  We consider perturbations of 2D CFTs by multiple relevant operators. The
massive phases of such perturbations can be labeled by conformal boundary
conditions. Cardy's variational ansatz approximates the vacuum state of the
perturbed theory by a smeared conformal boundary state. In this paper we study
the limitations and propose generalisations of this ansatz using both analytic
and numerical insights based on TCSA. In particular we analyse the stability of
Cardy's ansatz states with respect to boundary relevant perturbations using
bulk-boundary OPE coefficients. We show that certain transitions between the
massive phases arise from a pair of boundary RG flows. The RG flows start from
the conformal boundary on the transition surface and end on those that lie on
the two sides of it. As an example we work out the details of the phase diagram
for the Ising field theory and for the tricritical Ising model perturbed by the
leading thermal and magnetic fields. For the latter we find a pair of novel
transition lines that correspond to pairs of RG flows. Although the mass gap
remains finite at the transition lines, several one-point functions change
their behaviour. We discuss how these lines fit into the standard phase diagram
of the tricritical Ising model. We show that each line extends to a
two-dimensional surface $\xi_{\sigma,c}$ in a three coupling space when we add
perturbations by the subleading magnetic field. Close to this surface we locate
symmetry breaking critical lines leading to the critical Ising model. Near the
critical lines we find first order phase transition lines describing two-phase
coexistence regions as predicted in Landau theory. The surface $\xi_{\sigma,c}$
is determined from the CFT data using Cardy's ansatz and its properties are
checked using TCSA numerics.
