Scaling Relations from the Scaling Hypothesis

The Philosophy of Landau Theory: Order Parameters and Free Energy

At the heart of Landau's approach is the concept of an order parameter, which we've already encountered. The order parameter, which we will denote generically by $\varphi$, is a quantity that is zero in the high-temperature, symmetric phase and non-zero in the low-temperature, broken-symmetry phase. For an Ising-like ferromagnet, the order parameter is the average magnetization $m$. For a liquid-gas transition, it is the density difference ${\rho }_l-{\rho }_g$.

Landau's central idea is that close to a continuous phase transition, where the order parameter is small, the thermodynamic potential (specifically, the free energy density $f$) can be expanded as a Taylor series in powers of the order parameter. The equilibrium state of the system is then found by minimizing this free energy with respect to the order parameter.

This approach is analogous to analyzing the stability of a mechanical system by expanding its potential energy around an equilibrium point. Here, we are constructing a "potential landscape" for the order parameter, and the system's phase is determined by the location of the minima in this landscape.

Constructing the Free Energy: Symmetry as the Guiding Principle

Let's make this concrete by considering a system with the same symmetries as the Ising model. The order parameter is the magnetization, $m$.

1. The Expansion in Zero Field

First, consider the system in zero external magnetic field ($h=0$). The free energy density $f$ is a function of $m$ and temperature $T$. We can expand it as a power series in $m$:

Here, the coefficients $c_i(T)$ are, for now, unknown functions of temperature, and $f_0(T)$ represents the free energy of the symmetric phase ($m=0$).

Now, we impose the crucial constraint of symmetry. The Ising Hamiltonian in zero field is invariant under a spin flip transformation, $S_i\to -S_i$ for all spins. At the macroscopic level, this translates to an invariance of the free energy under the reversal of magnetization, $m\to -m$. Therefore, we must have:

This condition immediately forces all coefficients of odd powers of $m$ to be zero: $c_1=c_3=c_5=\cdots =0$. The expansion simplifies to:

where we have relabeled $c_2(T)$ as $a(T)$ and $c_4(T)$ as $b(T)$.

2. Phenomenological Coefficients and Stability

For the free energy to be physically sensible, it must be bounded from below as $m$ becomes large. This ensures the system is stable and doesn't collapse into a state of infinite magnetization. This stability condition requires that the highest-order term in our expansion must be positive. For a continuous transition, we can often truncate the series at the $m^4$ term, which requires that its coefficient, $b(T)$, must be positive near the critical temperature $T_c$. Since we assume the coefficients are smooth functions of temperature, we can approximate $b(T)$ by a positive constant, $b(T)\approx b(T_c)=b>0$.

The coefficient $a(T)$ is the most critical part of the model.

• In the high-temperature (disordered) phase ($T>T_c$), we know the equilibrium state is $m=0$. For $m=0$ to be a minimum of the free energy, the $m^2$ term must be positive, so $a(T)>0$.
• In the low-temperature (ordered) phase ($T<T_c$), a non-zero magnetization appears, meaning the minimum of the free energy shifts away from $m=0$. This can only happen if the curvature at the origin becomes negative, i.e., $a(T)<0$.

Thus, the coefficient $a(T)$ must change sign at the critical temperature $T_c$. The simplest possible assumption for a function that smoothly passes through zero is a linear one:

where $a_0$ is a positive constant.

Putting this together, and absorbing some numerical factors for later convenience, the Landau free energy density near a continuous phase transition is:

This is the canonical form of the Landau free energy. It is a phenomenological model, but one grounded in the powerful and general principle of symmetry.

Visualizing the Phase Transition

The elegance of Landau theory is most apparent when we visualize the free energy landscape. Let's plot $f(m,T)-f_0(T)$ as a function of $m$ for different temperatures.

A sketch of the Landau free energy density $f(m)$ as a function of the order parameter $m$ at different temperatures. (a) For $T>T_c$, the free energy has a single minimum at $m=0$, corresponding to the disordered phase. (b) At $T=T_c$, the minimum at $m=0$ becomes very flat. (c) For $T<T_c$, $m=0$ becomes an unstable maximum, and two new symmetric minima appear at non-zero values $m=\pm m_0$, representing the spontaneous emergence of order.

• Case (a): $T>T_c$
  The coefficient of $m^2$ is positive. The free energy is a simple parabola-like potential with a single minimum at $m=0$. The system is in its symmetric, disordered phase.

• Case (b): $T=T_c$
  The coefficient of $m^2$ is exactly zero. The potential becomes very flat near the origin, $f\propto m^4$. This is the critical point, where fluctuations (which we are currently ignoring) will be exceptionally large.

• Case (c): $T<T_c$
  The coefficient of $m^2$ is now negative. The origin $m=0$ is no longer a minimum but an unstable local maximum. The $+b{m}^4$ term ensures stability and creates two new, degenerate minima at non-zero values of $m$. The system must spontaneously choose one of these minima, breaking the $m\to -m$ symmetry and acquiring a non-zero magnetization. This "double-well" or "Mexican hat" potential is a hallmark of spontaneous symmetry breaking in many areas of physics, from condensed matter to quantum field theory.

Adding an External Field

What happens if we apply an external magnetic field, $h$? From thermodynamics, we know the appropriate potential to consider is the Gibbs free energy, which includes a term coupling the field to the order parameter. For a magnetic system, this term is $-mh$. Adding this to our expansion gives the full Landau free energy density:

The linear term in $m$ explicitly breaks the $m\to -m$ symmetry. If you visualize the potential landscape now, the field $h$ "tilts" the double-well potential, favoring one minimum over the other. If $h>0$, the minimum at positive $m$ becomes lower (more stable), and the minimum at negative $m$ becomes higher (metastable).

By finding the minimum of this full expression ($\partial f/\partial m=0$), we obtain the Landau equation of state:

This single equation, derived from symmetry principles, contains a remarkable amount of physics, describing the system's behavior both above and below the critical temperature, in and out of an external field.

Conclusion

In this lesson, we have constructed the Landau free energy, a cornerstone of the modern theory of phase transitions. We moved beyond the purely descriptive scaling hypothesis to a semi-microscopic model rooted in the universal language of symmetry.

Key Takeaways:

• Landau theory describes phase transitions by expanding a free energy density, $f$, as a polynomial in an order parameter, $m$.
• The form of this polynomial is dictated by the symmetries of the system. For systems with up-down symmetry (like the Ising model), only even powers of $m$ are allowed in zero field.
• The phase transition is driven by a change in the sign of the coefficient of the $m^2$ term, $a(T)$, which is assumed to be proportional to $T-T_c$.
• The resulting Landau free energy, $f(m,T,h)=\frac{a_0}{2}(T-T_c)m^2+\frac{b}{4}{m}^4-mh$, provides a universal model for continuous phase transitions.
• It's crucial to recognize that by using a single, spatially uniform order parameter $m$, Landau theory implicitly averages over all spatial fluctuations. It is therefore a mean-field theory.

The power of this construction is that it provides a concrete analytical model. In our next lesson, we will use this model to do what the scaling hypothesis could not: we will explicitly calculate the critical exponents $\alpha ,\beta ,\gamma ,\delta$ and show that they are exactly the mean-field exponents. This will firmly establish Landau theory's place as the canonical mean-field description of critical phenomena.