Dimensional crossover and effective exponents Original Research Article

We investigate the critical behavior of the γΦ4 theory defined on S1 × Rd having two finite length scales /gb, the circumference of S1, and k-1, the blocking scale introduced by the renormalization group transformation. By numerically solving the coupled differential RG equations for the finite-temperature blocked potential U/gβ,k(/gf) and the wave-function renormalization constant Zβ,k (/gf) , we demonstrate how the finite-size scaling variable β = βk determines whether the phase transition is (d + 1)- or d-dimensional in the limits β ⪢ 1 and β ⪡ 1, respectively. For the intermediate values of β, finite-size effects play an important role. We also discuss the failure of the polynomial expansion of the effective potential near criticality.