Epsilon Expansion for Multicritical Fixed Points and Exact Renormalisation Group Equations

TL;DR

This paper applies the Polchinski exact renormalisation group equations to analyze multicritical fixed points in scalar theories, comparing results with the epsilon expansion to test approximation methods and derive critical exponents.

Contribution

It introduces an alternative truncation of the RG equations that accurately computes critical exponents and anomalous dimensions, and constructs an exact marginal operator.

Findings

Linearisation of nonlinear equations at multicritical points validates epsilon expansion as a perturbative method.

The alternative truncation yields correct critical exponents to order epsilon and anomalous dimensions to order epsilon^2.

Comparison with standard epsilon expansion confirms the validity of the local potential and derivative expansions.

Abstract

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.