I have a question about Landau's theory of quantum phase transition. In his model, the free energy is assumed to be

\begin{equation}
F = f_0 + \alpha (T-T_c) \Delta^2 + \beta \Delta^4
\end{equation}

The ground state of the system depends strongly on the sign of $T-T_c$. In this way, we find that the scaling exponent near the critical point is $1/2$, which may be somewhat different from that in experiments -- as a results, we need renormalization group method to understand the discrepancy. This is a theory that has been accepted by this community.

OK, now my question is why in the second term the coefficient is $\alpha (T-T_c)$, instead of $\alpha (T-T_c)^\gamma$, where $\gamma$ is a constant, e.g., $\gamma = 3/5$ or $1/3$. This is perhaps a trivial problem, but has never been discussed explicitly in standard textbooks. The answer to this problem is not so straightforward for most of us.

A related question maybe like that: how to prove this point in experiments. Thanks very in advance.

This post imported from StackExchange Physics at 2014-05-04 11:12 (UCT), posted by SE-user Ming Gong