Mean-field approach to midspectrum eigenstates of long-range interacting quantum systems

We study the equilibrium properties of the spin-1/2 / 2 XY chain with an infinite-range transverse interaction. At zero temperature, competition between the XY- and z-ordered phases induced by the infinite-range interactions gives rise to a first-order transition upon increasing the transverse coupling. We show that the two gapless points of the XY model behave in fundamentally different ways: isotropic spin chains experience a first-order transition at finite coupling; maximal anisotropic chains overcome a nonalgebraic phase transition at zero coupling strength. The phase diagram depicts a first-order reentrant transition that turns second order along a tricritical line separating a paramagnetic phase from an ordered one at finite temperature. The mean-field approach captures the local properties of the eigenstates and reveals the appearance of a magnetization gap in the spectrum. Global properties, e.g., entanglement entropy, are well approximated only at spectral boundaries. The mean entanglement entropy and the level-spacing ratio deviate from the Gaussian results, revealing the interacting nature of the problem.