Ten facts to remember about BCS

(There was once a late-night comedian named David Letterman who was known for his "Top Ten" lists. This is like that but any humor is accidental.)

1. The variational ground state we found for a model pairing Hamiltonian,
   $$\begin{equation} |\Psi_{BCS}\rangle = \prod_k (u_k + v_k b^\dagger_k) |0\rangle, \quad {u_k}^2 = \frac{1}{2}\left(1 + {\epsilon_k - \mu \over E_k}\right), \quad {v_k}^2 = \frac{1}{2}\left(1 - {\epsilon_k - \mu \over E_k}\right), \end{equation}$$
   has a gap to fermionic excitations. The excitation at momentum $k$ has
   energy
   $$\begin{equation} E_k =\sqrt{(\epsilon_k - \mu)^2 + {\Delta_k}^2}. \end{equation}$$
   Trying to add or subtract an electron gives
   $$\begin{equation} c^\dagger_{p\uparrow} |\Psi_{BCS}\rangle = u_p \prod_{k \not = p} (u_k + v_k b^\dagger_k) |0\rangle = u_p |\psi_{p^\uparrow}\rangle,\quad c_{-p\downarrow} |\Psi_{BCS}\rangle = -v_p \prod_{k \not = p} (u_k + v_k b^\dagger_k) |0\rangle = -v_p |\psi_{p^\uparrow}\rangle. \end{equation}$$
   Hence we defined normalized, fermionic creation and annihilation operators
   $$\begin{equation} \gamma_{p\uparrow}^\dagger = {u_p} c_{p\uparrow}^\dagger - {v_p} c_{-p \downarrow}, \quad \gamma_{-p\downarrow} = {v_p} c_{p\uparrow}^\dagger + {u_p} c_{-p \downarrow}. \end{equation}$$
   Note that each of these adds momentum $p$ and spin up.

2. The particle number is uncertain in the BCS ground state as written above, and the Bogoliubov excitations do not have well-defined particle number (or charge). Schrieffer argues that one shouldn't be troubled by the uncertainty in the ground state because the relative fluctuations are small. A different interpretation is that the particle number of an isolated system would be well-defined: the particle number is only uncertain when a supercurrent flows, i.e., when it is possible to measure the superconducting phase.

3. The Ginzburg-Landau equation
   $$\begin{equation} {\hbar^2 n^* \over 2 m^*} \left[ \nabla + {i e^* \over \hbar c} {\bf A}({\bf r})\right]^2 \psi({\bf r}) + a(T) \psi({\bf r}) + b(T) |\psi({\bf r})|^2 \psi({\bf r}) = 0. \end{equation}$$
   describes the center-of-mass eigenstate of the Cooper pairs. In the BCS form written above, there are $N/2$ Cooper pairs that all have total momentum zero and zero spin. In a vector potential, say, the center-of-mass wavefunction would obey the Ginzburg-Landau eqn.

4. The main error in the unrealistic interaction model (the pairing form for the Coulomb term) is the neglect of long-range interactions, which will turn out to be responsible for the phonon gap in an s-wave superconductor in three dimensions (the Anderson-Higgs effect).

5. With a specific, even more unrealistic model for the pair potential,
   $$\begin{equation} V_{kk^\prime} = \begin{cases} V_0 & {\rm if}\ |\epsilon_k - \mu|\ {\rm and}\ |\epsilon_{k^\prime}-\mu| < \omega_c \cr 0&{\rm otherwise} \end{cases} \end{equation}$$
   we could exactly solve the gap equation and found $\Delta \approx 2 \omega_c e^{-1/(N(0) V)}$.

6. The gap, which is nonperturbative in the coupling strength $V$ even in more realistic models of the potential, is of the same order of magnitude as the superconducting transition temperature $T_c \approx (2 \Delta) / 3.5$. There is really just one "small" energy scale in the theory.

7. The gap is not necessary for superconductivity. For instance, suppose we tried to argue that $T_c$ and $\Delta$ had to be of the same order because once the temperature was larger than $2 \Delta$, it would be easy to thermally excite quasiparticles. The counterexample is that $p$-wave and $d$-wave superconductors, which have gapless quasiparticles, can have nonzero superconducting transition temperatures. Another way to create gapless superconductivity is in a disordered system. Magnetic impurities (which break time-reversal symmetry) tend to destroy superconducting order much more rapidly than nonmagnetic impurities.

8. Suppose we try to make a length scale associated with the gap. A choice with the right units is
   $$\begin{equation} \xi \sim {\hbar v_F \over \Delta}. \end{equation}$$
   The penetration depth is rather different and has no $\Delta$:
   $$\begin{equation} \lambda = \left({m c^2 \over 4 \pi n_s e^2}\right)^{1/2}. \end{equation}$$
   The physical significance of $\xi$, known as (Pippard's) coherence length, is that it corresponds to the size of a Cooper pair, in the sense that the BCS wavefunction can be rewritten in the real-space form
   $$\begin{equation} |\Psi_{BCS}\rangle = {\cal A} \phi(x_1 - x_2) \phi(x_3 - x_4) \phi(x_5 - x_6)\ldots, \end{equation}$$
   and the pair wavefunction $\phi$ falls off exponentially beyond the scale $\xi$ (with a considerable amount of algebra, it can be shown that $\phi(r) = \sum_k g_k e^{i k r}$.
   (This rewriting was noticed by Dyson some time after the original BCS theory.) You should remind yourself why the coherence length and penetration depth are very different scales (cf. Ashcroft and Mermin or some similar text): note in particular that the coherence length gets shorter as the gap gets larger and $T_c$ increases. The penetration depth is essentially a measure of the superfluid density, and can be determined from the Ginzburg-Landau equation, while the coherence length cannot. Note that high-temperature superconductors have short coherence lengths.

9. The pair wavefunction introduced in the previous item has a symmetry related to that of the gap $\Delta_k$. For instance, if $\Delta_k$ has p-like or d-like symmetry, then the pair wavefunction vanishes when the two points come together. For this reason, $p$-like and $d$-like symmetries are favored when the two-particle potential is repulsive at short distances but attractive at long distances, as for the neutral atoms of He$^3$.

10. The Ginzburg-Landau equation, like single-particle quantum mechanics, is invariant under the gauge transformation
    $$\begin{equation} V \rightarrow V - {1 \over c} {\partial \chi \over \partial t},\quad A \rightarrow A + \nabla \chi,\quad \Phi \rightarrow \Phi + {2 e \chi \over \hbar c}. \end{equation}$$
    This has fundamental consequences like the Josephson effect and the quantization of flux
    through a superconducting ring. The notion of Cooper pairing explained the known experimental fact that $2e$ appears here rather than $e$.

Recall that a superconductor is type I (only one critical field) essentially if $\xi > \lambda$, and type II if $\lambda > \xi$. A simple way to express the difference is that the penetration depth depends on the energy cost to vary the magnetic field, while the coherence length depends on the energy cost to bend the phase. For example, around a vortex core the wavefunction must twist by $2 \pi$, so the core size (the region of no superconductivity) is at least of order $\xi$.

Because there are two lengths, in a type II superconductor there are two critical fields: $H_{c1}$, when the field begins to penetrate macroscopically into the superconductor (leading to a "mixed state" of vortices), and $H_{c2}$, when the vortex cores begin to overlap and superconductivity is destroyed. We can estimate $H_{c2}$ by asking when there is one flux quantum through an area the size of the coherence length (here the numerical factor requires a more detailed calculation:
$$\begin{equation} H_{c2} = {\phi_0 \over 2 \pi \xi^2}, \phi_0 = {2 \pi \hbar c \over 2 |e|} = 2 \times 10^{-7} \rm{G\ cm^2}. \end{equation}$$

Note that the magnetic field in a type II superconductor near $H_{c2}$ is more uniform than the superfluid density. A good introductory reference for you to review is the superconductivity chapter of Ashcroft and Mermin.

Bonus example: The structure of a vortex in a Type II superconductor is interesting. The GL current density is (here $n$ is the total density of electrons)
$$\begin{equation} {j_x \over n} = - {i e \hbar \over 2 m} \left(\psi_1^* {\partial \psi \over \partial x} - \psi_1 {\partial \psi^* \over \partial x} \right) - {2 e^2 \over m c} A_x \psi_1^* \psi_1. \end{equation}$$
and similarly for $j_y$ and $j_z$. Consider a single hole containing magnetic flux in a large superconducting body. Far away from the hole, the supercurrent should be zero and the magnitude $|\psi|$ should be constant. For constant $|\psi|$, the current can be rewritten
$$\begin{equation} j = {e \hbar n_s \over 2 m} \left(\nabla \phi - {2 e \over \hbar c} {\bf A}\right) \end{equation}$$ .
Setting this equal to zero gives
$$\begin{equation} \nabla \phi = {2 e \over \hbar c} {\bf A}. \end{equation}$$
Now, around any closed loop the phase $\Phi$ should change by a multiple of $2 \pi$ for the wavefunction in the GL equation to be single-valued.
Then the line integral of $A$ around the loop must be a multiple of ${2 \pi \hbar c \over 2 e} = \Phi_0$. Finally, the line integral of $A$ around the loop is just the integrated magnetic flux through the loop
$$\begin{equation} \Phi = \int {\bf B} dS = n \Phi_0 = n {\pi \hbar c \over e} = {h c \over 2 e}. \end{equation}$$
In real units the flux quantum is $2 \times 10^{-7}$ G cm$^2$ as mentioned above.

The above vortex is a simple example of a "topological defect" in a field theory: on a length scale much larger than the vortex, the vortex appears as a point singularity around which the phase wraps by $2 \pi n$. Similar topological defects occur in magnets, in liquid crystals (described by a classical field theory) and in high-energy physics (for example, magnetic monopoles in gauge theories).

A key difference between superfluids and superconductors is that, in a superconductor, the winding of the gauge field compensates the winding of the condensate phase $\theta$ so that the supercurrent goes to zero, which is what we used above. In a superfluid, there is no such gauge field so the supercurrent decays to zero only slowly (there is a ``quantization of vorticity''). As a result, the interaction between superfluid vortices is long-ranged, which leads to a famous phase transition in thin films: the Kosterlitz-Thouless transition.