Eigenstate Thermalization and Spontaneous Symmetry Breaking in the   One-Dimensional Transverse-Field Ising Model with Power-Law Interactions
  We study eigenstate thermalization and related signatures of quantum chaos in the one-dimensional ferromagnetic transverse-field Ising model with power-law interactions. The presence of long-range interactions allows for a finite-temperature phase transition despite the one-dimensional geometry of the model. Unlike previous studies of eigenstate thermalization in non-disordered systems with finite temperature phase transitions, our model possesses sufficiently many energy eigenstates below the critical energy density to allow us to make a definitive statement about the presence of eigenstate thermalization and chaotic level statistics in the broken-symmetry phase. 
Keith R. Fratus and Mark Srednicki
cond-mat.stat-mech