Quantum Quenches of Conformal Field Theory with Open Boundary

We develop a method to derive the exact formula of entanglement entropy for generic inhomogeneous conformal field theory (CFT) quantum quenches with open boundary condition (OBC), which characterizes the generic boundary effect unresolved by analytical methods in the past. We identify the generic OBC quenches with Euclidean path integrals in complicated spacetime geometries, and we show that a special class of OBC quenches, including the M\"obius and sine-square-deformation quenches, have simple boundary effects calculable from Euclidean path integrals in a simple strip spacetime geometry. We verify that our generic CFT formula matches well with free fermion tight-binding model numerical calculations for various quench problems with OBC. Our method can be easily generalized to calculate any local quantities expressible as one-point functions in such quantum quench problems.