Conventional Superconductivity Derivations

This appendix collects derivations supporting the conventional-superconductivity background chapter. They are standard results, kept here so the background chapter can focus on the physical structure used later in the thesis.

Cooper Instability

Consider two electrons above a filled Fermi sea with opposite momenta $\pm k$ and energies

ε_{k} = \frac{ħ ^{2} k ^{2}}{2 m} .

Take the Hamiltonian

H = H_{0} + V, H_{0} = 2 k \sum ε_{k}, V = k, k^{'} \sum V_{k, k^{'}} .

Assume

V_{k, k^{'}} = {- V, 0, ε_{F} \leq ε_{k}, ε_{k^{'}} < ε_{F} + ħ ω_{D}, otherwise, V > 0.

Expand the eigenstate as

Ψ = k \sum g_{k} Ψ_{k} .

Then the self-consistency condition becomes

1 = V k \sum \frac{1}{2 ε _{k} - E} .

Approximating the density of states by $N (ε_{F})$ across the shell gives

1 \approx V N (ε_{F}) \int_{ε_{F}}^{ε_{F} + ħ ω_{D}} \frac{d ε}{2 ε - E} .

Define

Δ = 2 ε_{F} - E .

Then in the weak-coupling limit,

Δ \approx 2 ħ ω_{D} exp [- \frac{2}{V N ( ε _{F} )}] .

Thus an arbitrarily weak attractive interaction in the Cooper channel produces a bound state in the presence of the Fermi sea [1].

BCS Variational State and Pseudospins

For each pair $(k ↑, - k ↓)$ define the empty and occupied pair states

∣0 ⟩_{k}, ∣1 ⟩_{k} = ∣ k ↑, - k ↓ ⟩ .

A variational state at fixed $k$ is

∣Ψ ⟩_{k} = u_{k} ∣0 ⟩_{k} + v_{k} ∣1 ⟩_{k}, ∣ u_{k} ∣^{2} + ∣ v_{k} ∣^{2} = 1.

The BCS ground state is

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

Minimisation of $⟨ Φ_{BCS} ∣ H ∣ Φ_{BCS} ⟩$ yields the gap equation and the coherence factors [2, 1].

The same structure can be written in terms of Anderson pseudospins,

σ_{k}^{+} = c_{k ↑}^{†} c_{- k ↓}^{†}, σ_{k}^{-} = c_{- k ↓} c_{k ↑},

σ_{k}^{z} = \frac{1}{2} (c_{k ↑}^{†} c_{k ↑} + c_{- k ↓}^{†} c_{- k ↓} - 1) .

Then the reduced BCS interaction takes the schematic form

H_{int} = - k, k^{'} \sum V_{k, k^{'}} σ_{k}^{+} σ_{k^{'}}^{-} .

This representation makes explicit that superconductivity is a collective pseudospin-ordering problem rather than a sum of independent two-body bound states.

Geometric Form of GL Theory

A compact reformulation treats $Ψ$ as a section of a complex line bundle over the sample, with electromagnetic coupling described by a connection one-form $A$ . Gauge transformations act as

Ψ \mapsto e^{i χ} Ψ, A \mapsto A + \frac{ħ}{q} \nabla χ .

The covariant derivative

D Ψ = (\nabla - i q A / ħ) Ψ

is gauge-covariant, and the magnetic field is the curvature

B = \nabla \times A .

Flux quantisation can then be understood as a statement about holonomy and consistency around nontrivial loops [3].

References

M. Tinkham, Introduction to superconductivity. Mineola, NY: Dover Publications, 2004. (↩︎)
J. Bardeen, L. Cooper, and J. Schrieffer, Theory of superconductivity, Phys. Rev., vol. 108, no. 5, pp. 1175–1204, 1957. doi:10.1103/PhysRev.108.1175 (↩︎)
M. Nakahara, Geometry, topology and physics. Boca Raton: Taylor & Francis, 2003. (↩︎)

Cooper Instability

BCS Variational State and Pseudospins

Geometric Form of GL Theory

References

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