  We propose scaling theories for Kosterlitz-Thouless (KT) phase transitions on
the basis of the hallmark exponential growth of their correlation length.
Finite-size scaling, finite-entanglement scaling, short-time critical dynamics,
and finite-time scaling, as well as some of their combinations are studied.
Relaxation times of both a usual power-law and an unusual power-law with a
logarithmic factor are considered. Finite-size and finite-entanglement scaling
forms somehow similar to a frequently employed ansatz are presented. The
Kibble-Zurek scaling of topological defect density for a linear driving across
the KT transition point is investigated in detail. An implicit equation for a
rate exponent in the theory is derived and the exponent varies with the
distance from the critical point and the driving rate consistent with relevant
experiments. To verify the theories, we utilize the KT phase transition of a
one-dimensional Bose-Hubbard model. The infinite time-evolving-block-decimation
algorithm is employed to solve numerically the model for finite bond
dimensions. Both a correlation length and an entanglement entropy in imaginary
time and only the entanglement entropy in real-time driving are computed. Both
the short-time critical dynamics in imaginary time and the finite-time scaling
in real-time driving, both including the finite bond dimension, for the
measured quantities are found to describe the numerical results quite well via
surface collapses. The critical point is also estimated and confirmed to be
$0.302(1)$ at the infinite bond dimension on the basis of the scaling form.
