The series refutes its own claim ── and then sorts out what actually is similar, and on which layer
Four episodes in, we have lined up things tunneling explains. The Sun burns. Alpha nuclei break up. Current flows without voltage. By now one wants to say ── "aren't tunneling, fusion and superconductivity the same phenomenon described differently?" You can't. The first two are right and the last is wrong. The Josephson effect really is Cooper pairs tunneling, but the reason Cooper pairs exist at all ── superconductivity itself ── has nothing to do with walls. It is an instability of the Fermi surface, an entirely different mechanism. ── But it would be dull if that were the end of it. The BCS gap \(\Delta\sim e^{-1/N(0)V}\) and tunneling's \(e^{-S_E/\hbar}\) are eerily similar in form. That is not a coincidence. This episode opens with the refutation and then goes on to sort what is similar on which layer into three grades.
First, terminology. Remembering "superconductivity = zero electrical resistance" will send you off course. The property that defines superconductivity is the Meissner effect.
| Property | What it is | Discovery |
|---|---|---|
| zero resistance | current does not decay | 1911, Onnes (mercury, 4.2 K) |
| Meissner effect | magnetic field is expelled from the interior. A metal that merely happened to lose its resistance would not do this; the field would stay trapped inside | 1933, Meissner and Ochsenfeld |
| energy gap | exciting one electron costs at least \(\Delta\). So a small poke does not destroy it | 1950s (heat capacity, infrared absorption) |
The Meissner effect is decisive because it shows this is a thermodynamic state. It does not depend on history: cool first then apply the field, or apply the field then cool, and you land in the same state. Superconductivity is not "an unlikely event" but a phase that is simply there. Already at this point it is a different kind of thing from tunneling.
In 1956 Cooper showed something, in a terrifyingly short paper.
Add two electrons on top of a Fermi surface, with any attraction between them, however weak. Then those two necessarily form a bound state.
This is a startling claim. Normally a weak attraction produces no bound state (in three dimensions there is a threshold). The presence of a Fermi surface removes the threshold ── the very existence of a filled Fermi sea is what makes binding possible. So superconductivity is not a two-electron problem; it is a many-body problem from the outset.
The attraction comes from lattice vibrations (phonons). An electron passing through pulls the positive ions slightly toward it, leaving a puddle of positive charge a moment later. The next electron arrives there ── a weak attraction that dodges Coulomb repulsion by being delayed in time.
The gap \(\Delta\) is fixed by an equation that contains it (self-consistency). The standard weak-coupling form is
$$1=N(0)V\!\!\int_0^{\hbar\omega_D}\!\!\frac{d\xi}{\sqrt{\xi^2+\Delta^2}} =N(0)V\,\mathrm{arcsinh}\!\left(\frac{\hbar\omega_D}{\Delta}\right)$$Solve for \(\Delta\) to get \(\Delta=\hbar\omega_D/\sinh\!\big(1/N(0)V\big)\). At weak coupling (\(N(0)V\ll1\)),
$$\boxed{\ \Delta\approx 2\hbar\omega_D\,e^{-1/N(0)V}\ }$$\(N(0)\) is the density of states at the Fermi surface and \(V\) the strength of the attraction. Riding on the exponent is 1/(coupling constant). \(T_c\) comes out in the same form, and taking the ratio cancels the coupling ── \(2\Delta/k_BT_c=3.53\). The same number for different materials, one of BCS's signature predictions.
Side by side. This is the body of the episode.
| Tunneling | Superconductivity (BCS) | |
|---|---|---|
| Stage | a barrier in space | no barrier. the Fermi surface |
| What happens | it moves from A to B | the ground state reorganises (a phase transition) |
| Number of particles | one (or one variable) | many-body from the start. Undefinable for one |
| Meaning of the exponent | a probability (dimensionless, unlikeliness) | an energy (dimensionful, depth of binding) |
| What rides on the exponent | \(S_E/\hbar\) (action ÷ ℏ) | \(1/N(0)V\) (inverse coupling) |
| Temperature | essentially temperature-independent | a phase transition at \(T_c\) |
| The Meissner effect? | cannot explain it | follows from symmetry breaking |
The fourth row is especially telling. One is a probability, the other an energy ── they do not even share dimensions. They cannot possibly be "the same phenomenon."
One more. The Meissner effect follows from the photon acquiring a mass inside a superconductor (the Anderson–Higgs mechanism; see the sister series "Fields That Click" and "Mass That Clicks"). The field cannot get in because a massive field decays exponentially. Another exponential ── but this one is a screening length, not a tunneling exponent. Similar in form, different in origin ── this episode is full of that.
The refutation ends here. Now, what really is common?
Differentiate \(e^{-1/\lambda}\) with respect to \(\lambda\) as often as you like and every derivative vanishes as \(\lambda\to0^+\). Its Taylor expansion is identically zero. However many orders of perturbation theory you sum, this term never appears.
The same holds for \(e^{-S_E/\hbar}\) (non-analytic in \(\hbar\)). Tunneling and the BCS gap both enter through the same hole: "invisible to perturbation theory." That is the real commonality, and Episode 6 takes it head on.
The figure solves the BCS gap equation. Horizontal axis: coupling strength \(\lambda=N(0)V\). Vertical: \(\Delta/\hbar\omega_D\) (log).
The important curve is the grey line ── the prediction of perturbation theory. At first order, tenth order or hundredth order, the answer is exactly zero. In the figure it stays pinned to the bottom edge. The true answer (purple) is not zero, and perturbation theory never once detects it. Measured values for aluminium, niobium and lead are plotted too.
Pulling this together, the word "similar" has at least three grades. This distinction is the single most useful tool in the series.
| Grade | Meaning | Examples |
|---|---|---|
| Grade 1 family resemblance | the same form of equation, different origins | the BCS gap \(e^{-1/\lambda}\) and tunneling \(e^{-S_E/\hbar}\) the Meissner screening length and the decay length inside a barrier |
| Grade 2 different solutions of one framework | different solutions or saddle points of the same theory | instantons, bounces and the BCS mean field (all "non-trivial saddle points" ── Episode 6) |
| Grade 3 identity | one follows from the other | fusion ← tunneling (Episode 2) the Josephson effect ← tunneling (Episode 4) |
The opening intuition ── "tunneling, fusion and superconductivity are the same phenomenon" ── mixes grade 3 with grade 1. Fusion and Josephson are grade 3 (genuinely the same); superconductivity itself is grade 1 (merely a similar form) ── and, as Episode 6 will show, they are connected at grade 2.
When you say "similar," always say which grade. That is this series' house rule.
Established: that the Meissner effect (1933) is the thermodynamic property defining superconductivity; the Cooper instability (1956); BCS theory (1957, Nobel Prize 1972) and the gap equation \(\Delta=\hbar\omega_D/\sinh(1/N(0)V)\) with weak-coupling form \(\Delta\approx2\hbar\omega_De^{-1/N(0)V}\); the universal ratio \(2\Delta/k_BT_c=3.53\); the isotope effect \(T_c\propto M^{-1/2}\) (1950) as evidence for the phonon mechanism; the understanding of the Meissner effect via the Anderson–Higgs mechanism; and the non-analyticity of \(e^{-1/\lambda}\) at \(\lambda=0\) with identically vanishing Taylor coefficients. All established physics.
Points to note: (1) BCS is a weak-coupling mean-field theory. In strong-coupling materials such as lead and mercury, \(2\Delta/k_BT_c\) departs from 3.53 (about 4.3 for lead) and Eliashberg theory is required. The weak-coupling form in the figure also deviates for \(\lambda\gtrsim0.3\). (2) Cuprate high-temperature superconductors cannot be explained by the BCS-phonon mechanism. Nearly forty years after their 1986 discovery, the mechanism is still unsettled (bonus ⑥ takes this up). This episode concerns conventional superconductors. (3) The isotope effect is not \(M^{-1/2}\) in every superconductor (it is small in transition metals, and there are zero and negative cases). (4) "A Fermi surface removes the threshold" is an argument about the effectively two-dimensional density of states; strictly it is a logarithmic divergence that does the work (Episode 6 translates this into renormalization-group language). (5) The measured points in the figure are guides, with \(\lambda\) back-solved from \(T_c\) and the Debye temperature using the weak-coupling formula; they are not precision values. (6) The body's claim "not tunneling" is about the mechanism that produces superconductivity; the Josephson effect (Episode 4) and quasiparticle tunneling in superconductor–insulator–metal junctions are unambiguously tunneling.
"Tunneling, fusion and superconductivity are the same phenomenon" ── the first two are right and the last is wrong. What defines superconductivity is the Meissner effect, and that makes it a thermodynamic phase. The mechanism is the Cooper instability: with a Fermi surface present, any attraction however weak produces a bound state ── a many-body story with nothing to do with barriers in space. The exponents mean different things too ── tunneling's is a probability, the BCS gap is an energy. Not even the same dimensions.
The commonality that remains is real and essential. Both \(e^{-1/\lambda}\) and \(e^{-S_E/\hbar}\) are non-analytic at zero coupling (or zero \(\hbar\)). All Taylor coefficients vanish, so no order of perturbation theory produces them. Both enter through the same hole: "invisible to perturbation theory."
So we graded "similar" into three: grade 1 (same form) / grade 2 (different solutions of one framework) / grade 3 (one follows from the other). Fusion and Josephson are grade 3; superconductivity itself is grade 1. But Episode 6 will show they are also connected at grade 2. When you say "similar," say which grade.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, moving the coupling strength shows the gap between the exact solution and the weak-coupling form. Check that the perturbation-theory line (grey) never leaves the bottom edge, whatever you do. "See the answer" opens each solution.