Bipartite Fidelity and Loschmidt Echo of Bosonic Conformal Interface

We study the quantum quench problem for a class of bosonic conformal interfaces by computing the Loschmidt echo and the bipartite fidelity. The quench can be viewed as a sudden change of boundary conditions parameterized by $\theta$ when connecting two one-dimensional critical systems. They are classified by $S(\theta)$ matrices associated with the current scattering processes on the interface. The resulting Loschmidt echo of the quench has long time algebraic decay $t^{-\alpha}$, whose exponent also appears in the finite size bipartite fidelity as $L^{-\frac{\alpha}{2}}$. We perform analytic and numerical calculations of the exponent $\alpha$, and find that it has a quadratic dependence on the change of $\theta$ if the prior and post quench boundary conditions are in the same type of $S$, while remains $\frac{1}{4}$ otherwise. Possible physical realizations of these interfaces include for instance connecting different quantum wires (Luttinger liquids), quench of the topological phase edge states \etc and the exponent can be detected in a X-ray edge singularity type experiment.