The Cooper Instability and Pairing Channels

The zero-total-momentum pair channel of a Fermi liquid is special: if a codimension-one set of partner states remains degenerate and has a finite density of states, repeated pair scattering produces a logarithm, $N_{\mathrm{pair}}(0)\log(\Lambda/E)$. An arbitrarily weak attractive eigenvalue then drives the normal-state expansion to strong coupling at an exponentially small scale. This page derives that logarithm, classifies the independent pairing channels, and separates the resulting instability from a transition temperature, an ordered-state solution, or a microscopic mechanism.

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, especially its exceptional-channel diagram explains why the Cooper family retains angular phase space while a generic four-fermion process does not.

What becomes unstable?

Work in a normal Fermi liquid in $d\geq2$ with a smooth Fermi surface, coherent quasiparticles, a nonsingular short-range interaction, and an energy shell $\lvert\xi_{a\mathbf k}\rvert<\Lambda$. The label $a$ may include band and internal quantum numbers. Zero-momentum pairing requires a partner $\bar a$ such that

$$\xi_{a\mathbf k}=\xi_{\bar a,-\mathbf k}$$

on an extended part of the Fermi surface. Time-reversal symmetry supplies equal-energy partners related by $\Theta$; at a time-reversal-invariant momentum they form a Kramers pair when $\Theta^2=-1$. Inversion can instead supply the same $\mathbf k\leftrightarrow-\mathbf k$ energy degeneracy even when the two states are not literally time reverses. A mismatch from population imbalance, orbital splitting, or a pair-breaking field stops the argument at that mismatch scale.

After matching onto quasiparticles and absorbing their residues into the interaction vertex, write

$$G_a(\mathbf k,i\omega_n) =\frac{1}{i\omega_n-\xi_{a\mathbf k}}, \qquad \omega_n=(2n+1)\pi T.$$

The singular object is the normal-state pair susceptibility. Its growth says that perturbation theory around the unpaired state is losing control; it does not yet say which ordered saddle is realized.

Deriving the Cooper logarithm

For one degenerate pair of states, the static zero-momentum bubble is

$$\begin{aligned} \Pi_{pp}(T) &=T\sum_{\omega_n}\int_{-\Lambda}^{\Lambda} \mathrm d\xi\,N_{\mathrm{pair}}(\xi) G(\xi,i\omega_n)G(\xi,-i\omega_n)\\ &=T\sum_{\omega_n}\int_{-\Lambda}^{\Lambda} \mathrm d\xi\,N_{\mathrm{pair}}(\xi) \frac{1}{\omega_n^2+\xi^2}. \end{aligned}$$

Here $N_{\mathrm{pair}}$ counts one member of the pair; for a two-spin-degenerate band it is half the spin-summed density of states. The fermionic Matsubara sum is

$$T\sum_{\omega_n}\frac{1}{\omega_n^2+\xi^2} =\frac{\tanh(\xi/2T)}{2\xi},$$

so

$$\Pi_{pp}(T) =\int_0^\Lambda\frac{\mathrm d\xi}{2\xi} \bigl[N_{\mathrm{pair}}(\xi)+N_{\mathrm{pair}}(-\xi)\bigr] \tanh\frac{\xi}{2T}.$$

For a constant density of states and a sharp symmetric energy cutoff, this becomes

$$\Pi_{pp}(T) =N_{\mathrm{pair}}(0) \log\frac{2e^\gamma\Lambda}{\pi T} +O\!\left(N_{\mathrm{pair}}(0)e^{-\Lambda/T}\right).$$

The exact factor $2e^\gamma/\pi$ belongs to that cutoff model. A smooth energy-dependent density of states leaves the leading $N_{\mathrm{pair}}(0)\log(\Lambda/T)$ unchanged but adds finite, matching-dependent terms. This distinction matters whenever a quoted transition scale is compared across regulators or microscopic models. The weak-coupling finite-temperature integral is the same one that enters the BCS linearized gap equation; Bardeen, Cooper, and Schrieffer 1957, §II, pp. 1179–1183.

The geometry explains the logarithm. At total momentum zero, choosing any $\mathbf k$ on the pair-degenerate surface automatically places its partner at $-\mathbf k$, so the full surface label survives while the radial integral gives $\int_E^\Lambda\mathrm d\lvert\xi\rvert/\lvert\xi\rvert$. At generic total momentum $\mathbf Q$, the conditions $\xi_{a\mathbf k}=0$ and $\xi_{b,\mathbf Q-\mathbf k}=0$ intersect in lower dimension, and the full logarithmic phase space is lost unless nesting or another special degeneracy restores it. Shankar 1994, §IX, pp. 181–183 extends the argument to time-reversal-invariant noncircular Fermi surfaces.

Ladder resummation and the breakdown scale

In a separable channel with signed interaction $V_a<0$, repeated pair scattering forms a geometric series,

$$\Gamma_a(E) =\frac{V_a}{1+V_aN_{\mathrm{pair}}(0)\log(\Lambda/E)}.$$

The denominator vanishes at

$$E_\ast =\Lambda\exp\!\left[-\frac{1}{N_{\mathrm{pair}}(0)\lvert V_a\rvert}\right].$$

This pole is the scale where the normal-state weak-coupling expansion fails. In the ideal instantaneous BCS model it tracks the mean-field transition scale up to a regulator-dependent prefactor, but in general it is not a calculation of $T_c$, the zero-temperature gap, or the phase stiffness. The original two-particle bound-state argument is Cooper 1956, pp. 1189–1190; the many-body saddle is the next step on the BCS page.

Pairing as an eigenvalue problem

For an anisotropic or multiband Fermi surface, define the weighted surface measure on sheet $j$ by

$$\mathrm d\mu_j(\mathbf k) =\frac{\mathrm dS_{\mathbf k}}{(2\pi)^d v_{F,j}(\mathbf k)}.$$

The antisymmetrized interaction is diagonalized through

$$-\sum_j\oint_{\mathrm{FS}_j}\mathrm d\mu_j(\mathbf k')\, V_{ij}(\mathbf k,\mathbf k')\phi_{nj}(\mathbf k') =\lambda_n\phi_{ni}(\mathbf k).$$

If equilibrium reciprocity gives $V_{ij}(\mathbf k,\mathbf k')=V_{ji}^{\ast}(\mathbf k',\mathbf k)$, this operator is self-adjoint in the inner product

$$\langle\phi,\psi\rangle =\sum_i\oint_{\mathrm{FS}_i}\mathrm d\mu_i\, \phi_i^\ast\psi_i,$$

so the eigenvalues are real and the eigenfunctions can be chosen orthonormal. A positive $\lambda_n$ is attractive because of the explicit minus sign. To connect with the signed RG convention below, define

$$g_n=-\lambda_n: \qquad \lambda_n>0 \quad\Longleftrightarrow\quad g_n<0 \quad\Longleftrightarrow\quad \text{attraction}.$$

In a single static weak-coupling channel, $E_{\ast,n}\sim\Lambda e^{-1/\lambda_n}$. On a three-dimensional sphere the eigenfunctions are spherical harmonics $Y_{\ell m}$; on a two-dimensional circle they are Fourier harmonics $e^{im\theta}$. A crystal instead organizes them into irreducible representations of its point group. In simple settings, $s$ wave is fully symmetric, $p$ wave is odd under $\mathbf k\to-\mathbf k$, and $d$ wave changes sign in a quadrupolar pattern. These names are shorthand for transformation laws, not evidence for a mediator.

The universal exchange condition includes momentum, internal labels, and relative frequency:

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

The familiar singlet-even and triplet-odd rule further assumes even-frequency pairing, inversion symmetry, one symmetric orbital sector, and spin—or a valid Kramers pseudospin—as a good label. Without inversion, even- and odd-parity components can mix; orbital or odd-frequency antisymmetry also changes the mnemonic. The linearized equation fixes the critical subspace but not a direction within a multidimensional representation. Quartic terms generally select a real, chiral, or other combination immediately below the transition, while strain or disorder can split the degeneracy. Sigrist and Ueda 1991, §II, pp. 240–252 develops the crystal-symmetry classification and multicomponent Ginzburg–Landau selection.

Shell flow and physical stopping scales

Let $\ell=\log(\Lambda_0/\Lambda)$ and use the signed dimensionless eigenvalue $g_n=-\lambda_n$. The autonomous leading-log Cooper flow is

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

Thus $g_n>0$ decreases toward zero, while $g_n<0$ reaches a pole at $\ell_\ast=1/\lvert g_n(0)\rvert$. The identity $g_n(\ell)^{-1}=g_n(0)^{-1}+\ell$ is the differential form of the ladder denominator. In the separable normalization drawn below, $g_n=N_{\mathrm{pair}}(0)V_n=-\lambda_n$. The figure puts the two ingredients side by side vertically: the pair-degenerate shell supplies the logarithm, and the sign of one normalized eigenchannel determines its one-loop fate.

The Cooper channel combines a full surface of degenerate partner states with the radial integral $\int_E^\Lambda\mathrm d\lvert\xi\rvert/\lvert\xi\rvert$; the figure abbreviates $N_{\mathrm{pair}}(0)$ as $N(0)$. Representative signed eigenvalues $g_n(0)=+0.20$ and $-0.20$ then flow toward zero and a pole at $\ell_\ast=5$, respectively. The pole marks failure of the normal-state expansion, not $T_c$. Original schematic for a regular pair-degenerate two-dimensional Fermi liquid; not to scale.

The machine-readable figure record gives the assumptions, equations, encodings, and failure cases without relying on the image.

The displayed beta function is not a theorem that every repulsive interaction becomes harmless. Particle–hole corrections can generate attractive high-angular-momentum components—the Kohn–Luttinger mechanism—and nested or van-Hove systems generally require coupled parquet or functional-RG flows whose eigenvectors can rotate. Shankar 1994, §§VI.C–VI.E and Appendix B, pp. 171–173 and 190–191 derives the autonomous flow and its qualification; Kohn and Luttinger 1965 gives the classic repulsion-induced route.

Physical scales can stop the flow before its pole. Temperature, pair-breaking mismatch, finite system size, or loss of quasiparticle coherence replaces $E$ by the largest applicable infrared cutoff. Retardation requires a second matching step: a static eigenproblem is only the low-energy limit of a frequency-dependent kernel. In the elementary two-window model with electronic scale $W\gg\omega_B$ and screened dimensionless Coulomb repulsion $\mu$,

$$\mu^\ast =\frac{\mu}{1+\mu\log(W/\omega_B)}.$$

This Morel–Anderson reduction is a scale-separated constant-channel approximation, not a general replacement for the momentum–Matsubara eigenproblem. A general retarded calculation determines $\Delta(\mathbf k,i\omega_n)$ together with normal-state renormalization and identifies $T_c$ by an eigenvalue crossing. Morel and Anderson 1962 gives the original retardation matching; Migdal control and Eliashberg theory develop its domain of validity.

What the instability establishes—and what it does not

• Finite density of states is part of the result. At zero temperature, $\Pi_{pp}\sim\int_0^\Lambda\mathrm d\xi\,N(\xi)/\xi$. If $N(\xi)\propto\lvert\xi\rvert^r$, the ordinary logarithm occurs for $r=0$, a finite threshold replaces it for $r>0$, and the divergence is stronger for $r<0$. A logarithmic van-Hove density of states yields a $\log^2$ enhancement, but particle–hole channels are then often singular as well.
• Pair degeneracy can fail. Population imbalance, orbital mismatch, or a pair-breaking field cuts off the zero-momentum logarithm. Finite-momentum pairing may become competitive, but it is a separate eigenproblem rather than an automatic continuation of this one.
• Pairing is not phase coherence. A growing pair susceptibility says nothing by itself about the stiffness. In two dimensions, pair formation can lie above the vortex-unbinding transition discussed on the order, stiffness, and vortices page.
• A leading eigenfunction is not a mechanism. Self-energy and vertex corrections can reorder channels, and the same symmetry may arise from different microscopic interactions. Mechanism claims require discriminating observables, not merely a calculated gap shape.

The chapter-wide structure diagram locates this conclusion before the saddle, response, and evidence layers. In the convention used here, its “leading $\lambda$” is the positive attractive eigenvalue $\lambda_n=-g_n$.

An attractive eigenchannel licenses a normal-state pairing instability, not a microscopic mechanism, phase stiffness, or gauge-invariant observable. Those conclusions enter only at later, separately tested stages. Original schematic, not to scale.

The machine-readable structure record preserves every dependency, and the paired-matter claim test matrix gives the corresponding controls and decisive negative tests.

Common pitfalls

Calling $E_\ast$ the transition temperature. $E_\ast$ is a normal-state stopping scale. A controlled $T_c$ requires the matched finite-temperature kernel and, beyond mean field, the relevant phase fluctuations.

Equating symmetry with mechanism. An $s$-, $p$-, or $d$-wave eigenfunction describes how a pair transforms. It does not identify the boson, fluctuation, or electronic process that produced the kernel.

Treating $\Lambda$ as automatically equal to the bandwidth. $\Lambda$ is the upper scale of the effective channel after matching. For a retarded attraction it may be a boson scale, while electronic repulsion has already flowed over a wider window.

Dropping the hypotheses behind “arbitrarily weak.” The statement needs a finite pair density of states, coherent partners, and no larger infrared cutoff. Dirac points, pseudogaps, mismatch, or strong incoherence can replace the weak-coupling instability by a finite threshold.

Exercises

Recover the logarithm. Starting from the pair bubble with a constant $N_{\mathrm{pair}}(0)$, perform the Matsubara sum and identify the interval of $\xi$ that produces the logarithm.

Solution

The standard fermionic sum gives

$$T\sum_{\omega_n}\frac{1}{\omega_n^2+\xi^2} =\frac{1-2f(\xi)}{2\xi} =\frac{\tanh(\xi/2T)}{2\xi}.$$

Combining positive and negative energies therefore yields

$$\Pi_{pp}(T) =N_{\mathrm{pair}}(0)\int_0^\Lambda \frac{\mathrm d\xi}{\xi}\tanh\frac{\xi}{2T}.$$

For $\xi\ll T$, $\tanh(\xi/2T)\simeq\xi/(2T)$ removes the infrared singularity. For $T\ll\xi\ll\Lambda$, the hyperbolic tangent is approximately one, so that interval gives $\int_T^\Lambda\mathrm d\xi/\xi=\log(\Lambda/T)$. Evaluating the smooth crossover produces the hard-cutoff constant $2e^\gamma/\pi$.

Project two angular channels. On a circular Fermi surface with normalized measure $\mathrm d\theta'/(2\pi)$, let

$$V(\theta,\theta') =V_0+2V_2\cos\!\bigl[2(\theta-\theta')\bigr].$$

Find the eigenvalues of the constant function and of $\cos2\theta$ and $\sin2\theta$. Which channel is leading when $V_2<0<V_0$?

Solution

The constant function is an eigenfunction of $V$ with eigenvalue $V_0$. Orthogonality of the circular harmonics gives eigenvalue $V_2$ for both $\cos2\theta$ and $\sin2\theta$; they form the two components of the $m=\pm2$ sector. Because the gap equation diagonalizes $-V$, its attractive eigenvalues are $\lambda_s=-N_{\mathrm{pair}}(0)V_0$ and $\lambda_d=-N_{\mathrm{pair}}(0)V_2$. Thus $V_2<0<V_0$ gives a two-component attractive $d$-wave subspace and a repulsive $s$-wave channel. Quartic terms are needed to choose a real or chiral combination below the transition.

Add a pair-breaking cutoff. An attractive channel starts at $g_0=-0.20$ and $\Lambda_0$. First find its uncut stopping scale. Then suppose mismatch stops the Cooper logarithm at $E_{\mathrm{pb}}=0.02\Lambda_0$. Does the one-loop pole occur before the flow stops?

Solution

The uncut pole is at $\ell_\ast=1/\lvert g_0\rvert=5$, hence

$$E_\ast=\Lambda_0e^{-5} \simeq6.74\times10^{-3}\Lambda_0.$$

The mismatch stops the logarithm at $\ell_{\mathrm{pb}}=\log(\Lambda_0/E_{\mathrm{pb}})=\log50\simeq3.91<5$. Therefore the pole is not reached. The running value is $g(\ell_{\mathrm{pb}})\simeq-0.92$, already outside a comfortable weak-coupling regime, so the safe conclusion is only that this leading-log flow is strongly enhanced but cut off before its formal divergence.

Find the threshold for a vanishing density of states. Let $N_{\mathrm{pair}}(\xi)=N_0\lvert\xi/\Lambda\rvert^r$ with $r>0$. Evaluate the zero-temperature pair bubble and find the ladder threshold for a constant attractive $V$.

Solution

At $T=0$,

$$\Pi_{pp}(0) =N_0\int_0^\Lambda\frac{\mathrm d\xi}{\xi} \left(\frac{\xi}{\Lambda}\right)^r =\frac{N_0}{r}.$$

The susceptibility is finite. The denominator $1+V\Pi_{pp}(0)$ can vanish only when $\lvert V\rvert\geq\lvert V_c\rvert=r/N_0$. The phrase “arbitrarily weak attraction” therefore does not apply when $r>0$.

Estimate retardation of repulsion. In the two-window model, take $\mu=0.30$ and $W/\omega_B=100$. Compute $\mu^\ast$ and state why the number is not yet a prediction of $T_c$.

Solution

Using $\log100\simeq4.605$,

$$\mu^\ast =\frac{0.30}{1+0.30\log100} \simeq0.126.$$

High-energy flow has reduced the effective low-energy repulsion, but a transition still depends on the attractive frequency-dependent kernel, its momentum and band structure, quasiparticle renormalization, and the infrared solution. The pseudopotential alone neither establishes attraction nor fixes $T_c$.

Continue

The BCS saddle and gap equation continue from the normal-state instability to a self-consistent ordered solution. Migdal control and Eliashberg theory replace the static kernel by a retarded one. Unconventional and topological pairing develops crystal, multiorbital, and frequency-dependent symmetry beyond the simplest mnemonic.

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.
• Kohn, W., and Luttinger, J. M. (1965). “New mechanism for superconductivity.” Physical Review Letters 15, 524–526. doi:10.1103/PhysRevLett.15.524.
• Morel, P., and Anderson, P. W. (1962). “Calculation of the superconducting state parameters with retarded electron-phonon interaction.” Physical Review 125, 1263–1271. doi:10.1103/PhysRev.125.1263.
• Shankar, R. (1994). “Renormalization-group approach to interacting fermions.” Reviews of Modern Physics 66, 129–192. doi:10.1103/RevModPhys.66.129.
• Sigrist, M., and Ueda, K. (1991). “Phenomenological theory of unconventional superconductivity.” Reviews of Modern Physics 63, 239–311. doi:10.1103/RevModPhys.63.239.