  We derive a new renormalization group to calculate a non-trivial critical
exponent of the divergent correlation length which gives a universality
classification of essential singularities in infinite-order phase transitions.
This method resolves the problem of a vanishing scaling matrix. The exponent is
obtained from the maximal eigenvalue of a scaling matrix in this
renormalization group, as in the case of ordinary second-order phase
transitions. We exhibit several nontrivial universality classes in
infinite-order transitions different from the well-known
Berezinski\u\i-Kosterlitz-Thouless transition.
