Renormalization group for renormalization-group equations toward the universality classification of infinite-order phase transitions

We derive a renormalization group to calculate the nontrivial critical expo nent of the divergent correlation length, thereby providing a universality classification of essential singularities in infinite-order phase transitio ns. This method thus resolves the vanishing scaling matrix problem. The exp onent is obtained from the maximal eigenvalue of a scaling matrix in this r enormalization group, as in the case of ordinary second-order phase transit ions. We exhibit several nontrivial universality classes in infinite-order transitions different from the well known Berezinskii-Kosterlitz-Thouless t ransition. [S1063-651X(99)05010-2].