cylinder with two local interfaces between ${\rm CFT}_{\rm UV}$ and the perturbed theory with two external states of ${\rm CFT}_{\rm UV}$ attached. The perturbed theory can be driven towards ${\rm CFT}_{\rm IR}$ that results in having a system with two conformal interfaces. Taking the limit $L\to \infty$ results in the following asymptotic Z\_[0,i]{}(L) \~e\^[-LE\_[0]{}\^[IR]{}]{} 0|v\_[0]{}v\_[0]{}|i where $E_{0}^{\rm IR}=-\pi c_{\rm IR}/6R$ is the vacuum energy of ${\rm CFT}_{\rm IR}$. Removing the divergent exponential factor we obtain up to the overall factor $ \langle 0|v_{0}\rangle$ the overlaps $\langle v_{0}|i\rangle $ of the perturbed vacuum $|v_{0}\rangle$ with the unperturbed scaling states. Overlaps with excited states $|v_{i}\rangle$ can be obtained from the subleading terms in the $L\to \infty$ asymptotic.

To summarise the above discussion, in the Lagrangian formalism the TCSA overlaps (\[TCSA\_pairing\]) are described in terms of a local interface between the perturbed and unperturbed theories. This has implications that are not evident in the Hamiltonian picture (and thus in TCSA). Thus for flows to a trivial fixed point the RG interface is a conformal boundary condition (the RG boundary). In the far infrared the perturbed theory vacuum state is given by the conformal boundary state describing this boundary condition (this was previously observed 