  We study crosscap states in integrable field theories and spin chains in 1+1
dimensions. We derive an exact formula for overlaps between the crosscap state
and any excited state in integrable field theories with diagonal scattering. We
then compute the crosscap entropy, i.e. the overlap for the ground state, in
some examples. In the examples we analyzed, the result turns out to decrease
monotonically along the renormalization group flow except in cases where the
discrete symmetry is spontaneously broken in the infrared. We next introduce
crosscap states in integrable spin chains, and obtain determinant expressions
for the overlaps with energy eigenstates. These states are long-range entangled
and their entanglement entropy grows linearly until the size of the subregion
reaches half the system size. This property is reminiscent of pure-state black
holes in holography and makes them interesting for use as initial conditions of
quantum quench. As side results, we propose a generalization of Zamolodchikov's
staircase model to flows between D-series minimal models, and discuss the
relation to fermionic minimal models and the GSO projection.
