Exact thermal energy eigenstates of nonintegrable 1D spin systems

Energy eigenstates of many-body quantum systems play a crucial role in various fields, including the study of thermalization in isolated systems. However, commonly used methods, such as matrix product state representations and the Bethe ansatz, are ineffective for constructing thermal energy eigenstates of realistic systems because of their large amount of entanglement and the nonintegrability of such systems. Here, we write down analytically a thermal energy eigenstate of a nonintegrable one-dimensional spin system. We focus on a class of volume-law entangled states, which we call entangled antipodal pair (EAP) states, and identify all Hamiltonians having an arbitrary EAP state as an eigenstate. Some of the Hamiltonians are rigorously shown to be nonintegrable. Furthermore, it is easy to show that EAP states are thermal in the sense of ``microscopic thermal equilibrium,'' as they are indistinguishable from the Gibbs state (at infinite temperature) with respect to all local observables. Our findings provide exact examples of thermal eigenstates of nonintegrable systems without relying on numerical approaches, which often suffer from finite-size effects, or using theoretically unjustified assumptions.