Quantum chaos and ensemble inequivalence of quantum long-range Ising chains

We use large-scale exact diagonalization to study the quantum Ising chain in a transverse field with long-range power-law interactions decaying with exponent α. We numerically study various probes for quantum chaos and eigenstate thermalization on the level of eigenvalues and eigenstates. The level-spacing statistics yields a clear sign towards a Wigner-Dyson distribution and therefore towards quantum chaos across all values of ɑ>0. Yet, for ɑ<1 we find that the microcanonical entropy is nonconvex. This is due to the fact that the spectrum is organized in energetically separated multiplets for ɑ<1. While quantum chaotic behavior develops within the individual multiplets, many multiplets do not overlap and do not mix with each other, as we analytically and numerically argue. Our findings suggest that a small fraction of the multiplets could persist at low energies for ɑ<<1 even for large N, giving rise to ensemble inequivalence.