Logarithmic corrections to the free energy from sharp corners with angle $2\pi$

摘要: We study subleading corrections to the corner free energy in classical two-dimensional critical systems, focusing on a generic boundary perturbation by stress-tensor of underlying conformal field theory (CFT). In particular case an angle $2\pi$, we find that there is unusual correction form $L^{-1}\log L$, where $L$ typical length scale system. This also affects one-point function operator near corner. The prefactor can be seen as semi-universal, sense it depends \emph{single} non-universal quantity, extrapolation length. Once this ultraviolet cutoff known, term entirely fixed geometry system, and central charge CFT. Such appears for example bipartite fidelity one-dimensional quantum system at criticality, which allows several numerical checks fermions systems. present exact result XX Ising chains confirms analysis. Finally, consider applications time evolution (logarithmic) Loschmidt echo entanglement entropy following local quench. Due subtle issues analytic continuation, logarithmic imaginary transforms into time-dependent $L^{-2}$ entropy, $L^{-2}\log L$ echo.