Inhomogeneous Quantum Quenches of Conformal Field Theory with Boundaries

We develop a method to calculate generic time-dependent correlation functions for inhomogeneous quantum quenches in (1+1)-dimensional conformal field theory (CFT) induced by sudden Hamiltonian deformations that modulate the energy density inhomogeneously. Our work particularly focuses on the effects of spatial boundaries, which have remained unresolved by previous analytical methods. For generic post-quench Hamiltonian, we develop a generic method to calculate the correlations by mirroring the system, which otherwise are Euclidean path integrals in complicated spacetime geometries difficult to calculate. On the other hand, for a special class of inhomogeneous post-quench Hamiltonians, including the M\"obius and sine-square-deformation Hamiltonians, we show that the quantum quenches exhibit simple boundary effects calculable from Euclidean path integrals in a straightforward strip spacetime geometry. Applying our method to the time evolution of entanglement entropy, we find that for generic cases, the entanglement entropy shows discontinuities (shockwave fronts) propagating from the boundaries.In contrast, such discontinuities are absent in cases with simple boundary effects. We verify that our generic CFT formula matches well with numerical calculations from free fermion tight-binding models for various quench scenarios.