Stabilizer Thermal Eigenstates at Infinite Temperature

Abstract: Understanding how to analyze highly entangled thermal eigenstates is a central challenge in the study of quantum many-body systems. In this Letter, we introduce a stabilizer-based approach to construct analytically tractable energy eigenstates of nonintegrable many-body Hamiltonians. Focusing on zero-energy eigenstates at infinite temperature, we prove a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy $k$-body microscopic thermal equilibrium for any $k\ge4$. We further show that this bound is tight by explicitly constructing two-body nonintegrable Hamiltonians whose stabilizer eigenstates reproduce thermal expectation values for all two-body and all three-body observables. Finally, we identify the structural origin of this limitation by characterizing the conditions under which a stabilizer state can appear as a zero-energy eigenstate of a Hamiltonian, thereby revealing a fundamental constraint imposed by the few-body nature of interactions.