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

$$L=\frac{a}{2}\phi^2+\frac{\lambda}{4}\phi^4-h\phi.$$

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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  • $\begingroup$ You've probably found the answer already, but if you're working in $2$ dimensions, where $\lambda$ has negative mass dimension, the critical point is not described by that $L$. The actual critical point has to include all the powers of $\phi^2$, and it's that theory that is described by a CFT. $\endgroup$
    – user196574
    Sep 5 at 5:32

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