Epsilon Expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order $\epsilon$ and also for the field anomalous dimension to order $\epsilon^2$. An exact marginal operator for the full RG equations is also constructed.Comment: 40 pages, 12 figures version2: small corrections, extra references, final appendix rewritten, version3: some corrections to perturbative calculation

Optimization of Renormalization Group Flow

Renormalization group flow equations for scalar lambda Phi^4 are generated using three classes of smooth smearing functions. Numerical results for the critical exponent nu in three dimensions are calculated by means of a truncated series expansion of the blocked potential. We demonstrate how the convergence of nu as a function of the order of truncation can be improved through a fine tuning of the smoothness of the smearing functions.Comment: 23 pages, 7 figure