Critical exponents for higher order phase transitions: Landau theory and RG flow

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.