Consider a phenomenological modelling of a continuous phase transition, where the Lagrangian of the system is given by


Here $\phi$ is the order parameter, $h$ is the external field, $a=a_0(T-T_c)$, and $a_0>0$ and $\lambda>0$. I have not written the kinetic term here, but one can always put in that. At $T =T_c$ we have a second order phase transition.

An oft-quoted statement is that at $T=T_c$, we have self similarity at all scales. How do I see this statement? Is it the same statement that the $\beta$ function of the above theory vanishes? If so, is there a reference where this calculation for this model is performed in detail?

Assume that I am working in 2 dimensions, where $\lambda$ has negative mass dimension. In this case, is it correct to say that a CFT describes the critical point? What happens to the coupling with negative mass dimension here?



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