The Cooper Instability and Pairing Channels

A Fermi surface is unstable to an arbitrarily weak attraction between time-reversed quasiparticles because the pair susceptibility grows as a logarithm, $N(0)\log(\Lambda/E)$, when the infrared scale $E$ is lowered. The instability occurs in each symmetry-resolved eigenchannel of the two-particle interaction separately: a negative eigenvalue flows to strong attraction, while a positive one becomes weaker. This conclusion establishes a pairing tendency and its symmetry channel. It does not identify the microscopic mediator or prove that the resulting state is realized in a material.

Required background. Fermi-surface kinematics supplies the shell phase space and density-of-states convention. Momentum-shell RG supplies the elimination-and-rescaling argument.

Helpful background. Fermi-surface patch theory explains why the Cooper channel is exceptional among four-fermion interactions.

The singular pair channel

Consider a normal Fermi liquid in $d\geq2$ with a smooth Fermi surface and well-defined quasiparticles over an energy shell $\lvert\xi_{\mathbf k}\rvert<\Lambda$. For zero total momentum, the two propagators carry $(\mathbf k,i\omega_n)$ and $(-\mathbf k,-i\omega_n)$. At temperature $T$ the static pair bubble is

$$\Pi_{pp}(T) =T\sum_{\omega_n}\int_{-\Lambda}^{\Lambda} \mathrm d\xi\,N(\xi) \frac{1}{\omega_n^2+\xi^2} =N(0)\log\frac{2e^\gamma\Lambda}{\pi T}+O(T^2/\Lambda^2).$$

The logarithm is kinematic. Both particles can remain arbitrarily close to the Fermi surface while their total momentum vanishes. A generic particle–particle pair with nonzero total momentum does not retain the same phase space, and generic four-fermion vertices are not logarithmically enhanced in this way. The coefficient above uses $N(0)$ per paired species; changing to a spin-summed density of states changes the definition of the coupling, not the physical scale.

The ladder amplitude in one separable channel, with interaction $V<0$, is

$$\Gamma(E)=\frac{V}{1+V N(0)\log(\Lambda/E)}.$$

Its denominator vanishes at $E_\ast\sim\Lambda e^{-1/[N(0)\lvert V\rvert]}$. The pole means that perturbation theory about the normal state has failed. It is not by itself a controlled calculation of the ordered state; that requires the paired saddle and its fluctuations on the BCS page. The original two-particle version of this argument is due to Cooper 1956, pp. 1189–1190, while the many-body variational construction is developed in Bardeen, Cooper, and Schrieffer 1957, §§II–III.

Symmetry-resolved attractive eigenvalues

On a rotationally invariant Fermi surface, expand the antisymmetrized interaction on the surface,

$$V(\hat{\mathbf k},\hat{\mathbf k}') =\sum_{\ell m}V_\ell Y_{\ell m}(\hat{\mathbf k})Y_{\ell m}^{\ast}(\hat{\mathbf k}').$$

Each $V_\ell$ renormalizes independently at leading logarithmic order. In a crystal, spherical harmonics are replaced by basis functions $\phi_{\Gamma a}(\mathbf k)$ of irreducible representations of the point group. With multiple orbitals or spin–orbit coupling, $V$ is a matrix kernel on band, pseudospin, and Fermi-surface labels. The linearized gap equation is the eigenproblem

$$-\oint_{\mathrm{FS}} \frac{\mathrm dS_{\mathbf k'}}{(2\pi)^d v_F(\mathbf k')} V(\mathbf k,\mathbf k')\Delta(\mathbf k') =\lambda\Delta(\mathbf k).$$

A positive $\lambda$ in this sign convention is attractive and gives $T_c\sim\Lambda e^{-1/\lambda}$. Degenerate basis functions within one representation leave a lower-temperature question—quartic terms, strain, disorder, and other couplings select the realized combination. Fermi statistics impose

$$\Delta_{\alpha\beta}(\mathbf k) =-\Delta_{\beta\alpha}(-\mathbf k),$$

so an even-parity orbital factor pairs with an antisymmetric internal state in the simplest one-orbital setting, and an odd-parity factor pairs with a symmetric one. In multiorbital systems, exchanging orbital labels is also part of the antisymmetry test; the singlet/even and triplet/odd mnemonic is no longer exhaustive.

Shell flow and the physical cutoff

Let $g_a=N_a(0)V_a$ be a dimensionless eigenvalue, negative for attraction, and $\ell=\log(\Lambda_0/\Lambda)$. Eliminating a thin shell gives

$$\frac{\mathrm dg_a}{\mathrm d\ell}=-g_a^2, \qquad g_a(\ell)=\frac{g_a(0)}{1+g_a(0)\ell}.$$

Repulsion is marginally irrelevant; attraction diverges at $\ell_\ast=1/\lvert g_a(0)\rvert$. This is dimensional transmutation: a weak dimensionless coupling generates an exponentially small scale. The numerical prefactor is not universal because it depends on the cutoff, frequency dependence, self-energy, and matching convention.

For a retarded interaction, the shell cannot simply begin at the electronic bandwidth with a constant attraction. A phonon-mediated kernel changes across a characteristic boson scale $\omega_B$, while instantaneous Coulomb repulsion is renormalized over a different interval. The resulting two-stage matching is the entry point to Migdal control and Eliashberg theory. Shankar 1994, §§VI.B–VI.C gives the Fermi-surface RG derivation and its assumptions.

The chapter-wide structure diagram places this logarithm before the saddle, response, and evidence layers. Follow the solid arrows as logical dependencies; dashed boundaries mark conclusions that need independent tests.

The Cooper logarithm licenses an instability in an attractive eigenchannel, not a microscopic mechanism or a gauge-invariant observable. Those conclusions enter only at later, separately tested stages. Original schematic, not to scale.

The complete comparison of claims and failure tests is maintained in the paired-matter claim test matrix.

Checks and limits

• The logarithm disappears if the density of states vanishes sufficiently rapidly at the chemical potential; Dirac or pseudogap systems need a finite critical attraction.
• A population imbalance, orbital mismatch, or pair-breaking field cuts off the time-reversed logarithm and may favor finite-momentum pairing rather than the zero-momentum channel treated here.
• A negative eigenvalue says nothing about phase stiffness. In two dimensions, pair formation may occur above the vortex-unbinding transition.
• The channel with the largest eigenvalue at the adopted approximation need not survive self-energy and vertex corrections. Mechanism claims require multiple discriminating observables, not merely a computed gap function.

Exercise

Integrate the leading flow. An attractive eigenchannel has $g_0=-0.20$ at cutoff $\Lambda_0$. Find the scale where the one-loop coupling diverges and explain what is, and is not, determined by this result.

Solution

The solution is $g(\ell)=g_0/(1+g_0\ell)$, so the denominator vanishes at $\ell_\ast=5$. Hence $E_\ast=\Lambda_0e^{-5}\simeq6.74\times10^{-3}\Lambda_0$. The result establishes the breakdown scale of the normal-state weak-coupling expansion in that eigenchannel. It does not fix the prefactor of $T_c$, the nonlinear order-parameter combination, the stiffness, or the microscopic origin of $g_0$.

References

• Bardeen, J., Cooper, L. N., and Schrieffer, J. R. (1957). “Theory of superconductivity.” Physical Review 108, 1175–1204. doi:10.1103/PhysRev.108.1175.
• Cooper, L. N. (1956). “Bound electron pairs in a degenerate Fermi gas.” Physical Review 104, 1189–1190. doi:10.1103/PhysRev.104.1189.
• Shankar, R. (1994). “Renormalization-group approach to interacting fermions.” Reviews of Modern Physics 66, 129–192. doi:10.1103/RevModPhys.66.129.