  In this work, we define and calculate critical exponents associated with
higher order thermodynamic phase transitions. Such phase transitions can be
classified into two classes: with or without a local order parameter. For phase
transitions involving a local order parameter, we write down the Landau theory
and calculate critical exponents using the saddle point approximation. Further,
we investigate fluctuations about the saddle point and demarcate when such
fluctuations dominate over saddle point calculations by introducing the
generalized Ginzburg criteria. We use Wilsonian RG to derive scaling forms for
observables near criticality and obtain scaling relations between the critical
exponents. Afterwards, we find out fixed points of the RG flow using the
one-loop beta function and calculate critical exponents about the fixed points
for third and fourth order phase transitions.
