  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. 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 Page curve. To explain this observation we propose a
new measure of the effective spin interactions and obtain the corresponding
random matrix result. By this the deviations of the entanglement entropy can be
attributed to significantly different behavior of the $k$-spin interactions
compared with RMT.
