in [@TWunpub] and [@CEF] for particular models). The boundary state at hand must satisfy Cardy constraint which arises from locality. There are similar constraints for flows to a non-trivial fixed point with local conformal interfaces described by conformal boundary conditions in the tensor product theories as was already mentioned above.

All massive flows from a given ${\rm CFT}_{\rm UV}$ are then labeled by conformal boundary conditions. They appear to be natural labels of infrared phases of massive theories. The unstable manifold is then broken into regions of possibly different dimensions labeled by conformal boundary conditions with particular conditions on boundaries separating these regions. It seems to be an interesting enterprise to investigate this structure for particular two-dimensional CFTs. For each Virasoro minimal model there is a finite number of fundamental conformal boundary conditions, so that we expect their unstable manifolds to be broken into finitely many regions (massive phases). As we will show below superpositions of fundamental boundary conditions may also arise for RG flows in the presence of spontaneous symmetry breaking.

In this paper we investigate the RG boundaries and interfaces for the simplest minimal model - the critical Ising model. It has three primary fields: ${\bf 1}$, $\epsilon$, 