The Missing Small Parameter: Why Cuprate Pairing Resists Calculation — Epoche C2
The claim under review A common summary of the field runs like this. Superconductivity in the copper oxides is an engineering problem. The physics is the electron pairing that Bardeen, Cooper and Schrieffer described in 1957, only with a stronger glue, and what remains is to find the compound in which the glue is strongest. This review argues that the summary is wrong, and wrong for a structural reason rather than for want of effort. Forty years after Bednorz and Müller's discovery, the cuprates have no accepted pairing mechanism because the calculation BCS theory performs cannot be set up here at all. Energies below are quoted in eV and meV and lengths in ångströms, the convention of the condensed-matter literature cited; where electromagnetism enters, Gaussian CGS is understood. Why the conventional calculation converges BCS theory succeeds not because the attraction between electrons is understood in fine detail, but because it is weak — and weakness is a licence to expand in a small number. Two such numbers do the work. The first is the dimensionless electron–phonon coupling $\lambda$, the product of the electronic density of states at the Fermi energy and the phonon-mediated attraction. Solving the BCS gap equation in the limit $\lambda \ll 1$ gives $$k_B T_c = 1.13\,\hbar\omega_D\,\exp\!\left(-\frac{1}{\lambda}\right),$$ where $k_B$ is Boltzmann's constant, $T_c$ the transition temperature, and $\hbar\omega_D$ the Debye energy — roughly the largest lattice-vibration energy in the crystal. The exponential form is the signature of an instability that exists for any $\lambda > 0$: it cannot be reached by expanding in $\lambda$, which is exactly why the pairing had to be summed to all orders in the first place. Invert the formula for aluminium, with $T_c = 1.18$ K and Debye temperature $\Theta_D = 428$ K: $\exp(-1/\lambda) = 1.18/(1.13 \times 428) = 2.44\times10^{-3}$, so $\lambda = 0.17$. The number really is small. The second small number is Migdal's ratio $\hbar\omega_D/E_F$, the phonon energy divided by the Fermi energy — the energy of the highest occupied electron state measured from the band bottom. It is small because ions are heavy and slow compared with electrons. For aluminium, $\hbar\omega_D = k_B \times 428\ \mathrm{K} = 0.037$ eV against $E_F = 11.7$ eV, giving $3.2\times10^{-3}$. Migdal showed in 1958 that corrections to the electron–phonon vertex are suppressed by this ratio, which is what licenses Eliashberg's treatment even when $\lambda$ approaches unity, as in lead. The theory of conventional superconductivity is therefore a controlled theory: one can say what the next term is and why it is negligible. What is different in a copper oxide Reduce a CuO$_2$ plane to its minimal model and the picture inverts. The single-band Hubbard description carries a nearest-neighbour hopping $t \approx 0.4$ eV, so a bandwidth $W = 8t \approx 3.2$ eV, against an on-site Coulomb repulsion $U \approx 3.5$ eV — the energy cost of putting two electrons on the same copper site. The expansion parameter is $U/W \approx 1.1$. Set that against aluminium's $3.2\times10^{-3}$: the ratio of the two is $1.1/3.2\times10^{-3} \approx 3\times10^{2}$, two and a half orders of magnitude. There is nothing to expand in, in either direction — the interaction is neither a perturbation on the band structure nor the band structure a perturbation on the interaction. The second obstruction is worse, because it concerns the starting point rather than the expansion. Cooper's calculation asks what happens when an attraction is switched on in front of a filled Fermi sea of long-lived quasiparticles. Above $T_c$ the cuprates do not supply one. Near optimal doping the resistivity is linear in temperature from just above $T_c$ to well above room temperature, with a scattering rate close to the Planckian value $\hbar/\tau = k_B T$. At $T = 300$ K this gives $\tau = \hbar/(k_B T) = 1.05\times10^{-27}/(1.38\times10^{-16}\times300) = 2.5\times10^{-14}$ s, and with a nodal Fermi velocity $v_F \approx 2.5\times10^{7}$ cm s$^{-1}$ a mean free path $\ell = v_F\tau \approx 6.3\times10^{-7}$ cm $= 63$ Å — some sixteen lattice spacings, since the in-plane Cu–Cu distance is $a = 3.8$ Å, and shrinking as $1/T$ with no sign of saturating. The excitation decays in about the time $\hbar/k_BT$ set by its own thermal energy: the ratio $\hbar/(\tau k_B T)$ that would license treating it as long-lived is exactly one, not small. One cannot compute the instability of a state one has not solved. What the data actually pin down Observation Value What it excludes Gap symmetry, from phase-sensitive tricrystal and corner-junction experiments $\Delta(\mathbf{k}) = \tfrac{1}{2}\Delta_0(\cos k_x a - \cos k_y a)$ Any isotropic $s$-wave pairing; favours a repulsive interaction peaked near $\mathbf{Q}=(\pi/a,\pi/a)$ Gap ratio, optimally doped Bi-2212 ($T_c = 91$ K, antinodal $\Delta_0 \approx 35$ meV) $2\Delta_0/k_BT_c = 70/7.84 = 8.9$ Weak coupling: the $d$-wave BCS value is 4.28, so the measured ratio is 2.1 times larger Oxygen isotope exponent $\alpha$ in $T_c \propto M^{-\alpha}$, at optimal doping $\alpha \approx 0.02$–$0.05$ Phonons as the sole scale-setter, which would give $\alpha = 0.5$; but $\alpha$ rises towards 0.5 when underdoped, so phonons are not absent either Pseudogap onset $T^*$, terminating near hole doping $p^* \approx 0.19$ Suppression of low-energy spectral weight above $T_c$ A single-transition picture; whether this is competing order or precursor pairing is unsettled The $d$-wave form factor changes sign under a 90° rotation and vanishes along the zone diagonals. That sign change is what allows a repulsive interaction to bind: the pair wavefunction avoids the origin in real space, so a short-range repulsion costs nothing. This is the argument behind spin-fluctuation exchange, and it gets the symmetry right. But the interaction it uses is obtained by resumming diagrams in a coupling that must be strong for $T_c$ to be high —