Critical Exponents in Landau Theory

Welcome back. In our last lesson, we developed a powerful framework for classifying phase transitions by analyzing the structure of the Landau free energy potential. We saw how the symmetry and coefficients of a simple polynomial in the order parameter can distinguish between continuous (second-order) and discontinuous (first-order) transitions.

Today, we delve deeper into the quantitative power of this approach. Near a continuous phase transition, many physical quantities exhibit singular behavior, diverging or vanishing as power laws of the reduced temperature, $t = (T - T_{c}) / T_{c}$ . The exponents in these power laws, known as critical exponents, are remarkably universal—they are often identical for vastly different physical systems, from magnets to fluids. This lesson will equip you with the methodology to calculate the complete set of these exponents directly from the Landau-Ginzburg free energy, revealing the first and most fundamental universality class: that of mean-field theory.

The Zoology of Critical Exponents

Before we begin our calculations, it's essential to understand the physical meaning of the exponents we seek. They are the language used to describe the collective, scale-invariant physics at the critical point. The following video from an MIT OpenCourseWare lecture provides an excellent phenomenological introduction, defining the most common critical exponents by relating them to measurable quantities in a ferromagnet.

2. Lec 1 (continued); The Landau-Ginzburg Approach Part 1

This lecture by Prof. Mehran Kardar introduces the concept of universality and defines the critical exponents that characterize the singular behavior of thermodynamic quantities near a second-order phase transition.

Please watch the segment from "functional forms are the things that are universal". The lecturer defines the exponents \beta (for magnetization), \delta (for the critical isotherm), \gamma (for susceptibility), and \alpha (for specific heat). Don't worry about the subsequent derivation relating susceptibility to fluctuations; we will cover that later. The goal here is to grasp the physical definition of each exponent.

To summarize and complete the set of exponents we will calculate, here are the key definitions for a magnetic system with order parameter $η$ (magnetization), external field $h$ , and reduced temperature $τ = T - T_{c}$ :

Exponent	Definition	Physical Quantity
$α$	$C_h \sim	T-T_c
$β$	$η \sim (T_{c} - T)^{β}$ for $T < T_{c}, h = 0$	Order Parameter
$γ$	$\chi \sim	T-T_c
$δ$	$h \sim η^{δ}$ for $T = T_{c}$	Critical Isotherm
$ν$	$\xi \sim	T-T_c
$η_{e}$	$G (r) \sim r^{- (d - 2 + η_{e})}$ for $T = T_{c}$	Correlation Function

(Note: We use $η_{e}$ for the correlation function exponent to avoid confusion with the order parameter $η$ .)

Our task is to derive the values for these exponents using the standard symmetric Landau model.

Calculating the Mean-Field Exponents

Let's begin with the Landau free energy density for a system with up-down symmetry ( $η \to - η$ ) in the presence of a uniform external field $h$ :

f (η, T, h) = f_{0} (T) + \frac{A}{2} η^{2} + \frac{B}{4} η^{4} - h η

As established previously, we assume $B > 0$ for stability, and $A = a (T - T_{c})$ with $a > 0$ , which captures the temperature dependence that drives the transition. The system's equilibrium state is found by minimizing $f$ with respect to $η$ :

\frac{\partial f}{\partial η} = A η + B η^{3} - h = 0

This simple algebraic equation is the key to all the thermodynamic exponents. The following text provides a clear and direct calculation of $α, β, γ, δ$ , and $ν$ .

4.3: Landau's mean-field theory - Physics LibreTexts

This section from the "Essential Graduate Physics" textbook on Physics LibreTexts systematically derives the critical exponents from the Landau free energy. The notation is slightly different: they define \tau \propto (T_c-T), so that \tau > 0 below the critical temperature, and their coefficients a, b are defined as positive constants. The physics is identical.

Please read the entire section "the critical exponent calculation". Follow each derivation step-by-step, comparing the results with the definitions in our table. Pay special attention to the simple heuristic argument used to find the correlation length exponent \nu.

Let's consolidate and walk through these calculations to ensure the logic is perfectly clear.

Exponent $β$ (Order Parameter):
- Set $h = 0$ and consider $T < T_{c}$ (so $A < 0$ ).
- The minimum is at $A η + B η^{3} = 0 ⟹ η^{2} = - A / B = - a (T - T_{c}) / B = a (T_{c} - T) / B$ .
- Therefore, $η = \pm a / B (T_{c} - T)^{1/2}$ .
- Comparing with $η \sim (T_{c} - T)^{β}$ , we find $β = 1/2$ .
Exponent $α$ (Specific Heat):
- First, find the equilibrium free energy density for $h = 0$ .
- For $T > T_{c}$ , $η = 0$ , so $f_{e q} = f_{0} (T)$ .
- For $T < T_{c}$ , substitute $η^{2} = - A / B$ into the expression for $f$ :
  $f_{e q} = f_{0} + \frac{A}{2} (- \frac{A}{B}) + \frac{B}{4} (- \frac{A}{B})^{2} = f_{0} - \frac{A ^{2}}{4 B} = f_{0} - \frac{a ^{2} ( T - T _{c} ) ^{2}}{4 B}$
- The specific heat is $C_{h} = - T \frac{\partial ^{2} F}{\partial T ^{2}} \propto - \frac{\partial ^{2} f _{e q}}{\partial T ^{2}}$ .
- For $T > T_{c}$ , the specific heat comes only from the background term $f_{0}$ . The singular part is zero.
- For $T < T_{c}$ , the second derivative of the singular part $- \frac{a ^{2} ( T - T _{c} ) ^{2}}{4 B}$ is a non-zero constant, $- \frac{a ^{2}}{2 B}$ .
- Thus, the specific heat has a finite jump at $T_{c}$ , not a power-law divergence. This corresponds to $α = 0$ .
Exponent $γ$ (Susceptibility):
- The susceptibility is $χ = (\frac{\partial η}{\partial h})_{h = 0}$ . We differentiate the minimization condition $A η + B η^{3} - h = 0$ with respect to $h$ :
  $A \frac{\partial η}{\partial h} + 3 B η^{2} \frac{\partial η}{\partial h} - 1 = 0 ⟹ χ = \frac{1}{A + 3 B η ^{2}}$
- For $T > T_{c}$ , we evaluate at the equilibrium state $η = 0$ , giving $χ = 1/ A = 1/ (a (T - T_{c}))$ . Thus, $χ \sim (T - T_{c})^{- 1}$ , which means $γ = 1$ .
- For $T < T_{c}$ , we use $η^{2} = - A / B$ : $χ = 1/ (A + 3 B (- A / B)) = 1/ (A - 3 A) = 1/ (- 2 A) = 1/ (2 a (T_{c} - T))$ . Thus, $χ \sim (T_{c} - T)^{- 1}$ , which means $γ^{'} = 1$ .
Exponent $δ$ (Critical Isotherm):
- Set $T = T_{c}$ , which means $A = 0$ .
- The minimization condition becomes $B η^{3} - h = 0$ , or $h = B η^{3}$ .
- Comparing with the definition $h \sim η^{δ}$ , we find $δ = 3$ .

This completes the set of thermodynamic exponents.

Spatial Correlations and the Remaining Exponents

To find the exponents $ν$ and $η_{e}$ , which describe spatial properties, we must include the gradient term in the free energy, leading to the Ginzburg-Landau functional:

F [η (r)] = \int d^{d} r [\frac{A}{2} η (r)^{2} + \frac{B}{4} η (r)^{4} + \frac{C}{2} (\nabla η (r))^{2} - h (r) η (r)]

Here, $C > 0$ penalizes spatial variations of the order parameter.

5. Exponent $ν$ (Correlation Length):
As shown in the Physics LibreTexts reading, a powerful heuristic argument gives us $ν$ . Fluctuations with a characteristic length scale $ξ$ contribute a gradient term of order $C η^{2} / ξ^{2}$ . This term becomes important when it is comparable to the term driving the transition, $∣ A ∣ η^{2}$ . Equating them gives:

\frac{C}{ξ ^{2}} \approx ∣ A ∣ = a ∣ T - T_{c} ∣ ⟹ ξ^{2} \approx \frac{C}{a ∣ T - T _{c} ∣}

From this, we see $ξ \sim ∣ T - T_{c} ∣^{- 1/2}$ . This implies $ν = 1/2$ .

6. Exponent $η_{e}$ (Correlation Decay):
To find $η_{e}$ , we must properly calculate the two-point correlation function $G (r - r^{'}) = ⟨ η (r) η (r^{'}) ⟩_{c}$ . This requires the machinery of functional calculus and Fourier transforms, which is a standard technique in theoretical physics. The following resource shows this derivation.

1. Exact Solution of the Infinite Range Ising Model

These lecture notes by Gordon Semenoff provide a concise derivation of the Landau theory exponents, including a clear calculation of the correlation function and the exponents \nu and \eta_e.

Please start by reading the summary table in Section 2, "Table of Critical Exponents". Then, jump to Section 3, "Landau Theory", and read from the introduction of the Ginzburg-Landau functional (equation 34) to the end of the document. Focus on the derivation starting from equation (44), which shows how the correlation function is the Green's function of a differential operator. Follow the Fourier transform logic to see how G(k) \sim 1/(k^2+\tau), which leads directly to the values for \nu and \eta_e.

The key result from this derivation is that the correlation function in Fourier space behaves as:

G ~ (k) = \frac{1}{C k ^{2} + A}

At the critical point $T = T_{c}$ , $A = 0$ , so $G ~ (k) \sim 1/ k^{2}$ . The Fourier transform of this back to real space gives a correlation function that decays with distance $r$ as:

G (r) \sim \frac{1}{r ^{d - 2}}

Comparing this to the definition $G (r) \sim r^{- (d - 2 + η_{e})}$ , we immediately see that $η_{e} = 0$ .

Summary of Mean-Field Exponents

We have systematically derived the full set of critical exponents predicted by Landau's mean-field theory. Let's gather them in one place:

Exponent	Mean-Field Value
$α$	0 (jump)
$β$	1/2
$γ$	1
$δ$	3
$ν$	1/2
$η_{e}$	0

These values are a cornerstone of statistical mechanics. They are called "mean-field" because, as we will see, Landau theory is equivalent to models where fluctuations are averaged out or ignored. The fact that these values do not depend on the specific microscopic details of a model, only on general properties like symmetry, is a profound illustration of universality.

Conclusion

In this lesson, we have moved from a qualitative classification of phase transitions to a quantitative prediction of the universal exponents that govern critical phenomena. By systematically minimizing the Landau free energy functional, we derived the set of mean-field critical exponents. This process—defining a quantity of interest, relating it to the free energy, and performing an expansion around the critical point—is a paradigmatic method in theoretical physics.

Key Takeaways:

Continuous phase transitions are characterized by universal critical exponents that describe power-law scaling of physical quantities near the critical point.
The standard symmetric Landau free energy predicts the "mean-field" values for these exponents: $α = 0$ , $β = 1/2$ , $γ = 1$ , $δ = 3$ .
Including a spatial gradient term allows for the calculation of spatial exponents, yielding $ν = 1/2$ and $η_{e} = 0$ .
The derivation of these exponents relies only on the analytical structure of the free energy expansion, making them independent of microscopic details and thus universal within this class of theories.

While powerful, Landau theory is an approximation. It neglects the very fluctuations that become dominant at the critical point, as hinted by the divergence of the correlation length $ξ$ . In our next and final lesson in this module, we will confront this issue directly by deriving the Ginzburg criterion, which provides a precise condition for the validity of mean-field theory and tells us when and why these calculated exponents will fail to match experimental results.

Recommended resources

The Zoology of Critical Exponents

Calculating the Mean-Field Exponents

Spatial Correlations and the Remaining Exponents

Summary of Mean-Field Exponents

Conclusion