  The eigenstate thermalization hypothesis (ETH) explains why chaotic quantum
many-body systems thermalize internally if the Hamiltonian lacks symmetries. If
the Hamiltonian conserves one quantity ("charge"), the ETH implies
thermalization within a charge sector -- in a microcanonical subspace. But
quantum systems can have charges that fail to commute with each other and so
share no eigenbasis; microcanonical subspaces may not exist. Furthermore, the
Hamiltonian will have degeneracies, so the ETH need not imply thermalization.
We adapt the ETH to noncommuting charges by positing a non-Abelian ETH and
invoking the approximate microcanonical subspace introduced in quantum
thermodynamics. Illustrating with SU(2) symmetry, we apply the non-Abelian ETH
in calculating local observables' time-averaged and thermal expectation values.
In many cases, we prove, the time average thermalizes. However, we also find
cases in which, under a physically reasonable assumption, the time average
converges to the thermal average unusually slowly as a function of the
global-system size. This work extends the ETH, a cornerstone of many-body
physics, to noncommuting charges, recently a subject of intense activity in
quantum thermodynamics.
