Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 303 / 1 / d / Solution

The mean-field approximation suppresses correlated order-parameter fluctuations. Near a continuous phase transition, the correlation length diverges and long-wavelength fluctuations become increasingly important. The Ginzburg criterion therefore fails in sufficiently low dimension, and the interacting renormalization-group fixed point changes the mean-field exponents. For the tricritical $m^6$ theory the upper critical dimension is three: below $d=3$ the exponents are generally non-mean-field, while at $d=3$ one expects logarithmic corrections to mean-field scaling.