Effective Critical Exponents from Finite Temperature Renormalization Group

Effective critical exponents for the \lambda \phi^4 scalar field theory are calculated as a function of the renormalization group block size k_o^{-1} and inverse critical temperature \beta_c. Exact renormalization group equations are presented up to first order in the derivative expansion and numerical solutions are obtained with and without polynomial expansion of the blocked potential. For a finite temperature system in d dimensions, it is shown that \bar\beta_c = \beta_c k_o determines whether the d-dimensional (\bar\beta_c << 1) or (d+1)-dimensional (\bar\beta_c >> 1) fixed point governs the phase transition. The validity of a polynomial expansion of the blocked potential near criticality is also addressed.