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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.

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In principle, since Landau theory is phenomenological, one could imagine that the quadratic term could have a non trivial dependence on the microscopic parameter (for example $(T-T_c)^{0.3}$). However, there are strong arguments that show that it is not the case when one is trying to justify Landau theory microscopically.

The main argument is that this kind of singular behavior is possible only if fluctuations at all length scales are taken into account. Indeed, if only some fluctuations are included, then the partition is just the sum of a finite number of exponential, and it is thus analytical. Furthermore, picking by hand the correct exponent (using the OP's exponent $\gamma$) renders the theory useless, since one is putting by hand the correct behavior, which is then no better than the scaling hypothesis.

Also, all calculations made for deriving microscopically the Landau free energy show that the coefficient are analytical function of the microscopic parameters (though the dependence on $T-T_c$ might be quadratic and not linear depending on the symmetries of the problem). See for example the case of the Ising model or Gorkov's derivation of the Ginzburg-Landau free energy for a superconductor.

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As I remember, one of the most important problems of Landau theory that it can't describe critical points in a right manner, because simply it's not taking in account fluctuations, this answers your both questions. Moreover, when Landau proposed his theory, there was not suitable technologies to make accurately experiments to support or falsify landau theory very close to critical points, only letter it was understood that new theories as Izing model is required for right explanation of near crirical pount behaviour.

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