Abstract
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 $\alpha$. 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 $\alpha>0$. Yet, for $\alpha<1$ we find that the microcanonical entropy is nonconvex. This is due to the fact that the spectrum is organized in energetically separated multiplets for $\alpha<1$. While quantum chaotic behaviour develops within the individual multiplets, many multiplets don't overlap and don't 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 $\alpha\ll 1$ even for large $N$, giving rise to ensemble inequivalence.
Highlights
Thermalization in classical Hamiltonian systems is well understood in terms of chaotic dynamics and the related essentially ergodic exploration of the phase space [1,2,3]
From the quantum point of view the physical mechanism is quite different, with the eigenstates of the Hamiltonian behaving similar to the eigenstates of a random matrix with the additional property that they appear thermal from the point of view of local measurements
This is the paradigm of eigenstate thermalization (ETH) introduced in Refs. [4,5,6,7]
Summary
Thermalization in classical Hamiltonian systems is well understood in terms of chaotic dynamics and the related essentially ergodic exploration of the phase space [1,2,3]. A significant exception is [72], which showed quantum chaos at low energies for α = 1.5 in the clean ferromagnetic spin-1/2 Ising model with long-range power-law interactions The dynamics of this model has been intensively studied, mostly in connection with the persistence of long-range order in the asymptotic state of the dynamics [38,51,52,53,54,56,57,58,59,60], for different values of α and small transverse field, but it is not known if this asymptotic state is thermal.
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