  We propose using smeared boundary states $e^{-\tau H}|\cal B\rangle$ as
variational approximations to the ground state of a conformal field theory
deformed by relevant bulk operators. This is motivated by recent studies of
quantum quenches in CFTs and of the entanglement spectrum in massive theories.
It gives a simple criterion for choosing which boundary state should correspond
to which combination of bulk operators, and leads to a rudimentary phase
diagram of the theory in the vicinity of the RG fixed point corresponding to
the CFT, as well as rigorous upper bounds on the universal amplitude of the
free energy. In the case of the 2d minimal models explicit formulae are
available. As a side result we show that the matrix elements of bulk operators
between smeared Ishibashi states are simply given by the fusion rules of the
CFT.
