  In this study, we examined consequences of unconventional time development of
two-dimensional conformal field theory induced by the $L_{1}$ and $L_{-1}$
operators, employing the formalism previously developed in a study of
sine-square deformation. We discovered that the retainment of the Virasoro
algebra requires the presence of a cut-off near the fixed points. The
introduction of a scale by the cut-off makes it possible to recapture the
formula for entanglement entropy in a natural and straightforward manner.
