Numerical evidence for the non-Abelian eigenstate thermalization hypothesis

The eigenstate thermalization hypothesis (ETH) explains how generic quantum many-body systems thermalize internally. It implies that local operators' time-averaged expectation values approximately equal their thermal expectation values, regardless of microscopic details. The ETH's range of applicability therefore impacts theory and experiments. Murthy textit{et al.} recently showed that non-Abelian symmetries conflict with the ETH. Such symmetries have excited interest in quantum thermodynamics lately, as they are equivalent to conserved quantities that fail to commute with each other and noncommutation is a quintessentially quantum phenomenon. Murthy textit{et al.} proposed a non-Abelian ETH, which we support numerically. The numerics model a one-dimensional (1D) next-nearest-neighbor Heisenberg chain of up to 18 qubits. We represent local operators with matrices relative to an energy eigenbasis. The matrices bear out seven predictions of the non-Abelian ETH. We also prove analytically that the non-Abelian ETH exhibits a self-consistency property. The proof relies on a thermodynamic-entropy definition different from that in Murthy textit{et al.} This work initiates the observation and application of the non-Abelian ETH.