for $k=2,4,7,12$ spin interactions are shown in dark to light colors. For each interaction order 500 coefficients are used. The red dashed line shows the prediction (18) for the COE.

Figure 4

Distribution of $\tilde{\eta }=ln\left(\eta \right)$ for the kicked spin chain with $L=12$. Panel (a) shows the results for parameter set A and panel (b) for parameter set B, see Table 1. The red dashed lines show the COE result (A1).