Variational Derivation of the BCS Gap Equation and Ground State Energy

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Hello!

In our last lesson, we explored Landau's Fermi liquid theory, which provides a powerful description of the normal metallic state in terms of weakly interacting quasiparticles. We concluded by noting that this seemingly stable state is, in fact, susceptible to an instability where an attractive interaction, no matter how weak, can bind quasiparticles into pairs. This is the gateway to superconductivity.

Today, we will take that crucial step and study the theory of Bardeen, Cooper, and Schrieffer (BCS), which explains this remarkable phenomenon. Our goal is to understand how the superconducting ground state forms. Using a variational approach, you will derive the famous BCS gap equation, which determines the fundamental energy scale of the superconductor, and calculate the BCS ground state energy, which quantifies the stability of this new phase of matter.

1. The Cooper Problem: Why Pairing is Inevitable

The first conceptual breakthrough came from Leon Cooper in 1956. He considered a simplified problem: what happens to two electrons interacting just above a filled, inert Fermi sea? He discovered that if there is any attractive potential between them, they will form a bound state. This is counter-intuitive because in free space (3D), a weak potential well isn't always deep enough to hold a bound state. The presence of the Fermi sea changes everything.

The following video provides an elegant physical argument for why this is the case. It explains how the Pauli exclusion principle, by restricting the available states for scattering, fundamentally alters the nature of the problem.

20. Fermi gases, BEC-BCS crossover

Prof. Wolfgang Ketterle explains the essence of the Cooper problem. He shows how the presence of the Fermi sea effectively constrains the interacting particles to a two-dimensional energy-momentum space, where a bound state always forms for any attractive potential.

Please watch from 10:48 to 29:30. Pay close attention to the discussion of bound states in different dimensions and how the Fermi surface provides an effective two-dimensional density of states for the pairing problem. The key result is that the problem of two fermions interacting above the Fermi sea in 3D becomes mathematically analogous to two particles interacting in 2D.

The core insight from Cooper's analysis is that Pauli blocking forbids the two electrons from scattering into states inside the Fermi sea. They can only scatter to empty states above the Fermi energy $ε_{F}$ . When their total momentum is zero (i.e., they have momenta $k$ and $- k$ ), they are always scattering on the "surface" of the Fermi sphere. This restriction to a thin energy shell $\sim ε_{F}$ gives the density of states an effectively constant, 2D-like character.

As you saw in the video (and may recall from your QM courses), a bound state is always possible for an arbitrarily weak attractive potential in 1D and 2D, but not necessarily in 3D. The Fermi sea cleverly provides the conditions for this 2D-like binding to occur. These bound pairs are what we call Cooper pairs.

2. The BCS Variational Ground State

Cooper showed that pairs will form. The next step, taken by Bardeen, Cooper, and Schrieffer, was to construct a many-body ground state for a whole system of such pairs. The crucial idea is that if all electrons form pairs, the ground state can be thought of as a condensate of Cooper pairs.

A brilliant way to construct this state is to use a variational wavefunction. We are looking for a state that minimizes the system's energy. The BCS ansatz for the ground state is:

∣ Ψ_{BCS} ⟩ = k \prod (u_{k} + v_{k} c_{k ↑}^{†} c_{- k ↓}^{†}) ∣0 ⟩

Let's break this down:

$∣0 ⟩$ is the vacuum state (no electrons).
$c_{k σ}^{†}$ is the creation operator for an electron with momentum $k$ and spin $σ$ .
The term $c_{k ↑}^{†} c_{- k ↓}^{†}$ creates a Cooper pair with zero total momentum and spin singlet configuration.
$v_{k}$ and $u_{k}$ are complex variational parameters, representing the probability amplitudes for the pair state $(k ↑, - k ↓)$ to be occupied or empty, respectively.
The product $\prod_{k}$ runs over all relevant momentum states (typically those in a shell around the Fermi surface).
For the state to be normalized, we must have $∣ u_{k} ∣^{2} + ∣ v_{k} ∣^{2} = 1$ . This means $∣ v_{k} ∣^{2}$ is the probability that the pair state $k$ is occupied, and $∣ u_{k} ∣^{2}$ is the probability that it's empty.

This wavefunction is a coherent superposition of states with different numbers of Cooper pairs. It is mathematically analogous to the coherent state of photons in a laser beam, but built from fermionic pairs. Our task is to find the values of $u_{k}$ and $v_{k}$ that minimize the total energy of the system.

3. The Gap Equation and Condensation Energy

To perform the variational calculation, we need the Hamiltonian. For a system with a pairwise attractive interaction, the simplified BCS Hamiltonian can be written in second quantization as:

H_{BCS} = k, σ \sum ξ_{k} c_{k σ}^{†} c_{k σ} + k, k^{'} \sum V_{k k^{'}} c_{k^{'} ↑}^{†} c_{- k^{'} ↓}^{†} c_{- k ↓} c_{k ↑}

Here:

$ξ_{k} = ε_{k} - μ$ is the single-particle energy measured relative to the chemical potential $μ$ .
The first term is the kinetic energy.
The second term describes the scattering of a Cooper pair from $(k ↑, - k ↓)$ to $(k^{'} ↑, - k^{'} ↓)$ with an interaction potential $V_{k k^{'}}$ . For superconductivity, this potential must be attractive, $V_{k k^{'}} < 0$ .

The next step is to calculate the expectation value of this Hamiltonian with respect to the BCS wavefunction, $⟨ Ψ_{BCS} ∣ H_{BCS} ∣ Ψ_{BCS} ⟩$ , and then minimize it with respect to $u_{k}$ and $v_{k}$ .

Superconductivity and Low Temperature Physics I

The following lecture notes from WMI provide a clear, step-by-step derivation of the BCS gap equation and the ground state energy using the variational method we've just described.

This is the core of today's lesson. Please study the following sections carefully: Pages 56-59 (up to the definition of the self-consistent gap equation): Follow the minimization procedure. Start with the expectation value of the energy (p. 56), and follow the variation with respect to the amplitudes (using the parameterization with heta_k on p. 57) to arrive at the expressions for u_k^2, v_k^2, and the gap equation on p. 59. This is the main derivation. Page 63 (Section 4.2.1): See how the general gap equation is solved for the simplified model where the attractive potential is a constant, -V_0, within an energy shell \hbar\omega_D around the Fermi surface. This gives the famous explicit formula for the gap \Delta. Pages 64-66 (Section 4.2.2): Follow the calculation of the condensation energy—the energy difference between the superconducting ground state and the normal Fermi sea. This shows mathematically why the BCS state is stable.

Let's summarize the key results from this derivation.

The BCS Gap Equation

The minimization procedure naturally leads to the introduction of a crucial quantity, the gap parameter $Δ_{k}$ :

Δ_{k} = - k^{'} \sum V_{k k^{'}} u_{k^{'}} v_{k^{'}}

The variational parameters that minimize the energy are then found to be:

∣ v_{k} ∣^{2} = \frac{1}{2} (1 - \frac{ξ _{k}}{E _{k}}) and ∣ u_{k} ∣^{2} = \frac{1}{2} (1 + \frac{ξ _{k}}{E _{k}})

where we have defined a new energy scale $E_{k} = ξ_{k}^{2} + ∣ Δ_{k} ∣^{2}$ .

Substituting the expression for $u_{k} v_{k}$ back into the definition of $Δ_{k}$ gives the celebrated self-consistent BCS gap equation:

Δ_{k} = - k^{'} \sum V_{k k^{'}} \frac{Δ _{k^{'}}}{2 E _{k^{'}}}

This equation is self-consistent because the gap $Δ$ appears on both sides. It's not a formula you solve directly, but an equation that must hold true for the system to be in its lowest energy state. A non-zero solution for $Δ$ signifies the existence of the superconducting state.

For the simple model of a constant attractive potential $- V_{0}$ in an energy shell of width $ℏ ω_{D}$ around the Fermi surface, the gap at zero temperature becomes:

Δ (0) \approx 2ℏ ω_{D} exp (- \frac{1}{N ( 0 ) V _{0}})

where $N (0)$ is the density of states at the Fermi level. This non-perturbative result shows that the energy gap is exponentially small for weak coupling ( $N (0) V_{0} ≪ 1$ ), which is characteristic of conventional superconductors.

The Condensation Energy

With the optimal $u_{k}$ and $v_{k}$ , we can calculate the ground state energy of the superconducting state, $E_{S}$ , and compare it to the energy of the normal state, $E_{N}$ (a filled Fermi sea). The difference is the condensation energy:

E_{cond} = E_{S} - E_{N} = - \frac{1}{2} N (0) Δ^{2} (0)

Since $N (0)$ and $Δ^{2}$ are both positive, the condensation energy is negative. This proves that the BCS state is energetically more favorable than the normal metallic state, explaining why the transition to superconductivity occurs. The energy gain is the driving force for the formation of the condensate.

Test your understanding!

Looking at the formula for the condensation energy, $E_{cond} = - \frac{1}{2} N (0) Δ^{2}$ , what would happen to the stability of the superconducting state if the density of states at the Fermi level, $N (0)$ , were very small? How does this relate to the materials that tend to be good superconductors?

Show answer

If $N (0)$ were very small, the energy gain from forming the superconducting state would be very small, making the state less stable and the critical temperature very low. A large density of states at the Fermi level means there are many electrons available near $ε_{F}$ to participate in pairing. This leads to a larger energy gap $Δ$ (from the gap equation) and a much larger condensation energy, resulting in a more robust superconducting state with a higher transition temperature. This is why materials with a high density of states at the Fermi level, such as certain transition metals and their compounds, are often good candidates for superconductivity.

Conclusion

In this lesson, we have constructed the theoretical foundation for superconductivity. Building upon the instability of the Fermi liquid, we have seen how a new, collective ground state emerges.

Key Takeaways:

Cooper Pairing: The Pauli exclusion principle restricts scattering to states near the Fermi surface, making it possible for an arbitrarily weak attractive interaction to bind two electrons into a Cooper pair.
BCS Wavefunction: The superconducting ground state is described by a variational wavefunction that represents a coherent condensate of Cooper pairs, parameterized by the occupation probabilities $∣ v_{k} ∣^{2}$ .
BCS Gap Equation: Minimizing the system's energy with this wavefunction leads to a self-consistent equation for the gap parameter $Δ$ . A non-zero solution for $Δ$ signals the formation of the superconducting state.
Condensation Energy: The BCS ground state has a lower energy than the normal metallic state. This energy gain, proportional to $Δ^{2}$ , is what stabilizes the superconductor.

Preview of the Next Lesson:

We have established the ground state and the existence of the gap $Δ$ . But what is the physical meaning of the quantity $E_{k} = ξ_{k}^{2} + Δ^{2}$ that appeared in our derivation? It turns out this is the energy of the elementary excitations—the quasiparticles—above the new superconducting ground state. In our next lesson, we will analyze this quasiparticle spectrum and explore its profound consequences for the thermodynamic and transport properties of a superconductor, finally revealing the tangible, measurable meaning of the "energy gap".