Superfluid Helium-4: A Landau Theory Approach

Introduction

In our last lesson, we saw how Landau theory, with a real scalar order parameter, can model both continuous and first-order phase transitions by carefully choosing the terms in the free energy expansion. This worked well for systems like Ising magnets where the order parameter can be represented by a single real number. Today, we advance this framework to describe one of the most fascinating phenomena in condensed matter physics: the superfluid transition in liquid Helium-4.

This transition, marked by the famous $λ$ -line on the He-4 phase diagram, cannot be described by a simple scalar. The ordered phase, the superfluid state, possesses a macroscopic quantum coherence that requires an order parameter with not just a magnitude but also a phase. Our primary goal is to apply Landau theory to this transition by introducing a complex order parameter. We will see how the fundamental symmetries of the system dictate the form of the Landau free energy and how minimizing this energy naturally leads to the emergence of the superfluid state.

The phase diagram of Helium-4. The transition from the normal liquid (He I) to the superfluid (He II) occurs along the $\lambda$-line, a line of continuous phase transitions that we will model using Landau theory.

The Complex Order Parameter for Superfluidity

The transition from the normal liquid He I to the superfluid He II below about 2.17 K is a manifestation of Bose-Einstein condensation in an interacting liquid. In the superfluid state, a macroscopic fraction of the helium atoms occupies the same single-particle quantum state. This collective behavior is captured by a macroscopic wavefunction, which serves as our order parameter.

This order parameter, denoted by $ψ (r)$ , is a complex field:

ψ (r) = ∣ ψ (r) ∣ e^{i θ (r)}

The components of this field have direct physical interpretations:

The amplitude squared, $∣ ψ ∣^{2}$ , is proportional to the density of condensed atoms, known as the superfluid density $ρ_{s}$ . Below the critical temperature $T_{c}$ , $ρ_{s}$ grows from zero, acting as the order parameter for the transition.
The phase, $θ (r)$ , describes the quantum coherence of the condensate. While a uniform phase characterizes the ground state, spatial gradients of the phase are associated with dissipationless flow. The superfluid velocity is directly related to the gradient of the phase: $v_{s} \propto \nabla θ$ .

From a more formal perspective, which your background in quantum mechanics will make accessible, the existence of a non-zero order parameter corresponds to off-diagonal long-range order (ODLRO).

[PDF] An introduction to the Ginzburg-Landau theory of phase transitions ...

This document provides a formal definition of the superfluid order parameter.

Please read the short section titled Superfluid. It defines the complex order parameter \psi(\mathbf{r}) in terms of the long-distance behavior of the two-point correlation function (or one-body density matrix) \rho(\mathbf{r}, \mathbf{r}') = \langle a^\dagger(\mathbf{r})a(\mathbf{r}')\rangle. A non-zero value of \psi signifies ODLRO and a Bose condensate.

Constructing the Free Energy for a Complex Order Parameter

Now, we construct the Landau free energy density, $f$ , as a power series expansion in $ψ$ and its complex conjugate $ψ^{*}$ . For now, let's consider a spatially uniform system, so $ψ$ is constant. The construction is governed by fundamental symmetries.

Analyticity: We assume $f$ can be expanded in a Taylor series in $ψ$ and $ψ^{*}$ .
Reality: The free energy must be a real quantity. This means $f = f^{*}$ .
Symmetry: This is the most crucial constraint. The underlying Hamiltonian is invariant under a global change of phase of the wavefunction, $ψ \to ψ e^{i ϕ}$ . This is a global U(1) gauge symmetry. The free energy, being a macroscopic consequence of this Hamiltonian, must also respect this symmetry.

This U(1) symmetry severely restricts the allowed terms in the expansion. A term like $ψ$ is forbidden, as it transforms to $ψ e^{i ϕ}$ . Similarly, $ψ^{2}$ is forbidden. The only combinations that remain invariant are powers of $ψ^{*} ψ = ∣ ψ ∣^{2}$ .

The following reading from A.J. Leggett's notes formalizes this argument and presents the resulting free energy expansion.

[PDF] Ginzburg-Landau theory

This resource derives the Landau free energy for a complex order parameter based on symmetry principles and then finds the equilibrium value of the order parameter as a function of temperature.

First, read the brief introduction to the order parameter in this paragraph. Then, focus on the section describing the free energy. Pay close attention to the argument that phase invariance forces the free energy to be a function of |\psi|^2 only, leading to the standard expansion. Follow the subsequent minimization procedure.

As derived in the reading, the Landau free energy density takes the form:

f (ψ, T) = f_{0} (T) + a (T) ∣ ψ ∣^{2} + \frac{1}{2} b (T) ∣ ψ ∣^{4} + O (∣ ψ ∣^{6})

This potential is often called the "sombrero potential" or "wine-bottle potential". For stability, we require $b (T) > 0$ , which we can approximate as a constant $b$ . As is standard in Landau theory, we assume the coefficient $a (T)$ drives the transition, with the linear approximation $a (T) = α (T - T_{c})$ and $α > 0$ .

Minimization and Spontaneous Symmetry Breaking

The equilibrium state of the system is found by minimizing $f (ψ, T)$ with respect to $ψ$ .

Above $T_{c}$ (Normal Phase)

For $T > T_{c}$ , the coefficient $a (T)$ is positive. Since $b$ is also positive, the free energy is minimized when $∣ ψ ∣^{2} = 0$ , which means $ψ = 0$ . This corresponds to the normal liquid phase (He I), where the superfluid density $ρ_{s}$ is zero. The U(1) symmetry is preserved.

Below $T_{c}$ (Superfluid Phase)

For $T < T_{c}$ , the coefficient $a (T)$ becomes negative. The minimum of $f$ is no longer at the origin. Minimizing with respect to $∣ ψ ∣$ gives:

\frac{\partial f}{\partial ∣ ψ ∣} = 2 a ∣ ψ ∣ + 2 b ∣ ψ ∣^{3} = 0

This yields a non-trivial solution for the magnitude of the order parameter:

∣ ψ_{0} ∣^{2} = - \frac{a ( T )}{b} = \frac{α}{b} (T_{c} - T)

The key insight here is that the minimum of the free energy does not correspond to a single value of $ψ$ , but to an entire circle of values in the complex plane with radius $∣ ψ_{0} ∣$ . The system must "choose" a specific phase $θ_{0}$ to settle into, even though all phases are energetically equivalent. This is a classic example of spontaneous symmetry breaking of a continuous symmetry (the U(1) phase symmetry).

Since the superfluid density is proportional to the order parameter squared, $ρ_{s} \propto ∣ ψ_{0} ∣^{2}$ , this model predicts that:

ρ_{s} (T) \propto (T_{c} - T) for T ≲ T_{c}

This linear dependence is the characteristic mean-field result, corresponding to a critical exponent $β = 1/2$ if we define $ρ_{s}$ as the order parameter (note that typically the order parameter itself is considered, giving $∣ ψ ∣ \propto (T_{c} - T)^{1/2}$ ). Comparing this prediction to the experimental curve in the "Normal and superfluid density" figure, you can see it provides a reasonable qualitative description near $T_{c}$ .

The breaking of a continuous symmetry has profound physical consequences. According to Goldstone's theorem, it implies the existence of a gapless excitation mode, the Goldstone mode. In the case of superfluidity, this mode manifests as a propagating temperature/entropy wave known as second sound, a phenomenon unique to the superfluid phase. Furthermore, the dynamics of the phase $θ$ are deeply connected to the chemical potential $μ$ through the Josephson relation, $ℏ \frac{\partial θ}{\partial t} = μ$ , which governs many remarkable superfluid phenomena.

Conclusion

In this lesson, we have successfully extended Landau's phenomenological theory to describe the superfluid transition in Helium-4. By recognizing that the order parameter must be a complex field $ψ$ to capture the macroscopic quantum coherence of the superfluid state, we were able to construct and analyze the appropriate free energy functional.

Key Takeaways:

The superfluid transition is described by a complex order parameter, $ψ = ∣ ψ ∣ e^{i θ}$ , where $∣ ψ ∣^{2}$ is proportional to the superfluid density $ρ_{s}$ .
The U(1) phase invariance of the system dictates that the Landau free energy can only be a function of $∣ ψ ∣^{2}$ , leading to the "sombrero" potential: $f \approx a ∣ ψ ∣^{2} + \frac{1}{2} b ∣ ψ ∣^{4}$ .
Below $T_{c}$ , the system spontaneously breaks the continuous U(1) symmetry, choosing a ground state with a non-zero $∣ ψ ∣$ but an arbitrary phase $θ$ .
The theory predicts that the superfluid density grows linearly with $(T_{c} - T)$ just below the transition, which is the standard mean-field result.

So far, we have assumed a spatially uniform order parameter. This is sufficient to describe the onset of the ordered phase but cannot capture phenomena involving fluid flow or topological defects like vortices. In our next lesson, we will remedy this by adding a term to the free energy that penalizes spatial variations of the order parameter. This will lead us to the full Landau-Ginzburg free energy functional, a cornerstone of modern condensed matter theory.

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