  A strongly non-integrable system is expected to satisfy the eigenstate
thermalization hypothesis, which states that the expectation value of an
observable in an energy eigenstate is the same as the thermal value. This must
be revised if the observable is an order parameter for a spontaneously broken
symmetry, which has multiple thermal values. We propose that in this case the
system is unstable towards forming nearby eigenstates which yield each of the
allowed thermal values. We provide strong evidence for this from a numerical
study of the 2D transverse-field quantum Ising model.
