Lecture 19: Cooper Instability and Pairing Mean Field

The magnetic lectures showed that interactions may demand a new reference state, but a metallic Fermi surface supports a distinct instability whose phase space is concentrated near that surface. This lecture tests the normal state’s stability against pairing and constructs the self-consistent field that replaces it. The connection is a shared logic of competing reference states and stability, not a claim that superconductivity follows from magnetic order. A weak microscopic attraction can produce a nonperturbative reorganization when an extended set of low-energy states cooperates, so coupling strength alone does not measure the importance of an instability.

An elemental superconductor such as aluminum or tin supplies the bounded physical realization. Its normal state has a Fermi surface, lattice vibrations provide a retarded attractive contribution over a characteristic phonon energy window, and cooling produces a coherent superconducting state. The reduced BCS model replaces the detailed momentum- and frequency-dependent electron–phonon interaction by a constant attraction and a sharp cutoff. It is trusted for the weak-coupling logarithmic instability and its parameter dependence, not for a precision transition temperature or for unconventional pairing.

The Fermi surface changes the binding question: why can an arbitrarily weak attraction reorganize states in its vicinity even when the same attraction need not bind two particles in vacuum? The Cooper logarithm and pair mean field answer that question and organize superconducting gaps, phase coherence, and the interpretation of pair-sensitive experiments.

We will isolate the attractive Cooper shell, derive its logarithmic instability, and promote the unstable pair amplitude to a self-consistent mean field whose assumptions and solution can be checked.

The transition temperature and its isotope trend can constrain the phonon scale; tunneling and heat capacity later test the excitation gap and thermodynamic state. This lecture stops at the instability and pair mean field. It does not yet claim a tunneling spectrum, Meissner response, vortex physics, or a complete material-specific interaction.

Compare a pair continuum whose available states begin at the Fermi surface with a vacuum continuum whose low-energy density of states vanishes. Before calculating, decide whether the same weak attraction must have the same binding threshold in both cases. The logarithmic integral will locate the difference.

Sketch occupied and available states around the Fermi surface, write a zero-center-of-mass time-reversed pair operator, state the units and spin convention of ${N_0}$ , and recall why a self-consistent mean-field root still needs free-energy selection. Recover the Fermi surface from Lecture 5, pair operators from Lecture 8, or the stability logic from Lecture 17 if any answer is missing. The new calculation changes the integration domain to the Pauli-blocked shell and then asks for self-consistent closure. The microscopic origin of the attraction is a declared model input rather than an assumed phonon derivation.

The reduced attractive-shell model

The reduced attractive shell is a deliberately scale-limited model, not a universal microscopic force. The Cooper equation is exact within that kernel and exposes the Fermi-surface logarithm. Passing to BCS mean field changes the description: it selects a quasifree variational state with a pairing amplitude and is not controlled merely because its equation is self-consistent. Microscopic retardation, Coulomb repulsion, phase fluctuations, and strong coupling are the first corrections that can change the pairing scale or the nature of the paired state.

Cooper-shell kinematics

Work first in a finite periodic box with discrete momenta ${\mathcal K}$ , one-particle energies ${\xi_{\mathbf k}=\varepsilon_{\mathbf k}-\mu}$ , and canonical fermion operators on the Fock space ${\mathfrak F}$ . The Cooper pair-label set is $\mathcal K_{\mathrm C} =\{\mathbf k\in\mathcal K:0<\xi_{\mathbf k}<\hbar\omega_{\mathrm c}\},$ whereas the many-body reduced model uses the two-sided shell ${\mathcal K_{\mathrm p} =\{\mathbf k\in\mathcal K:|\xi_{\mathbf k}|<\hbar\omega_{\mathrm c}\}}$ . Each ${\mathbf k\in\mathcal K_{\mathrm p}}$ labels the oriented pair orbital ${(\mathbf k{\uparrow},-\mathbf k{\downarrow})}$ ; a fixed total order on ${\mathcal K_{\mathrm p}}$ fixes all product-state signs. The finite-box single-spin pair-orbital density of states ${N_0}$ has units ${\mathrm{energy}^{-1}}$ , and the discrete interaction matrix elements have energy units, so ${gN_0}$ is dimensionless. A per-volume convention is equivalent only if the momentum sum and interaction normalization are changed together.

The Cooper calculation concerns an instability of a prepared Fermi surface in a selected pairing channel. The filled Fermi sea, Pauli-blocked shell, and channel projection determine its kernel and binding criterion. The gap introduced later is a collective field in a mean-field closure. Its phase changes with the fermion gauge convention, whereas spectra and gauge- covariant observables do not. Exact number conservation and the convenient broken-symmetry representation are compatible descriptions; the cutoff, density of states, and weak-coupling window will delimit where that representation is quantitative.

Lattice deformation can mediate a retarded attraction between electrons over an energy window set, in the simplest model, by a Debye scale. We denote the effective sharp cutoff by ${\hbar\omega_{\mathrm c}}$ , which may be of order ${\hbar\omega_{\mathrm D}}$ but is model data here. Coulomb repulsion and retardation are not derived. The constant attractive shell is a controlled pedagogical idealization, not a universal microscopic force law.

The filled sea blocks all pair states below the Fermi surface, so the available continuum begins at ${\xi=0}$ . If the pair kernel contains ${1/(2\xi+E_{\mathrm b})}$ , predict whether the integral stays finite or diverges as the proposed binding energy ${E_{\mathrm b}}$ approaches zero. What would that behavior imply about a threshold attraction?

Let ${|\mathrm{FS}\rangle}$ be an inert filled Fermi sea and define ${\hat b_{\mathbf k}^{\dagger} =\hat c_{\mathbf k\uparrow}^{\dagger} \hat c_{-\mathbf k\downarrow}^{\dagger}}$ . The Cooper pairing problem is the eigenvalue problem obtained by restricting the two-electron Hamiltonian above that sea to $\mathcal H_{\mathrm C} =\operatorname{span}\{ \hat b_{\mathbf k}^{\dagger}|\mathrm{FS}\rangle: \mathbf k\in\mathcal K_{\mathrm C}\} \cong\ell^2(\mathcal K_{\mathrm C}).$ Its amplitudes are functions ${\varphi:\mathcal K_{\mathrm C}\to\mathbb C}$ . The sea, energy reference, time-reversed pair basis, cutoff shell, and interaction kernel are part of the definition. This restricted two-particle problem is not the many-body BCS state and does not by itself establish superconducting order.

For two electrons above an inert filled sea, a zero-center-of-mass pair state has amplitudes ${\varphi_{\mathbf k}}$ : $|\Psi\rangle=\sum_{\mathbf k}^{\mathrm{shell}} \varphi_{\mathbf k} \hat c_{\mathbf k\uparrow}^\dagger \hat c_{-\mathbf k\downarrow}^\dagger|\mathrm{FS}\rangle.$

A constant attraction ${-g}$ inside ${0<\xi_{\mathbf k}<\hbar\omega_{\mathrm c}}$ gives, after projecting ${\hat H|\Psi\rangle=\mathcal E|\Psi\rangle}$ onto one pair basis state, $(2\xi_{\mathbf k}-\mathcal E)\varphi_{\mathbf k} =g\sum_{\mathbf k'}\varphi_{\mathbf k'}.$ If ${C=\sum_{\mathbf k'}\varphi_{\mathbf k'}}$ , then ${\varphi_{\mathbf k}=gC/(2\xi_{\mathbf k}-\mathcal E)}$ . Summing this relation over ${\mathbf k}$ and canceling the nonzero common amplitude ${C}$ produces the scalar condition. For a bound state ${\mathcal E=-E_{\mathrm b}<0}$ , $1=g\sum_{\mathbf k\in\mathcal K_{\mathrm C}} \frac{1}{2\xi_{\mathbf k}+E_{\mathrm b}}.$ Approximating the single-spin density of states by ${N_0}$ converts the sum to $1\simeq gN_0\int_0^{\hbar\omega_{\mathrm c}} \frac{\mathrm d\xi}{2\xi+E_{\mathrm b}} =\frac{gN_0}2 \ln\!\frac{2\hbar\omega_{\mathrm c}+E_{\mathrm b}}{E_{\mathrm b}}.$ For finite ${\mathcal K_{\mathrm C}}$ , the corresponding normalized amplitudes are $\varphi_{\mathbf k} =\frac{\mathcal N}{2\xi_{\mathbf k}+E_{\mathrm b}}, \qquad {|\mathcal N|}^{-2} =\sum_{\mathbf k\in\mathcal K_{\mathrm C}} \frac{1}{{(2\xi_{\mathbf k}+E_{\mathrm b})}^2}.$ This formula is also the finite-volume regularization, but “arbitrarily weak” binding relative to the Fermi edge requires the continuum limit ${\xi_{\min}\to0}$ before ${g\to0^+}$ . At fixed level spacing, the ${E_{\mathrm b}=0}$ sum is finite. In the constant-DOS continuum, the logarithmic right side decreases continuously from infinity to zero as ${E_{\mathrm b}}$ increases from ${0^+}$ to infinity, proving existence and uniqueness. Pauli blocking is essential: it makes the Fermi surface the lower edge of the available pair continuum and concentrates the logarithm near ${\xi=0}$ .

The logarithmic instability and bound state

For Definition 19.1 with ${V_{\mathbf k\mathbf k'}=-g}$ , ${g>0}$ , and a constant single-spin density ${N_0>0}$ on ${0<\xi<\hbar\omega_{\mathrm c}}$ , there is exactly one bound solution for every ${g>0}$ . Its amplitude ${\varphi_{\mathbf k}\propto{(2\xi_{\mathbf k}+E_{\mathrm b})}^{-1}}$ depends on ${\mathbf k}$ only through ${\xi_{\mathbf k}}$ , so it is even under ${\mathbf k\to-\mathbf k}$ whenever ${\xi_{-\mathbf k}=\xi_{\mathbf k}}$ . Its binding energy is $E_{\mathrm b} =\frac{2\hbar\omega_{\mathrm c}}{\exp[2/(gN_0)]-1} \simeq2\hbar\omega_{\mathrm c}\exp[-2/(gN_0)] \quad(gN_0\ll1).$ The statement is conditional on the inert sea, constant attractive channel, finite shell, nonzero edge density of states, and absence of pair-breaking terms in the reduced problem; it is not a proof of superconductivity for every attractive interaction.

A Zeeman mismatch, magnetic pair breaking, or another term that mixes the declared pair subspace changes the kernel or its available low-energy phase space. The theorem then no longer licenses the phrase “arbitrarily weak,” even if an attractive matrix element remains elsewhere in the Hamiltonian.

Scale and cutoff dependence

For the restricted Cooper problem, take one-spin shell DOS ${N_0>0}$ , attractive strength ${g>0}$ , cutoff ${E_{\mathrm c}=\hbar\omega_{\mathrm c}>0}$ , and binding energy ${E_{\mathrm b}>0}$ . Evaluate ${1=gN_0\int_0^{E_{\mathrm c}}\mathrm d\xi/(2\xi+E_{\mathrm b})}$ , solve for ${E_{\mathrm b}}$ , and test weak coupling. Then use the already introduced changed DOS ${N(\xi)=N_0\xi/E_{\mathrm c}}$ to decide whether arbitrarily weak attraction binds.

The shell amplitude equation has ${a_{\mathbf k}\propto {(2\xi_{\mathbf k}+E_{\mathrm b})}^{-1}}$ , since both particles lie above the filled sea. The factor two is their kinetic energy, not a spin degeneracy. Integration gives ${1=(gN_0/2)\ln[(2E_{\mathrm c}+E_{\mathrm b})/E_{\mathrm b}]}$ , hence ${E_{\mathrm b}=2E_{\mathrm c}/[\mathrm e^{2/(gN_0)}-1]}$ .

For ${E_{\mathrm b}\ll\hbar\omega_{\mathrm c}}$ , $E_{\mathrm b}\simeq 2\hbar\omega_{\mathrm c} \exp[-2/(gN_0)].$ The exponential nonanalyticity means no finite-order power series in ${g}$ produces the scale. If the shell width is doubled, the prefactor doubles but the essential exponential remains. If the density of states vanishes at the Fermi level, the logarithmic argument must be reconsidered.

The exponent is dimensionless because ${gN_0}$ is dimensionless under the declared single-spin DOS convention. Including both spin labels in ${N_0}$ without changing the interaction sum would change the factor two in the exponent; that is a convention change, not new physics.

Let ${N(\xi)=N_0\xi/(\hbar\omega_{\mathrm c})}$ on the same shell. Divide ${\xi/(2\xi+E_{\mathrm b})=1/2-E_{\mathrm b}/[2(2\xi+E_{\mathrm b})]}$ before integrating. Then $1=gN_0\left[ \frac12-\frac{E_{\mathrm b}}{4\hbar\omega_{\mathrm c}} \ln\!\frac{2\hbar\omega_{\mathrm c}+E_{\mathrm b}}{E_{\mathrm b}} \right].$ As ${E_{\mathrm b}\to0^+}$ , the right side approaches ${gN_0/2}$ instead of diverging. A bound solution therefore exists only for ${gN_0>2}$ ; equality is the zero-binding threshold. The Cooper conclusion depends on finite low-energy phase space, not on attraction alone.

The binding tends to zero as ${g\to0^+}$ in the constant-DOS case, and scales linearly with ${E_{\mathrm c}}$ at fixed ${gN_0}$ , checking two independent limits. The restricted two-particle state is not a self-consistent many-pair condensate; that change of state space follows next.

In Figure 19.1, identify the lower edge responsible for the logarithm, the upper cutoff that affects its prefactor, and the time-reversed partner of one marked state. Then decide which region would be removed if the filled sea were absent.

Figure 19.1. The Pauli-blocked Cooper continuum, exact integration bounds, and matched edge-density comparison distinguish a logarithmic instability from a finite threshold.

Panel (a) fixes the pair and shell geometry, panel (b) displays the exact constant-DOS kernel, panel (c) changes only the edge density of states, and panel (d) shows that the integrated constant-DOS kernel diverges as ${E_{\mathrm b}\to0^+}$ while the linear-DOS one tends to ${1/2}$ ; the right side of the binding condition then tends to ${gN_0/2}$ , so binding needs ${gN_0>2}$ .

The figure separates infrared mechanism from ultraviolet model choice. The Fermi-surface edge produces the logarithm; the cutoff terminates the reduced model and fixes its prefactor. States deep in the sea are Pauli blocked, while states far outside the shell are excluded by assumption. Problem 2 preserves those bounds and changes ${N(\xi)}$ , so the reader must recompute rather than reuse the constant-DOS conclusion.

From one pair to a coherent product state

The one-pair result does not yet describe overlapping pair orbitals. The many-body model and its simplest variational family must be declared separately.

On the shell Fock space ${\mathfrak F_{\mathrm p} =\mathfrak F(\operatorname{span}\{ \vert{}\mathbf k{\uparrow}\rangle,\vert{}{-\mathbf k}{\downarrow}\rangle: \mathbf k\in\mathcal K_{\mathrm p}\})}$ , define ${\hat b_{\mathbf k} =\hat c_{-\mathbf k\downarrow}\hat c_{\mathbf k\uparrow}}$ . The reduced BCS Hamiltonian is the self-adjoint operator $\hat H_{\mathrm{red}} =\sum_{\mathbf k\in\mathcal K_{\mathrm p}} \xi_{\mathbf k} \left(\hat n_{\mathbf k\uparrow} +\hat n_{-\mathbf k\downarrow}\right) +\sum_{\mathbf k,\mathbf k'\in\mathcal K_{\mathrm p}} V_{\mathbf k\mathbf k'} \hat b_{\mathbf k}^{\dagger}\hat b_{\mathbf k'} .$ The pair-label convention counts each oriented pair block once. The entries ${V_{\mathbf k\mathbf k'}}$ are the matrix elements of a Hermitian linear operator ${\hat V:\ell^2(\mathcal K_{\mathrm p})\to \ell^2(\mathcal K_{\mathrm p})}$ with energy units. They obey ${V_{\mathbf k\mathbf k'}=V_{\mathbf k'\mathbf k}^*}$ . The operator ${\hat V}$ is attractive when $\sum_{\mathbf k,\mathbf k'} f_{\mathbf k}^*V_{\mathbf k\mathbf k'}f_{\mathbf k'}\le0 \quad\text{for every }f\in\ell^2(\mathcal K_{\mathrm p}).$ The constant-shell model ${V_{\mathbf k\mathbf k'}=-g}$ , ${g>0}$ , is a rank-one negative-semidefinite special case. The exact operator commutes with the total number ${\hat N}$ and fermion parity. Its cutoff and instantaneous kernel are effective-model data; microscopic retardation is outside this definition.

Write the declared finite pair-label set as ${\mathbf k_1\prec\cdots\prec\mathbf k_{L_{\mathrm p}}}$ . A BCS variational state is the normalized even-parity Fock-space vector $|\Psi_{\mathrm{BCS}}[u,v]\rangle =\prod_{r=1}^{L_{\mathrm p}} \left(u_{\mathbf k_r} +v_{\mathbf k_r}\hat b_{\mathbf k_r}^{\dagger}\right)|0\rangle, \qquad {|u_{\mathbf k_r}|}^2+{|v_{\mathbf k_r}|}^2=1 .$ The index order fixes the product convention. Each factor is normalized, and distinct even pair factors commute, so the full vector is normalized. Except at number-sharp endpoints (every ${u_{\mathbf k}v_{\mathbf k}=0}$ ) or after number projection (Section 19.4), it is not an eigenvector of ${\hat N}$ , although it has definite even fermion parity. A global phase convention may make one nonzero pair amplitude ${u_{\mathbf k}^*v_{\mathbf k}}$ real; the relative phases of different pair amplitudes remain physical.

Pairing mean field and its constant term

Let ${\hat\rho}$ be a density operator on ${\mathfrak F_{\mathrm p}}$ , and write ${F_{\mathbf k}[\hat\rho] =\operatorname{Tr}(\hat\rho\hat b_{\mathbf k})}$ .

For Definition 19.2 and a declared state ${\hat\rho}$ , the BCS gap parameter is the ordinary complex function $\Delta[\hat\rho]:\mathcal K_{\mathrm p}\longrightarrow\mathbb C, \qquad \Delta_{\mathbf k}[\hat\rho] =-\sum_{\mathbf k'\in\mathcal K_{\mathrm p}} V_{\mathbf k\mathbf k'}F_{\mathbf k'}[\hat\rho].$ This equation defines a state-dependent order parameter; it is not yet a self-consistency equation. For ${V_{\mathbf k\mathbf k'}=-g}$ , ${\Delta=g{\langle\hat B\rangle}_{\hat\rho}}$ , where ${\hat B=\sum_{\mathbf k}\hat b_{\mathbf k}}$ . Under the active convention ${\hat c_{\mathbf k\sigma}\mapsto \mathrm e^{\mathrm i\alpha}\hat c_{\mathbf k\sigma}}$ , both ${F_{\mathbf k}}$ and ${\Delta_{\mathbf k}}$ transform by ${\mathrm e^{2\mathrm i\alpha}}$ . Thus ${\Delta}$ is gauge covariant, not gauge invariant. A global phase convention may make one nonzero component ${\Delta_{\mathbf k}}$ real; relative phases remain physical when the gap has more than one independent component.

Write ${\hat b_{\mathbf k}=F_{\mathbf k}+\delta\hat b_{\mathbf k}}$ . The product then splits exactly into four terms; mean field discards only the last, the product of two fluctuations: $\begin{aligned} \hat b_{\mathbf k}^{\dagger}\hat b_{\mathbf k'} &=\hat b_{\mathbf k}^{\dagger}F_{\mathbf k'}+F_{\mathbf k}^{*}\hat b_{\mathbf k'} -F_{\mathbf k}^{*}F_{\mathbf k'}+\delta\hat b_{\mathbf k}^{\dagger}\delta\hat b_{\mathbf k'}\\ &\simeq\hat b_{\mathbf k}^{\dagger}F_{\mathbf k'}+F_{\mathbf k}^{*}\hat b_{\mathbf k'} -F_{\mathbf k}^{*}F_{\mathbf k'}. \end{aligned}$ The subtracted constant prevents the mean interaction energy from being counted twice.

Definition 19.4 gives ${\sum_{\mathbf k'}V_{\mathbf k\mathbf k'}F_{\mathbf k'}=-\Delta_{\mathbf k}}$ , and ${V_{\mathbf k\mathbf k'}=V_{\mathbf k'\mathbf k}^*}$ gives ${\sum_{\mathbf k}F_{\mathbf k}^*V_{\mathbf k\mathbf k'}=-\Delta_{\mathbf k'}^*}$ . The quadratic mean-field Hamiltonian is therefore $\hat H_{\mathrm{MF}} =\sum_{\mathbf k\in\mathcal K_{\mathrm p}}\xi_{\mathbf k} \left(\hat n_{\mathbf k\uparrow} +\hat n_{-\mathbf k\downarrow}\right) -\sum_{\mathbf k\in\mathcal K_{\mathrm p}} \left(\Delta_{\mathbf k}\hat b_{\mathbf k}^{\dagger} +\Delta_{\mathbf k}^{*}\hat b_{\mathbf k}\right) +C_{\mathrm{MF}}.$

Its scalar constant is $C_{\mathrm{MF}}=-\sum_{\mathbf k,\mathbf k'}F_{\mathbf k}^*V_{\mathbf k\mathbf k'}F_{\mathbf k'}.$ Hermiticity makes this scalar real. It is nonnegative only under the attractive-kernel hypothesis in Definition 19.2: if ${V}$ is negative semidefinite, then ${F^\dagger V F\leq0}$ and hence ${C_{\mathrm{MF}}\geq0}$ . For the constant kernel this is ${C_{\mathrm{MF}}={|\Delta|}^2/g}$ . The constant leaves the block eigenvectors unchanged and remains essential to Lecture 17’s free-energy comparison. A self-consistent root is only stationary; stability still requires the lower trial free energy.

Self-consistency has a precise just-in-time thermal meaning. At a chosen temperature ${T>0}$ , set ${\beta={(k_{\mathrm B}T)}^{-1}}$ . The same quadratic ${\hat H_{\mathrm{MF}}(\Delta)}$ defines the normalized trial density operator $\hat\rho_{\mathrm{MF}}(\Delta) =\frac{\mathrm e^{-\beta\hat H_{\mathrm{MF}}(\Delta)}} {\operatorname{Tr}\mathrm e^{-\beta\hat H_{\mathrm{MF}}(\Delta)}}.$ The exponential supplies equilibrium weights, and the denominator makes their trace one; no prior statistical-mechanics course is assumed. The expectation in Definition 19.4 must be evaluated with this same operator, turning the definition into a fixed-point condition.

Linearized thermal instability and a conventional-material trend

The thermal pair amplitude is derived rather than imported. For uniform ${\Delta}$ , omit the shared constant temporarily and use the ordered basis ${\mathcal B_{\mathbf k} =(|0\rangle,|\uparrow\rangle,|\downarrow\rangle,|p\rangle)}$ , where ${|p\rangle=\hat b_{\mathbf k}^{\dagger}|0\rangle}$ . Then ${[\hat H_{\mathbf k}]}_{\mathcal B_{\mathbf k}} {=} \begin{pmatrix} 0&0&0&-\Delta^*\\ 0&\xi_{\mathbf k}&0&0\\ 0&0&\xi_{\mathbf k}&0\\ -\Delta&0&0&2\xi_{\mathbf k} \end{pmatrix}.$ Only the even states contribute to ${\operatorname{Tr}(\hat b_{\mathbf k} \mathrm e^{-\beta\hat H_{\mathbf k}})}$ . All four states contribute to the normalization. In the even subspace, ${\hat H_{\mathbf k}=\xi_{\mathbf k}\hat I+ \mathbf h_{\mathbf k}\cdot\hat{\boldsymbol\sigma}}$ , where ${{(\mathbf h_{\mathbf k}\cdot\hat{\boldsymbol\sigma})}^2 =E_{\mathbf k}^2\hat I}$ and $E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+{|\Delta|}^2}.$ The identity ${\mathrm e^{-\beta\mathbf h\cdot\hat{\boldsymbol\sigma}} =\cosh(\beta E)\hat I -(\sinh(\beta E)/E)\mathbf h\cdot\hat{\boldsymbol\sigma}}$ gives $Z_{\mathbf k} =2\mathrm e^{-\beta\xi_{\mathbf k}} \left[1+\cosh(\beta E_{\mathbf k})\right], \qquad \operatorname{Tr}\!\left( \hat b_{\mathbf k}\mathrm e^{-\beta\hat H_{\mathbf k}}\right) =\mathrm e^{-\beta\xi_{\mathbf k}} \frac{\Delta}{E_{\mathbf k}}\sinh(\beta E_{\mathbf k}).$ The first quantity normalizes the four mutually exclusive block states; the second retains the phase-coherent even-sector contribution. Their ratio is

(19.1)

${\langle\hat b_{\mathbf k}\rangle}_{\mathrm{MF}} =\frac{\Delta}{2E_{\mathbf k}} \tanh\!\left(\frac{\beta E_{\mathbf k}}2\right).$

A proposed pairing solution is closed only if the same pair-label set, kernel, gap sign, and counting convention are used in Definition 19.2, Definition 19.4, the anomalous terms, the positive constant, and the normalized thermal trace. It must satisfy ${\Delta_{\mathbf k}=\Delta_{\mathbf k} [\hat\rho_{\mathrm{MF}}(\Delta)]}$ ; stability then requires the appropriate trial free energy, not stationarity alone. A field inserted without this fixed-point evaluation is an external source, not a self-consistent gap.

The onset follows without solving the nonlinear gap below the transition. As ${\Delta\to0}$ , Equation (19.1) becomes $\langle\hat b_{\mathbf k}\rangle \simeq\frac{\Delta}{2|\xi_{\mathbf k}|} \tanh\!\left(\frac{\beta|\xi_{\mathbf k}|}{2}\right).$ Substitution into ${\Delta=g\sum_{\mathbf k} \langle\hat b_{\mathbf k}\rangle}$ , followed by the symmetric-shell single-spin DOS replacement, gives $1=gN_0\int_0^{\hbar\omega_{\mathrm c}} \frac{\mathrm d\xi}{\xi} \tanh\!\left(\frac{\xi}{2k_{\mathrm B}T_{\mathrm c}}\right).$ The negative- ${\xi}$ half of the shell supplies the factor that cancels ${1/2}$ . Since ${\tanh[\xi/(2k_{\mathrm B}T)]/\xi\to1/(2k_{\mathrm B}T)}$ , the integrand is finite at the origin. In weak coupling, $k_{\mathrm B}T_{\mathrm c} \simeq\frac{2\mathrm e^{\gamma}}{\pi} \hbar\omega_{\mathrm c} \exp\!\left[-\frac{1}{gN_0}\right],$ with ${\gamma}$ the Euler–Mascheroni constant. Lecture 20 will obtain the same onset as the ${\Delta\to0}$ boundary of the full thermal self-consistency equation and will calculate the excitation spectrum.

For the aluminum/tin-like model, increasing ${gN_0}$ increases the instability scale exponentially, while increasing the phonon cutoff increases the prefactor. In the simplest harmonic picture, characteristic phonon frequencies scale with ionic mass approximately as ${M^{-1/2}}$ , so an isotope dependence of ${T_{\mathrm c}}$ supports a lattice contribution. The inference is qualified: isotope substitution can also affect coupling parameters, and Coulomb pseudopotential, retardation, strong coupling, anisotropy, or multiple bands are absent from this formula. Pressure can alter the cutoff, density of states, and interaction simultaneously, so the reduced model predicts a trend only after those changes are specified.

A measured isotope-scale check

The two independent reports received on 24 March 1950 found that lighter mercury isotope mixtures had a higher zero-field superconducting transition temperature [Max50; RSWN50]. Maxwell obtained ${T_{\mathrm c}=4.156\,\mathrm K}$ for natural mercury with average mass number ${200.6}$ , and ${4.177\,\mathrm K}$ for a sample that was approximately ${98\%}$ ${{}^{198}\mathrm{Hg}}$ , by extrapolating magnetic critical-field curves to zero field [Max50, p. 477]. The later complete Rutgers susceptibility study used seven isotope mixtures; its endpoint entries were ${(M,T_{\mathrm c})=(199.5,4.185\,\mathrm K)}$ and ${(203.4,4.146\,\mathrm K)}$ , and its least-squares log–log fit gave ${\alpha=0.504}$ in ${M^\alpha T_{\mathrm c}=\text{constant}}$ [RSN51, Table I and Fig. 5]. The readouts were related but not identical. Maxwell detected the loss of superconductivity magnetically with a ballistic-galvanometer method. The Rutgers apparatus drove a mutual-inductance primary at ${20\,\mathrm{cps}}$ , took the peak change in susceptibility as the critical field at each temperature, and fitted the resulting critical-field curves before extracting ${T_{\mathrm c}}$ [Max50, p. 477; RSN51, pp. 691–693].

The endpoint ratio gives a compact check of the scale rather than a new fit: $\frac{4.185}{4.146}=1.00941, \qquad \left(\frac{203.4}{199.5}\right)^{1/2}=1.00973.$ The source reports fitted transition temperatures but no pointwise standard uncertainties for Table I. The displayed digits are therefore historical source values, not a modern uncertainty budget. Their mass trend is consistent with a lattice-frequency scale; it does not derive the attractive interaction, establish the reduced Hamiltonian, or make the isotope exponent universal.

Gauge covariance and exact number conservation

The gap phase changes under the electron gauge convention because the pair amplitude carries twice the one-fermion ${U(1)}$ phase. A broken-symmetry mean-field state selects one phase and is not an eigenstate of total particle number. By contrast, every interaction term of Definition 19.2 creates and annihilates the same number of electrons, so ${[\hat H_{\mathrm{red}},\hat N]=0}$ , and a finite exact eigenstate may be chosen with definite ${N}$ .

These statements are compatible. Number projection of the BCS state restores definite ${N}$ , in which case the non-number-conserving expectation ${\langle\hat b_{\mathbf k}\rangle}$ vanishes. Pair order is then encoded in number-conserving two-particle correlations and their macroscopic eigenvalue. Mean field chooses a phase to represent the same thermodynamic coherence efficiently; it does not claim microscopic charge nonconservation.

For one pair orbital, use ${|\psi\rangle=u|0\rangle+v\hat c_\uparrow^\dagger\hat c_\downarrow^\dagger|0\rangle}$ with ${|u|^2+|v|^2=1}$ . Calculate mean electron number and ${\langle\hat c_\downarrow\hat c_\uparrow\rangle}$ , and compare with the incoherent mixture having the same empty/paired probabilities.

In the declared order, ${\hat c_\downarrow\hat c_\uparrow|p\rangle=|0\rangle}$ and it annihilates the vacuum. Thus the only nonzero coherence matrix element connects the paired ket to the empty bra; no extra fermion sign appears.

The number operator counts two electrons in the paired component, so ${\langle\hat N\rangle=2|v|^2}$ , whereas the mean pair number is ${|v|^2}$ . The operator action above gives ${\langle\hat c_\downarrow\hat c_\uparrow\rangle=u^*v}$ . The pair amplitude vanishes when either component is absent and has maximum magnitude ${1/2}$ for equal empty and paired weights. An incoherent mixture with the same probabilities ${{|u|}^2}$ and ${{|v|}^2}$ has the same mean electron and pair numbers but zero ${u^*v}$ -type coherence. Pair occupancy and pair amplitude therefore answer different questions.

Normalization bounds ${2|uv|\le |u|^2+|v|^2=1}$ , checking the maximum amplitude independently of the number expectation. The endpoints ${u=0}$ and ${v=0}$ have definite electron number and zero anomalous amplitude ${u^*v}$ , while deleting off-diagonal density-matrix elements keeps the number but removes that amplitude. These are different tests of the distinction. The next example combines two such orbitals and keeps the mean-field energy constant explicitly.

Let ${\hat B=\hat b_1+\hat b_2}$ , use interaction ${-g\hat B^\dagger\hat B}$ , and the real product ${\prod_{j=1}^2(u_j+v_j\hat b_j^\dagger)|0\rangle}$ , with ${u_j^2+v_j^2=1}$ for each factor in the paired subspace of Example 19.2. For the course convention ${\Delta=g\sum_j u_jv_j}$ , derive the decoupled energy including its constant. In one fixed energy unit set ${g=1}$ , ${\xi_1=-1/2,\xi_2=1/2}$ , and ${(u_1,v_1)=(1/2,\sqrt3/2)}$ , ${(u_2,v_2)=(\sqrt3/2,1/2)}$ . Compare the decoupled energy with the exact expectation of this finite interaction in the same trial state.

Writing ${\hat B=\langle\hat B\rangle+\delta\hat B}$ and dropping ${\delta\hat B^\dagger\delta\hat B}$ gives $\hat H_{\mathrm{MF}}=2\sum_j\xi_j\hat b_j^\dagger\hat b_j -\Delta(\hat B^\dagger+\hat B)+\frac{\Delta^2}{g},\qquad \langle\hat H_{\mathrm{MF}}\rangle=2\sum_j\xi_jv_j^2-\frac{\Delta^2}{g}.$ For the specified amplitudes, ${\sum u_jv_j=\sqrt3/2}$ , so ${\Delta=\sqrt3/2}$ , the kinetic energy is ${-1/2}$ , and ${\langle\hat H_{\mathrm{MF}}\rangle=-5/4}$ . Dropping the positive constant would instead give ${-2}$ . The constant leaves eigenvectors at fixed ${\Delta}$ unchanged but is essential to the energy functional and its stationarity when ${\Delta}$ varies.

There is a separate finite-model audit. In the original product state, ${\langle\hat b_j^\dagger\hat b_j\rangle=v_j^2}$ , while distinct levels factorize. Thus $\langle\hat B^\dagger\hat B\rangle= {\left(\sum_j u_jv_j\right)}^2+\sum_j v_j^4 =\frac34+\frac58=\frac{11}{8}.$ The exact trial expectation is ${-1/2-11/8=-15/8}$ , not ${-5/4}$ . The difference ${-g\sum v_j^4}$ is the fluctuation contribution discarded by this decoupling. Direct occupation counting and the explicit constant therefore check different approximations. A decoupled finite-model energy must not be called the exact variational expectation of the original Hamiltonian. Phase and number bookkeeping next use this same state without requiring that approximation.

Reconstruct the lower edge and factor two in the Cooper logarithm, then write all three terms in the pairing decoupling. For the two-level state, recompute ${\Delta}$ and the positive ${\Delta^2/g}$ constant. Before the phase/number example, predict whether ${\operatorname{Var}(\hat N)}$ vanishes and whether number projection removes every pair correlation; then check these predictions against that calculation. This separates Cooper binding, pair occupancy, anomalous coherence, and exact number conservation before the Lecture 20 handoff.

Identify the logarithmic step in the Cooper equation, state why deep filled states do not participate in the reduced model, and calculate the pair amplitude of one normalized ${u|0\rangle+v|\uparrow\downarrow\rangle}$ factor.

The ideal Cooper bound state is not yet the full BCS condensate. The reduced attraction shell is a model assumption, and pairing mean field does not imply that the exact finite-system Hamiltonian destroys electrons.

Reuse the specific two-pair-level state of Example 19.3, with ${(u_1,v_1)=(1/2,\sqrt3/2)}$ and ${(u_2,v_2)=(\sqrt3/2,1/2)}$ . Under the simultaneous field/operator transformation ${\hat c\mapsto \mathrm e^{\mathrm i\alpha}\hat c}$ , determine how ${\Delta}$ transforms and whether the anomalous Hamiltonian is invariant. Calculate the number variance, project the state onto ${N=2}$ , and find one number-conserving pair correlator.

Since ${\hat b=\hat c_\downarrow\hat c_\uparrow\mapsto \mathrm e^{2\mathrm i\alpha}\hat b}$ , ${\Delta=g\langle\hat B\rangle\mapsto \mathrm e^{2\mathrm i\alpha}\Delta}$ . Then ${\Delta\hat b^\dagger}$ , its conjugate and ${|\Delta|^2/g}$ are unchanged: the gap is covariant while the matched Hamiltonian expression is invariant. This is not invariance under rotating the fermions while holding an externally fixed gap phase unchanged. Each independent level contains zero or two electrons, with variance ${4u_j^2v_j^2}$ . Hence $\langle\hat N\rangle=2(v_1^2+v_2^2)=2,\qquad \operatorname{Var}(\hat N)=4\sum_j u_j^2v_j^2=\frac32.$ The ${N=2}$ component has amplitudes ${v_1u_2=3/4}$ and ${u_1v_2=1/4}$ , so its norm squared is ${10/16}$ and $|\Psi_{N=2}\rangle=\frac{3|10\rangle+|01\rangle}{\sqrt{10}}.$ It has ${\operatorname{Var}(\hat N)=0}$ , ${\langle\hat b_j\rangle=0}$ , but ${\langle\hat b_1^\dagger\hat b_2\rangle=3/10}$ , because that operator takes ${|01\rangle}$ to ${|10\rangle}$ . Unit norm and exact number are one check; invariant pair correlations under a common phase rotation are another. Projection removes the anomalous one-point expectation, not all pair coherence. This finite state does not establish spontaneous symmetry breaking. Lecture 20 uses the unprojected mean-field representation to construct its canonical quasiparticles, with the same gap sign.

From the Cooper logarithm to a pairing field

Lecture 19 began with a reduced attraction acting only in the declared shell. Pauli blocking turned the Fermi surface into the lower edge of the pair continuum, and a constant low-energy density of states produced the Cooper logarithm and its nonanalytic bound-state scale. Extending one pair to many orbitals required a coherent pair amplitude, a mean-field decoupling with its double-counting constant, and a free-energy stability test. The broken-symmetry state carries a gauge-covariant phase, while the exact finite Hamiltonian remains number conserving; the two descriptions encode the same pair coherence differently. The logarithmic conclusion can fail when low-energy phase space vanishes, and mean field remains an approximation even when it is self-consistent. Which canonical excitations diagonalize its pair-creation and pair-annihilation terms?

The larger-picture change is an instability of the reference state itself. The Fermi surface is no longer only phase space for long-lived excitations; it amplifies a weak attractive channel into coherent order. The mechanism, onset scale, excitation spectrum, and electromagnetic consequences remain distinct claims. The reduced conventional model earns the first two and prepares the third; it does not establish the fourth or license the same mechanism in every superconductor.

Pairing mean field mixes electron creation and annihilation. Which canonical fermionic operators diagonalize that quadratic Hamiltonian, and how do their thermal occupations feed back on the gap?

Reconstruct the Cooper logarithm, identify the anomalous average and mean-field constant term, and explain why the assumed pair amplitude must satisfy a self-consistency condition. The pairing argument must retain its normal-state reference, selected channel, energy shell, density of states, and weak-coupling scale. The logarithmic instability is a many-fermion statement; the self-consistent gap is a collective field whose phase depends on gauge convention while spectra and gauge-covariant responses do not. Exact number conservation changes the diagnostic language to pair correlations, not the existence of the channel instability established by the calculation.

Problems

1. Cooper logarithm

Use the constant-shell Cooper problem with $N(\xi)=N_0,\qquad 0<\xi<\hbar\omega_{\mathrm c},\qquad gN_0=0.25,\qquad \hbar\omega_{\mathrm c}=20.0\,\mathrm{meV}.$

Project the eigenvalue equation, eliminate the common amplitude, and derive the exact logarithmic equation before taking the weak-coupling limit.
Solve for the exact ${E_{\mathrm b}}$ , obtain its weak-coupling form, and report both values in ${\mathrm{meV}}$ to four significant figures.
Write the normalized discrete amplitudes ${\varphi_{\mathbf k}}$ , and check the exponent’s units, the ${g\to0^+}$ limit, and the role of the lower integration edge.

2. Change cutoff or density of states

Compare the following three Cooper models, all using the conventions of Problem 1: $\begin{array}{c|c|c} \text{case}&\hbar\omega_{\mathrm c}&gN_0\\ \hline A&20.0\,\mathrm{meV}&0.25\\ B&40.0\,\mathrm{meV}&0.15 \end{array}$ Cases ${A}$ and ${B}$ have constant ${N_0}$ . Case ${C}$ instead has $N_C(\xi)=N_0\frac{\xi}{\hbar\omega_{\mathrm c}}, \qquad 0<\xi<\hbar\omega_{\mathrm c} .$

Compute the exact ${E_{\mathrm b}}$ for ${A}$ and ${B}$ . State why doubling the cutoff does not determine which binding scale is larger.
Evaluate the case- ${C}$ pair integral analytically. Determine its ${E_{\mathrm b}\to0^+}$ limit and prove that a bound state requires ${gN_0>2}$ .
Identify precisely which hypothesis of Theorem 19.1 fails in case ${C}$ ; do not describe the change as a mere prefactor correction.

3. Mean-field bookkeeping

Let $\hat H_{\mathrm{int}}=-g\hat B^\dagger\hat B,\qquad \hat B=\hat b_1+\hat b_2,\qquad g=1.00\,\mathrm{meV},$ and use the real product state in Example 19.3 with $(u_1,v_1)=\left(\frac12,\frac{\sqrt3}{2}\right),\qquad (u_2,v_2)=\left(\frac{\sqrt3}{2},\frac12\right).$

With ${\Delta=g\langle\hat B\rangle}$ , derive the mean-field interaction, including its constant. Compute ${\Delta}$ , the constant, and ${\operatorname{Var}(\hat N)}$ .
Under ${\hat c\mapsto\mathrm e^{\mathrm i\theta}\hat c}$ , find the transformations of ${\hat B}$ and ${\Delta}$ , and verify that the anomalous mean-field terms are covariant.
Prove ${[\hat H_{\mathrm{int}},\hat N]=0}$ and explain why the mean-field anomalous terms need not commute with ${\hat N}$ .
Project the supplied state onto exactly one pair. Compute the normalized state, ${\langle\hat b_j\rangle}$ , and ${\Gamma_{12}=\langle\hat b_1^\dagger\hat b_2\rangle}$ . State which number-conserving evidence of coherence survives.

(advanced): A Number-Conserving View of the Paired State

Let ${\mathcal I=\{1,\ldots,L\}}$ label ${L}$ disjoint pair orbitals, so ${{(\hat b_i^\dagger)}^2=0}$ and distinct pair creators commute. For ${\phi=(\phi_1,\ldots,\phi_L)\in\mathbb C^L}$ , define $\hat B_\phi^\dagger=\sum_{i=1}^L\phi_i\hat b_i^\dagger, \qquad e_M({|\phi|}^2) =\sum_{\substack{I\subset\mathcal I\\|I|=M}} \prod_{i\in I}{|\phi_i|}^2 .$ Hard-core nilpotency removes repeated indices, while the ${M!}$ permutations of each subset give ${(\hat B_\phi^\dagger)}^M|0\rangle =M!\sum_{|I|=M}\left(\prod_{i\in I}\phi_i\right)|I\rangle.$

For ${0\le M\le L}$ and ${e_M({|\phi|}^2)>0}$ , the fixed-pair-number collective state is $|\Phi_M(\phi)\rangle =\frac{{(\hat B_\phi^\dagger)}^M|0\rangle} {M!\sqrt{e_M({|\phi|}^2)}} .$ It is a normalized vector in the exact ${2M}$ -electron sector. Its global phase is immaterial, and ${\langle\Phi_M|\hat b_i|\Phi_M\rangle=0}$ because ${\hat b_i}$ changes particle number.

For any number-conserving density operator ${\hat\rho}$ on these orbitals, the pair-density matrix is the matrix representation ${\Gamma_{\hat\rho}=[\Gamma_{ij}]}$ of the positive linear operator ${\hat\Gamma_{\hat\rho}:\mathbb C^L\to\mathbb C^L}$ , where $\Gamma_{ij} =\operatorname{Tr}(\hat\rho\,\hat b_i^\dagger\hat b_j).$ It is Hermitian and positive semidefinite because ${x^\dagger\Gamma_{\hat\rho}x =\operatorname{Tr}[\hat\rho\,\hat B_x^\dagger\hat B_x]\ge0}$ , and ${\operatorname{Tr}\hat\Gamma_{\hat\rho}}$ is the mean pair number.

Under a unitary coordinate change in the span of the pair operators, ${\Gamma}$ transforms by conjugation, so its eigenvalues are basis independent. “Macroscopic eigenvalue” means a largest eigenvalue proportional to ${L}$ along a stated thermodynamic sequence. State that scaling sequence explicitly.

For the uniform choice ${\phi_i=1}$ , Definition 19.5 is the equal superposition of all ${M}$ -pair subsets. Direct counting gives $\Gamma_{ii}=\frac ML,\qquad \Gamma_{ij}=\frac{M(L-M)}{L(L-1)}\quad(i\ne j).$ Hence the uniform vector has eigenvalue $\lambda_{\max} =\frac{M(L-M+1)}L,$ while every orthogonal vector has eigenvalue ${M(M-1)/[L(L-1)]}$ . At fixed filling ${\nu=M/L\in(0,1)}$ , ${\lambda_{\max}=\nu(1-\nu)L+O(1)}$ , even though every anomalous one-point function vanishes. Number-conserving off-diagonal pair coherence therefore survives exact number projection. The broken-symmetry phase is an efficient thermodynamic representation of this order; it is not microscopic charge nonconservation.

Finite check and broken-symmetry comparison. For ${L=4}$ and ${M=2}$ , $\Gamma_{ii}=\frac12,\qquad \Gamma_{ij}=\frac13\ (i\ne j),\qquad \operatorname{spec}(\hat\Gamma) =\left\{\frac32,\frac16,\frac16,\frac16\right\}.$ The trace is ${2=M}$ , as required. A uniform unprojected product with ${{|u|}^2={|v|}^2=1/2}$ instead has ${\langle\hat b_i\rangle=u^*v}$ and off-diagonal ${\Gamma_{ij}={|u^*v|}^2=1/4}$ . At fixed filling and large ${L}$ , both descriptions approach the same off-diagonal value ${\nu(1-\nu)}$ , while only the broken-symmetry representative has a nonzero one-pair expectation.

Physical use and boundary. Exact number is important in an isolated mesoscopic superconducting grain, where even–odd energy differences and pair-transfer matrix elements can reveal pairing without a literal ${\langle\hat b_i\rangle}$ . In a macroscopic sample weak coupling to an environment or a phase-sensitive measurement supplies an effective phase reference, making the broken-symmetry state economical. The extensive eigenvalue of ${\hat\Gamma}$ is the number-conserving statement common to both descriptions.

This calculation does not by itself predict a spectral gap, Meissner screening, or a material transition temperature. It assumes a chosen set of hard-core pair orbitals and diagnoses coherence within that set. Removing this appendix leaves the Cooper logarithm, mean-field closure, problems, and the Lecture 20 interface intact; the appendix only strengthens the interpretation of exact number conservation.

References

[Max50] Maxwell, E. (1950). “Isotope effect in the superconductivity of mercury.” Physical Review 78 (4), 477–477. https://doi.org/10.1103/PhysRev.78.477.

[RSN51] Reynolds, C. A., Serin, B., and Nesbitt, L. B. (1951). “The isotope effect in superconductivity. I. mercury.” Physical Review 84 (4), 691–694. https://doi.org/10.1103/PhysRev.84.691.

[RSWN50] Reynolds, C. A., Serin, B., Wright, W. H., and Nesbitt, L. B. (1950). “Superconductivity of isotopes of mercury.” Physical Review 78 (4), 487–487. https://doi.org/10.1103/PhysRev.78.487.

Lecture 19: Cooper Instability and Pairing Mean Field

19.1 The reduced attractive-shell model

19.1.1 Cooper-shell kinematics

19.1.2 The logarithmic instability and bound state

19.1.3 Scale and cutoff dependence

19.2 From one pair to a coherent product state

19.3 Pairing mean field and its constant term

19.3.1 Linearized thermal instability and a conventional-material trend

19.3.2 A measured isotope-scale check

19.4 Gauge covariance and exact number conservation

19.5 From the Cooper logarithm to a pairing field

19.6 Problems