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

Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories