  We study non-invertible topological symmetry operators in massive quantum
field theories in (1+1) dimensions. In phases where this symmetry is
spontaneously broken we show that the particle spectrum often has degeneracies
dictated by the non-invertible symmetry and we deduce a procedure to determine
the allowed multiplets. These degeneracies are robust predictions and do not
require integrability or other special features of renormalization group flows.
We exhibit these conclusions in examples where the spectrum is known,
recovering soliton and particle degeneracies. For instance, the Tricritical
Ising model deformed by the subleading Z2 odd operator flows to a gapped phase
with two degenerate vacua. This flow enjoys a Fibonacci fusion category
symmetry which implies a threefold degeneracy of its particle states, relating
the mass of solitons interpolating between vacua and particles supported in a
single vacuum.
