Editing Landau theory (section)

===Applied fields===
In many systems, one can consider a perturbing field <math>h</math> that couples linearly to the order parameter. For example, in the case of a classical [[Magnetic dipole|dipole moment]] <math>\mu</math>, the energy of the dipole-field system is <math>-\mu B</math>. In the general case, one can assume an energy shift of <math>-\eta h</math> due to the coupling of the order parameter to the applied field <math>h</math>, and the Landau free energy will change as a result:
:<math>F(T,\eta) - F_0 = a_0 (T-T_c) \eta^2 + \frac{b_0}{2} \eta^4 - \eta h</math>
In this case, the minimization condition is 
:<math>\frac{\partial F}{\partial \eta} = 2 a(T) \eta + 2b_0 \eta^3 - h = 0 </math>
One immediate consequence of this equation and its solution is that, if the applied field is non-zero, then the magnetization is non-zero at any temperature. This implies there is no longer a spontaneous symmetry breaking that occurs at any temperature. Furthermore, some interesting thermodynamic and universal quantities can be obtained from this above condition. For example, at the critical temperature where <math>a(T_c)=0</math>, one can find the dependence of the order parameter on the external field: 
:<math>\eta(T_c) = \left( \frac{h}{2b_0} \right)^{1/3} \propto h^{1/\delta} </math>
indicating a critical exponent <math>\delta = 3</math>. 
[[File:LandauTheorySusceptibility.svg|thumb|Zero-field susceptibility as a function of temperature near the critical temperature]]
Furthermore, from the above condition, it is possible to find the zero-field susceptibility <math>\chi\equiv \partial \eta/\partial h|_{h=0}</math>, which must satisfy 
:<math>0 = 2 a \frac{\partial \eta}{\partial h} + 6b \eta^2 \frac{\partial \eta}{\partial h} - 1</math>
:<math>[2 a + 6b \eta^2] \frac{\partial \eta}{\partial h} = 1</math>
In this case, recalling in the zero-field case that <math>\eta^2 = -a/b</math> at low temperatures, while <math>\eta^2=0</math> for temperatures above the critical temperature, the zero-field susceptibility therefore has the following temperature dependence: 
:<math>\chi(T,h\to 0) = \begin{cases} \frac{1}{2a_0(T-T_c)}, & T>T_c \\ \frac{1}{-4a_0(T-T_c)}, & T<T_c \end{cases} \propto |T-T_c|^{-\gamma}</math>
which is reminiscent of the [[Curie-Weiss law]] for the temperature dependence of magnetic susceptibility in magnetic materials, and yields the mean-field critical exponent <math>\gamma=1</math>.

It is noteworthy that although the critical exponents so obtained are incorrect for many models and systems, they correctly satisfy various exponent equalities such as the [[Rushbrooke inequality|Rushbrooke equality]]: <math> \alpha + 2\beta + \gamma = 2</math>.