Lecture 20: Bogoliubov Excitations and Thermal Self-Consistency

A pairing field declares that the normal reference state has been reorganized, but it does not yet identify the independent excitations, their occupations, or what spectroscopy observes. This lecture completes the superconducting construction by changing to canonical quasiparticle modes and closing the thermal and observable self-consistency loop. It thereby joins the course’s earlier excitation, equilibrium, and spectral languages in one setting. Diagonalization is physically meaningful only when the transformed operators preserve the algebra and when energies, state preparation, and measurement weights are carried through the same change of variables.

Use a hypothetical clean one-band conventional superconductor, with aluminum as a qualitative weak-coupling reference, in a normal-metal–insulator–superconductor junction. At low temperature, its differential conductance is strongly suppressed over a voltage interval about zero bias and rises near the superconducting gap edge. The same sample has an exponentially small quasiparticle heat capacity well below its transition and a gap that closes as the transition is approached. These observations must be explained by one spectrum, one set of coherence factors, and one self-consistent ${\Delta(T)}$ —not by unrelated fitting gaps.

The common explanation requires an excitation that is neither pure electron addition nor pure electron removal. What state is created when a superconducting pair field mixes the two operations? Bogoliubov spectra and coherence factors answer the question and control tunneling, heat capacity, pair-breaking thresholds, and the distinction between full and nodal gaps.

We will construct the canonical Bogoliubov transformation, evaluate the anomalous average, close the finite-temperature gap equation, and connect its solution to a declared observable while distinguishing gapped from nodal spectra.

The clean uniform ${s}$ -wave model is intentionally narrower than the experiment. Junction matrix elements, lifetime broadening, disorder, anisotropy, multiple bands, strong coupling, vortices, and phase fluctuations are omitted. The ideal density-of-states edge is therefore a theoretical singularity, not a prediction of an infinite measured conductance.

At ${\xi_{\mathbf k}=0}$ , predict whether a nonzero ${\Delta_{\mathbf k}}$ preserves the normal-state crossing or opens a minimum excitation energy. Also predict whether the positive-energy excitation should be purely electron-like, purely hole-like, or an equal mixture there. An electron-only basis cannot settle both questions because the pair field couples particle addition to partner-hole creation.

Before introducing Nambu notation, recover (i) the eigenvalues of a generic traceless Hermitian ${2\times2}$ matrix from its trace and determinant, (ii) one fermionic self-anticommutator and one cross-anticommutator, and (iii) normalized probabilities for two states of energies ${0}$ and ${E}$ . The last item, not a prior statistical-mechanics course, will supply the Fermi occupation and thermal blocking factor locally.

Nambu representation and its redundancy

Bogoliubov quasiparticles are exact eigenmodes of the quadratic mean-field Hamiltonian, not exact excitations of the original interacting electron problem by construction. Nambu doubling is a representation with a redundancy that must not be mistaken for twice as many physical states. The canonical diagonalization below is exact within the BdG operator; the parent pairing field and gap equation remain variational, while the Fermi occupation factors are exact conditional on that effective equilibrium Hamiltonian. Phase fluctuations, disorder, and finite lifetimes are the first corrections that can change thermal or spectroscopic claims.

Work in the finite periodic box and paired Fock space ${\mathfrak F_{\mathrm p}}$ of Lecture 19. Let ${\mathcal K_{\mathrm p}}$ remain the full shell of oriented pair labels: ${\mathbf k}$ labels ${(\mathbf k{\uparrow},-\mathbf k{\downarrow})}$ . For generic ${\{\mathbf k,-\mathbf k\}}$ , the two blocks ${(\mathbf k{\uparrow},-\mathbf k{\downarrow})}$ and ${(-\mathbf k{\uparrow},\mathbf k{\downarrow})}$ contain four distinct spin orbitals. Retain both momentum labels; each block supplies two physical fermionic modes, so no additional half-momentum restriction is made. The Nambu component space is ${\mathbb C^2}$ ; a Nambu vector is a bookkeeping vector whose entries are operators on ${\mathfrak F_{\mathrm p}}$ , not a two-component state vector. The inherited gap is the ordinary function ${\Delta:\mathcal K_{\mathrm p}\to\mathbb C}$ from Lecture 19’s BCS gap definition. Under a gauge change it transforms covariantly; on the discrete label set it remains an ordinary function, not a distribution. All coefficient matrices below are ordinary linear maps on ${\mathbb C^2}$ and therefore carry no hats.

The Nambu spinor combines annihilation and creation coordinates so that the quadratic pairing problem can be written as a matrix eigenproblem. It does not increase the physical Hilbert space or the number of independent quasiparticle modes. An adjoint operator is fixed by its annihilation operator, so adding it as another coordinate does not supply a new mode. This relation is operator bookkeeping, not automatically a microscopic symmetry of the solid. Gauge choices and Nambu ordering change components; positive-energy modes, spectra, and consistently transformed probe matrix elements carry the physical prediction.

The mean-field Hamiltonian couples an electron annihilation operator to the creation operator of its time-reversed partner. A one-electron basis therefore cannot diagonalize it. Freeze the Nambu operator ordering $\hat\Psi_{\mathbf k}= \begin{pmatrix} \hat c_{\mathbf k\uparrow}\\ \hat c_{-\mathbf k\downarrow}^\dagger \end{pmatrix}.$ This spinor is a bookkeeping device in operator space, not a two-component one-particle wave function with two independent physical states. For ${\mathbf k\in\mathcal K_{\mathrm p}}$ , the numerical BdG coefficient matrix ${h_{\mathrm{BdG}}(\mathbf k):\mathbb C^2\to\mathbb C^2}$ is

(20.1)

$h_{\mathrm{BdG}}(\mathbf k)= \begin{pmatrix} \xi_{\mathbf k}&-\Delta_{\mathbf k}\\ -\Delta_{\mathbf k}^*&-\xi_{\mathbf k} \end{pmatrix}, \qquad E_{\mathbf k}=\sqrt{\xi_{\mathbf k}^2+{|\Delta_{\mathbf k}|}^2}.$

The minus signs reproduce Lecture 19’s ${-\Delta_{\mathbf k}\hat b_{\mathbf k}^{\dagger} -\Delta_{\mathbf k}^{*}\hat b_{\mathbf k}}$ . They are not an independent BdG convention that may be changed without simultaneously redefining the gap.

The BdG matrix has zero trace and determinant ${-\xi_{\mathbf k}^2-{|\Delta_{\mathbf k}|}^2}$ . Predict the relation between its two eigenvalues and whether a nonzero ${\Delta_{\mathbf k}}$ can leave a zero-energy excitation at ${\xi_{\mathbf k}=0}$ .

Canonical Bogoliubov diagonalization

The characteristic equation gives the pair of eigenvalues ${\pm E_{\mathbf k}}$ . The canonical coordinates will be ${(\hat\gamma_{\mathbf k\uparrow},\hat\gamma_{-\mathbf k\downarrow}^{\dagger})}$ : the negative matrix eigenvalue belongs to the creation component of the second positive-energy physical mode. Indeed, ${-E_{\mathbf k}\hat\gamma_{-\mathbf k\downarrow}\hat\gamma_{-\mathbf k\downarrow}^{\dagger} =E_{\mathbf k}\hat\gamma_{-\mathbf k\downarrow}^{\dagger}\hat\gamma_{-\mathbf k\downarrow}-E_{\mathbf k}}$ . Thus each pair block has two positive-energy partner modes, each counted once. Adding a branch copy of either mode would duplicate it; retaining both oriented momentum blocks does not.

Write ${\Delta_{\mathbf k}=|\Delta_{\mathbf k}|\mathrm e^{\mathrm i\chi_{\mathbf k}}}$ . Choose parameters ${u_{\mathbf k}}$ and ${v_{\mathbf k}}$ , up to one common quasiparticle phase, as $u_{\mathbf k} =\sqrt{\frac12\left(1+\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right)}, \qquad v_{\mathbf k} =\mathrm e^{\mathrm i\chi_{\mathbf k}} \sqrt{\frac12\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right)}.$

With the Nambu ordering and BdG matrix frozen above, the corresponding positive-energy column is ${{(u_{\mathbf k}^{*},-v_{\mathbf k}^{*})}^{\mathsf T}}$ . The first row of its eigenvalue equation reads ${(\xi_{\mathbf k}-E_{\mathbf k})u_{\mathbf k}^{*}+\Delta_{\mathbf k}v_{\mathbf k}^{*}=0}$ ; complex conjugation gives the equivalent ${(\xi_{\mathbf k}-E_{\mathbf k})u_{\mathbf k} +\Delta_{\mathbf k}^{*}v_{\mathbf k}=0}$ , while ${E_{\mathbf k}^2-\xi_{\mathbf k}^2={|\Delta_{\mathbf k}|}^2}$ supplies the normalization.

For ${\mathbf k\in\mathcal K_{\mathrm p}}$ with ${E_{\mathbf k}>0}$ , define the matrix-valued ordinary function $U:\mathcal K_{\mathrm p}\longrightarrow U(2), \qquad U_{\mathbf k}= \begin{pmatrix} u_{\mathbf k}&-v_{\mathbf k}\\ v_{\mathbf k}^{*}&u_{\mathbf k}^{*} \end{pmatrix}, \qquad {|u_{\mathbf k}|}^2+{|v_{\mathbf k}|}^2=1.$ The Bogoliubov quasiparticle annihilation operators are the first component and the adjoint of the second component of ${\hat\Gamma_{\mathbf k}=U_{\mathbf k}\hat\Psi_{\mathbf k}}$ : $\begin{pmatrix} \hat\gamma_{\mathbf k\uparrow}\\ \hat\gamma_{-\mathbf k\downarrow}^{\dagger} \end{pmatrix} {=} \begin{pmatrix} u_{\mathbf k}&-v_{\mathbf k}\\ v_{\mathbf k}^{*}&u_{\mathbf k}^{*} \end{pmatrix} \begin{pmatrix} \hat c_{\mathbf k\uparrow}\\ \hat c_{-\mathbf k\downarrow}^{\dagger} \end{pmatrix}.$ They act on the same Fock space ${\mathfrak F_{\mathrm p}}$ as the electron operators. The definition is canonical only when the full two-mode transformation preserves the canonical anticommutation relations. The negative Nambu branch contains the creation operator of the second positive-energy partner; it is not an additional copy of that mode.

The coefficient matrix is unitary when ${{|u_{\mathbf k}|}^2+{|v_{\mathbf k}|}^2=1}$ . Unitarity preserves the two self-anticommutators and the cross-anticommutators; checking only one ${\{\hat\gamma,\hat\gamma^\dagger\}}$ can miss a partner-phase error. The canonical checks are explicit: $\{\hat\gamma_{\mathbf k\uparrow}, \hat\gamma_{\mathbf k\uparrow}^\dagger\} ={|u_{\mathbf k}|}^2+{|v_{\mathbf k}|}^2=1,$

$\{\hat\gamma_{-\mathbf k\downarrow}, \hat\gamma_{-\mathbf k\downarrow}^\dagger\}=1, \qquad \{\hat\gamma_{\mathbf k\uparrow}, \hat\gamma_{-\mathbf k\downarrow}\} =u_{\mathbf k}v_{\mathbf k}-v_{\mathbf k}u_{\mathbf k}=0.$

The remaining cross anticommutator with one adjoint vanishes because it contains anticommutators of distinct original modes. The inverse transform is the adjoint coefficient matrix,

$\begin{pmatrix} \hat c_{\mathbf k\uparrow}\\ \hat c_{-\mathbf k\downarrow}^\dagger \end{pmatrix} {=} \begin{pmatrix} u_{\mathbf k}^*&v_{\mathbf k}\\ -v_{\mathbf k}^*&u_{\mathbf k} \end{pmatrix} \begin{pmatrix} \hat\gamma_{\mathbf k\uparrow}\\ \hat\gamma_{-\mathbf k\downarrow}^\dagger \end{pmatrix}.$

These equations localize a phase error before the Hamiltonian is substituted. The coherence-factor magnitudes are therefore ${|u_{\mathbf k}|}^2=\frac12\left(1+\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right), \qquad {|v_{\mathbf k}|}^2=\frac12\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right).$ Far above the Fermi surface the positive excitation is electron-like; far below it is hole-like; at ${\xi=0}$ , the weights are equal.

For the ordered Nambu vector, pair-label domain, and sign convention above, suppose ${E_{\mathbf k}>0}$ . Then $U_{\mathbf k}h_{\mathrm{BdG}}(\mathbf k)U_{\mathbf k}^{\dagger} =\operatorname{diag}(E_{\mathbf k},-E_{\mathbf k}), \qquad u_{\mathbf k}^{*}v_{\mathbf k} =\frac{\Delta_{\mathbf k}}{2E_{\mathbf k}}.$ The transformation in Definition 20.1 preserves every canonical anticommutator and supplies exactly the two positive-energy partner modes of one physical pair sector. At a simultaneous zero ${\xi_{\mathbf k}=\Delta_{\mathbf k}=0}$ , the eigenvalue is degenerate and the coherence-factor phase is not fixed by this theorem.

Reuse the nonduplicated paired-subspace convention of the two-pair-level example. For a real ${\Delta>0}$ , diagonalize ${h=\bigl(\begin{smallmatrix}\xi&-\Delta\\-\Delta&-\xi\end{smallmatrix}\bigr)}$ and construct the canonical two-mode transformation for the Nambu column ${\hat\Psi}$ with entries ${\hat c_\uparrow}$ and ${\hat c_\downarrow^\dagger}$ . Verify a self- and a cross-anticommutator and account for the scalar constant when the Nambu expression is converted back to physical operators.

Squaring gives ${h^2=(\xi^2+\Delta^2)I}$ , so ${E=\sqrt{\xi^2+\Delta^2}>0}$ . The positive eigenvector is ${{(u,-v)}^{\mathsf T}}$ , with ${v/u=(E-\xi)/\Delta}$ and $u^2=\frac{E+\xi}{2E},\quad v^2=\frac{E-\xi}{2E},\quad uv=\frac{\Delta}{2E},\quad U=\begin{pmatrix}u&-v\\v&u\end{pmatrix}.$ Then ${UhU^{\mathsf T}=\operatorname{diag}(E,-E)}$ and ${\hat\gamma_\uparrow=u\hat c_\uparrow-v\hat c_\downarrow^\dagger}$ , ${\hat\gamma_\downarrow=v\hat c_\uparrow^\dagger+u\hat c_\downarrow}$ . The self-anticommutator is ${u^2+v^2=1}$ , while ${\{\hat\gamma_\uparrow,\hat\gamma_\downarrow\}=uv-vu=0}$ ; the latter would detect an inconsistent partner sign.

For one nonduplicated physical pair, the original block is ${\hat H_{\mathrm{pair}}=\hat\Psi^\dagger h\hat\Psi+(\xi+C)\hat I}$ . Indeed ${-\xi \hat c_\downarrow \hat c_\downarrow^\dagger =\xi \hat n_\downarrow-\xi\hat I}$ . After rotation, $\hat H_{\mathrm{pair}}=E(\hat n_{\gamma\uparrow}+\hat n_{\gamma\downarrow})+(\xi-E+C)\hat I.$ The scalar ${C}$ is the assigned mean-field constant, not another mode. At ${\xi=0}$ , ${u=v=1/\sqrt2}$ and the gap is ${\Delta}$ ; as ${\Delta\to0}$ away from the crossing the electron/hole weights return to zero or one. These spectral limits and the anticommutators are independent checks. Counting an annihilator and its adjoint as separate physical species would fail the state count. Thermal averaging next uses the two positive-energy partner occupations, not a duplicated Hilbert space.

At ${\xi=0}$ , the eigenvalues are ${\pm|\Delta|}$ and ${{|u|}^2={|v|}^2=1/2}$ : pairing avoids the normal-state crossing and makes the excitation an equal particle–hole mixture. The DOS calculation below will turn that block-level result into a spectral prediction and will identify when gap symmetry can restore zero-energy excitations.

Checking the Bogoliubov transformation before thermal averaging

Stop before introducing thermal occupations. For one declared ${\xi}$ and complex ${\Delta}$ :

use trace and determinant to recover ${\pm E}$ ;
normalize the full two-mode transformation and check one cross-anticommutator;
predict the electron-like and hole-like limits of the positive branch; and
state why the negative Nambu branch is not an additional excitation.

This recovery separates matrix eigenvalues, canonical operator algebra, and physical state counting before the gap equation combines them.

Just-in-time statistical mechanics: thermal blocking

For one fermionic Bogoliubov mode of energy ${E\ge0}$ , let ${\hat n=\hat\gamma^\dagger\hat\gamma}$ and ${\hat H_E=E\hat n}$ on the two-state mode Fock space ${\operatorname{span}\{|0\rangle,|1\rangle\}}$ . At ${T>0}$ , the normalized equilibrium density operator, with ${\beta={(k_{\mathrm B}T)}^{-1}}$ , is $\hat\rho_{E,T} =\frac{\mathrm e^{-\hat H_E/(k_{\mathrm B}T)}} {\operatorname{Tr}\mathrm e^{-\hat H_E/(k_{\mathrm B}T)}}.$ The two allowed occupations ${n=0,1}$ consequently have probabilities $p_0=\frac{1}{1+\mathrm e^{-\beta E}}, \qquad p_1=\frac{\mathrm e^{-\beta E}}{1+\mathrm e^{-\beta E}}.$ The mean occupation is therefore the ordinary function $f(E)\equiv\langle\hat n\rangle=0p_0+1p_1 =\frac{1}{\mathrm e^{\beta E}+1}.$

The thermal pair-blocking factor is the ordinary scalar function $B:(0,\infty)\times[0,\infty)\longrightarrow[0,1), \qquad B(T,E)=p_0-p_1=1-2f(E) =\tanh\!\left(\frac{E}{2k_{\mathrm B}T}\right).$ Thus ${B(T,0)=0}$ , ${\lim_{T\to0^+}B(T,E)=1}$ for ${E>0}$ , and ${\lim_{T\to\infty}B(T,E)=0}$ at fixed ${E}$ . It is neither a probability amplitude nor a distribution; it is a difference of two normalized occupation probabilities and measures thermal blocking of coherent pair amplitude.

Anomalous average from quasiparticle occupations

The blocking factor enters the pair amplitude through the inverse canonical transformation. With the ordering fixed above,

$\hat c_{\mathbf k\uparrow} =u_{\mathbf k}^{*}\hat\gamma_{\mathbf k\uparrow} +v_{\mathbf k}\hat\gamma_{-\mathbf k\downarrow}^{\dagger}, \qquad \hat c_{-\mathbf k\downarrow} =-v_{\mathbf k}\hat\gamma_{\mathbf k\uparrow}^{\dagger} +u_{\mathbf k}^{*}\hat\gamma_{-\mathbf k\downarrow}.$

In equilibrium, terms with two creation or two annihilation operators average to zero, and both positive-energy partner modes have mean occupation ${f(E_{\mathbf k})}$ . The surviving terms give

$\langle\hat c_{-\mathbf k\downarrow} \hat c_{\mathbf k\uparrow}\rangle =u_{\mathbf k}^{*}v_{\mathbf k} \left[1- \langle\hat\gamma_{\mathbf k\uparrow}^{\dagger} \hat\gamma_{\mathbf k\uparrow}\rangle -\langle\hat\gamma_{-\mathbf k\downarrow}^{\dagger} \hat\gamma_{-\mathbf k\downarrow}\rangle\right].$

The BdG eigenvector convention gives ${u_{\mathbf k}^{*}v_{\mathbf k} =\Delta_{\mathbf k}/(2E_{\mathbf k})}$ . Therefore

$\langle\hat c_{-\mathbf k\downarrow} \hat c_{\mathbf k\uparrow}\rangle =\frac{\Delta_{\mathbf k}}{2E_{\mathbf k}} \tanh\!\left(\frac{\beta E_{\mathbf k}}{2}\right).$ At ${T=0}$ , both positive-energy modes are empty, quasiparticle blocking is absent, and ${B\to1}$ . When thermal quasiparticles populate either partner, the empty-versus-paired coherence is reduced. This is the required thermal meaning: ${1-2f}$ is not a new interaction and not the probability of a Cooper pair.

Substitution into the gap definition ${\Delta_{\mathbf k}=-\sum_{\mathbf k'}V_{\mathbf k\mathbf k'}\langle\hat c_{-\mathbf k'\downarrow}\hat c_{\mathbf k'\uparrow}\rangle}$ of Lecture 19 yields

(20.2)

$\Delta_{\mathbf k} =-\sum_{\mathbf k'}V_{\mathbf k\mathbf k'} \frac{\Delta_{\mathbf k'}}{2E_{\mathbf k'}} \tanh\!\left(\frac{\beta E_{\mathbf k'}}{2}\right).$

Gap integral and its thermal limits

Return to the constant-attraction shell of Lecture 19. Set ${V_{\mathbf k\mathbf k'}=-g}$ , use the single-spin density of states ${N_0}$ , and let ${|\xi|<\hbar\omega_{\mathrm c}}$ . In the simplest phonon cutoff model one may identify ${\omega_{\mathrm c}=\omega_{\mathrm D}}$ , but the core mathematics needs only the declared cutoff. For an isotropic nonzero gap, divide the gap equation by ${\Delta}$ ; only ${|\Delta|}$ remains, through ${E}$ . Use the symmetry between positive and negative ${\xi}$ :

$1=gN_0 \int_0^{\hbar\omega_{\mathrm c}} \frac{\mathrm d\xi}{\sqrt{\xi^2+{|\Delta|}^2}} \tanh\!\left[ \frac{\beta\sqrt{\xi^2+{|\Delta|}^2}}{2} \right].$

Dividing by ${\Delta}$ analyzes only the paired branch; the algebraic solution ${\Delta=0}$ must be retained separately and compared by free energy. At ${T=0}$ , the thermal factor is one and the integral becomes

$\operatorname{arsinh}\frac{\hbar\omega_{\mathrm c}}{|\Delta|} \simeq\ln\frac{2\hbar\omega_{\mathrm c}}{|\Delta|}$

in weak coupling. It gives

$|\Delta(0)|\simeq2\hbar\omega_{\mathrm c} \mathrm e^{-1/(gN_0)}.$ At ${T=T_{\mathrm c}}$ , take ${\Delta\to0}$ only after retaining the ${\tanh}$ factor; thermal blocking regularizes the lower end of the integral. The result is $\frac1{gN_0} =\int_0^{\hbar\omega_{\mathrm c}} \frac{\mathrm d\xi}{\xi} \tanh\!\left(\frac{\xi}{2k_{\mathrm B}T_{\mathrm c}}\right) \simeq \ln\!\left( \frac{2\mathrm e^{\gamma}}{\pi} \frac{\hbar\omega_{\mathrm c}}{k_{\mathrm B}T_{\mathrm c}} \right),$

where ${\gamma}$ is Euler’s constant. The logarithm follows by scaling ${x=\xi/(2k_{\mathrm B}T_{\mathrm c})}$ and subtracting the large- ${x}$ integrand ${1/x}$ ; the remaining convergent universal integral evaluates to ${\ln(4\mathrm e^\gamma/\pi)}$ . Thus $k_{\mathrm B}T_{\mathrm c}\simeq1.134\hbar\omega_{\mathrm c} \mathrm e^{-1/(gN_0)}, \qquad \frac{2|\Delta(0)|}{k_{\mathrm B}T_{\mathrm c}}\simeq3.53.$ The shared exponential makes the ideal ratio independent of ${g}$ and the cutoff, but the numerical constants belong only to this isotropic weak-coupling model and its single-spin density-of-states convention. The stable branch is the free-energy minimum; self-consistency alone does not show that the paired solution is preferred, and ${\Delta=0}$ becomes unstable only below ${T_{\mathrm c}}$ .

Before reading Figure 20.1, interpret the gap equation as an intersection rather than as a closed formula for ${\Delta}$ . Predict what must happen to the nonzero intersection as ${T}$ approaches ${T_{\mathrm c}}$ . Both panels of the figure measure gaps in units of the zero-temperature gap ${\Delta_0=|\Delta(0)|}$ of the evaluated model.

Figure 20.1. A nonzero superconducting gap is a fixed point of the thermal self-consistency map; in the weak-coupling interpolation it decreases continuously to zero at the critical temperature.

Panel (a) makes clear that solving the equation and selecting the stable state are distinct operations. Panel (b) is explicitly a weak-coupling BCS interpolation, not an exact universal temperature law. Its role is to connect the two limiting calculations above and to expose the branch that disappears at ${T_{\mathrm c}}$ .

One parameter set now connects the spectrum, coherence factors, transition temperature, and density of states. Take ${gN_0=0.25}$ and ${\hbar\omega_{\mathrm c}=20\,\mathrm{meV}}$ . Then $|\Delta(0)|=40\mathrm e^{-4}\,\mathrm{meV}=0.733\,\mathrm{meV}, \qquad k_{\mathrm B}T_{\mathrm c} =22.68\mathrm e^{-4}\,\mathrm{meV}=0.415\,\mathrm{meV},$ so ${T_{\mathrm c}=4.82\,\mathrm K}$ using ${k_{\mathrm B}=0.08617\,\mathrm{meV\,K^{-1}}}$ , and ${2|\Delta(0)|/(k_{\mathrm B}T_{\mathrm c})=3.53}$ . At the normal-state energy ${\xi=|\Delta(0)|}$ , the quasiparticle has $E=\sqrt2|\Delta(0)|,\qquad {|u|}^2=0.8536,\qquad {|v|}^2=0.1464, \qquad |u^*v|=0.3536.$ At the observable energy ${E=2|\Delta(0)|}$ , the density-of-states formula below gives ${N_{\mathrm s}(E)/N_0=2/\sqrt3=1.155}$ . These numbers use one cutoff and degeneracy convention; changing only one of them would create an inconsistent example rather than a new physical effect.

Before the density-of-states section, reconstruct in one line why ${1-2f(E)=\tanh(\beta E/2)}$ , and point to where this combination entered the gap equation’s thermal factor. If the identity does not come out, the repair is the two-level occupation algebra ${f(E)=1/(\mathrm e^{\beta E}+1)}$ — not the gap integral itself.

Use the real positive-gap canonical transformation of Example 20.1 with equal partner occupations ${f(E)}$ , constant attraction ${-g}$ , and single-spin DOS ${N_0}$ on ${|\xi|<E_{\mathrm c}}$ . Derive the anomalous average and gap integral, then compare ${\Delta(0)}$ and ${T_{\mathrm c}}$ for ${gN_0\ll1}$ . For the thermal logarithm use, with Euler’s constant ${\gamma}$ , the convergent identity $\lim_{A\to\infty}\left[\int_0^A\frac{\tanh x}{x}\,\mathrm dx-\ln A\right] =\ln(4\mathrm e^{\gamma}/\pi).$

The inverse transformation gives ${\hat c_\uparrow=u^*\hat\gamma_\uparrow+v\hat\gamma_\downarrow^\dagger}$ and ${\hat c_\downarrow=-v\hat\gamma_\uparrow^\dagger+u^*\hat\gamma_\downarrow}$ . Cross expectations vanish in the diagonal thermal state. Thus $\langle\hat c_\downarrow\hat c_\uparrow\rangle =u^*v[(1-f)-f]=\frac{\Delta}{2E}\tanh\frac{E}{2k_{\mathrm B}T}.$ With ${\Delta=-\sum V\langle\hat b\rangle}$ , the two symmetric halves of the shell cancel the factor two in this denominator: $\frac1{gN_0}=\int_0^{E_{\mathrm c}}\frac{\mathrm d\xi}{\sqrt{\xi^2+\Delta^2}} \tanh\frac{\sqrt{\xi^2+\Delta^2}}{2k_{\mathrm B}T}.$ At zero temperature, substitute ${\xi=\Delta\sinh s}$ : the integral is ${\operatorname{arsinh}(E_{\mathrm c}/\Delta)}$ , giving the exact shell result ${\Delta(0)=E_{\mathrm c}/\sinh[1/(gN_0)]\simeq2E_{\mathrm c}\mathrm e^{-1/(gN_0)}}$ . At ${T_{\mathrm c}}$ , put ${\Delta=0}$ after retaining the thermal factor and set ${x=\xi/(2k_{\mathrm B}T_{\mathrm c})}$ . The supplied identity gives ${k_{\mathrm B}T_{\mathrm c}=(2\mathrm e^{\gamma}/\pi)E_{\mathrm c}\mathrm e^{-1/(gN_0)}}$ in weak coupling. Consequently ${2\Delta(0)/(k_{\mathrm B}T_{\mathrm c})=2\pi/\mathrm e^{\gamma}\simeq3.53}$ .

At ${T=0}$ the blocking factor is one; for ${E\ll k_{\mathrm B}T}$ it is ${E/(2k_{\mathrm B}T)}$ , making the thermal lower limit finite. These checks test occupation and convergence separately. The ratio has no units or cutoff dependence at this order. Dividing by ${\Delta}$ excluded the normal branch; free energy, not this equation alone, decides stability. The following state count and near-critical calculation test different consequences of the same self-consistent spectrum.

Fully gapped and nodal observables

For a constant real gap ${\Delta_0>0}$ and flat single-spin normal DOS ${N_0}$ , calculate the electron-addition DOS at ${E>0}$ . Include both roots of ${E=\sqrt{\xi^2+\Delta_0^2}}$ with the coherence weights of Example 20.1. Integrate to the image of a symmetric normal-state cutoff ${|\xi|<\xi_c}$ , ${\xi_c>0}$ , to check state conservation.

An ideal ${s}$ -wave gap ${\Delta_{\mathrm s}(\mathbf k)=\Delta_0}$ is nonzero everywhere on a simple Fermi surface. For a constant normal-state density of states, conservation of states under ${E=\sqrt{\xi^2+\Delta_0^2}}$ gives the superconducting density of states

$N_{\mathrm s}(E)=N_0 \frac{|E|}{\sqrt{E^2-\Delta_0^2}} \Theta(|E|-|\Delta_0|).$

Here ${\Theta:\mathbb R\to\{0,1\}}$ is the ordinary Heaviside step function away from the origin; its assigned value at zero does not change any integral used here. The ${\delta}$ introduced below is instead the Dirac distribution on ${\mathbb R}$ , defined by ${\int_{\mathbb R}\delta(E-E_0)\varphi(E)\,\mathrm dE=\varphi(E_0)}$ for smooth compactly supported test functions ${\varphi}$ . The spectrum has no states for ${|E|<|\Delta_0|}$ and an ideal integrable square-root edge at the gap. Scattering and instrumental resolution broaden the measured curve, so the divergence is a property of the ideal model rather than a literal infinite signal.

For ${E>0}$ , the change of variables includes both roots ${\xi_\pm=\pm\sqrt{E^2-\Delta_0^2}}$ and their electron-addition coherence weights:

$N_{\mathrm s}(E) =N_0\int\mathrm d\xi\,{|u_\xi|}^2 \delta\!\left(E-\sqrt{\xi^2+\Delta_0^2}\right).$

Since ${|\mathrm dE/\mathrm d\xi|=|\xi|/E}$ and ${{|u_{\xi_+}|}^2+{|u_{\xi_-}|}^2=1}$ , the two roots give exactly the displayed ${N_0E/\sqrt{E^2-\Delta_0^2}}$ , not an extra factor of two. With a matched normal-state cutoff ${|\xi|<\xi_{\mathrm c}}$ ,

$\int_{\Delta_0}^{\sqrt{\xi_{\mathrm c}^2+\Delta_0^2}} N_{\mathrm s}(E)\,\mathrm dE =N_0\xi_{\mathrm c},$

so the edge singularity redistributes states but does not create them.

The derivative of ${\sqrt{E^2-\Delta_0^2}}$ is ${E/\sqrt{E^2-\Delta_0^2}}$ , so the cutoff integral explicitly returns ${N_0\xi_c}$ . Independently, as ${E/\Delta_0\to\infty}$ , the DOS tends to ${N_0}$ , not ${2N_0}$ . The edge behaves as ${{(E-\Delta_0)}^{-1/2}}$ and is integrable; no extra state is created at the divergence. This is an electron spectral DOS, not an unrestricted count of redundant BdG branches. Tunneling next requires its own matrix-element and thermal-convolution assumptions.

From the ideal density of states to tunneling and heat capacity

The density of states is not itself the reading on a voltmeter. For a normal-metal–insulator–superconductor (NIS) junction with normal-state resistance ${R_{\mathrm T}}$ , a momentum-independent tunneling matrix element, and a flat normal-electrode density of states, the current is

$I(V)=\frac{1}{eR_{\mathrm T}}\int_{-\infty}^{\infty}\mathrm dE\, \frac{N_{\mathrm s}(E)}{N_0} \bigl[f(E-eV)-f(E)\bigr].$

Here ${V\in\mathbb R}$ is the applied voltage and ${f:\mathbb R\to[0,1]}$ is the Fermi function introduced above; the two Fermi factors already broaden the current thermally. Differentiating with ${G_{\mathrm N}=1/R_{\mathrm T}}$ gives ${G/G_{\mathrm N}=\int\mathrm dE\,[N_{\mathrm s}(E)/N_0]\,[-\partial_Ef(E-eV)]}$ , where ${G=\mathrm dI/\mathrm dV}$ : the measured conductance is ${N_{\mathrm s}}$ averaged over the thermal window ${-\partial_Ef}$ , of full width ${4\ln(1+\sqrt2)\,k_{\mathrm B}T\approx3.5\,k_{\mathrm B}T}$ at half maximum, not a pointwise copy of it. Only in the ideal ${T\to0}$ limit does

$\frac{G(V)}{G_{\mathrm N}} =\frac{\mathrm dI/\mathrm dV}{1/R_{\mathrm T}} \longrightarrow \frac{N_{\mathrm s}(eV)}{N_0}.$

Before reading Figure 20.2, predict where an ideal fully gapped junction develops conductance edges and why the recorded derivative is not, at finite temperature, a pointwise density of states.

Figure 20.2. A mounted N–I–S junction, enlarged crossed-film overlap and computed lock-in conductance connect the density of states to a tunneling record.

Panel (a) shows the mounted package; panel (b) exposes the junction layers with the declared vertical exaggeration: the insulating barrier lies between the normal and superconducting films at their overlap. The circuit in panel (c) separates the applied bias and measured current from the inferred differential conductance: the transimpedance amplifier (TIA) converts input current to an output voltage, and the lock-in amplifier (LIA) uses the dashed bias reference. The separate dotted stage-to-temperature-record branch denotes thermometry of the cold stage, not a direct measurement of junction-electron temperature; its sensor hardware is omitted.

Panel (d) compares the normalized phenomenologically broadened DOS ${N_\Gamma/N_0}$ (dashed) with the computed lock-in conductance ${G_1/G_{\mathrm N}}$ (solid), not measured data. Here ${G_1=I_{1,\mathrm{peak}}/V_{\mathrm{ac,peak}}}$ is the quasistatic first-harmonic current divided by the peak bias modulation. The model first convolves ${N_\Gamma}$ with the Fermi derivative at ${k_{\mathrm B}T_{\mathrm e}/\Delta=0.08}$ , then averages over the finite modulation ${eV_{\mathrm{ac,peak}}/\Delta=0.06}$ ; ${N_\Gamma}$ is the ideal ${N_{\mathrm s}}$ with its gap edge broadened by the selected width ${\Gamma/\Delta=0.055}$ . Here ${T_{\mathrm e}}$ is an assumed equilibrium junction-electron temperature, not the stage thermometer reading. The gap is fixed, the normal DOS and tunnel matrix element are flat, and heating, series resistance and environmental exchange are omitted. Only as temperature and modulation vanish does the solid response reduce to the chosen ${N_\Gamma/N_0}$ . The package is not a reconstruction of Giaever’s apparatus. The symmetric edge scale tracks ${|eV|\simeq\Delta}$ only under the flat-normal-DOS and constant-matrix-element assumptions stated above. Thermal convolution, phenomenological broadening, series resistance, and an energy-dependent tunneling matrix element can alter the line shape without changing the algebraic definition of ${N_{\mathrm s}}$ .

The preceding trace is illustrative. Before reading the historical record in Figure 20.4, predict how the low-bias current should change when the Pb electrode is driven from its normal control into the zero-field superconducting state, and keep the author’s derived relative slope distinct from both raw current and pointwise density of states.

Figure 20.3. Giaever’s source-specific crossed-film system and control logic. The project-original exploded view shows vapor-deposited Al, a 15–20 angstrom Al2O3 barrier and Pb on glass; vertical separations and film thicknesses are schematic. The readout compares curve 1 and curve 5 at 1.6 K before forming the paper’s relative slope. [Gia60, pp. 147–148]

Panel (a) supplies the historical sample rather than a generic modern package: Al remains the normal electrode at ${1.6\,\mathrm K}$ , while a field parallel to the films toggles Pb between the normal control and zero-field superconducting state. Panel (b) preserves the evidence hierarchy: applied voltage and measured current come first; the relative slope is the paper’s derived comparison, not a separate raw detector channel.

See the cited paper for this experimental plot; it is not included in this distribution.

Figure 20.4. Giaever’s measured normal-control and superconducting tunneling records. Panel (a) replots visible marker centers from curves 1 and 5; panel (b) replots visible markers from the paper’s author-derived slope of curve 5 relative to curve 1. Lines only guide the eye. No new differentiation, smooth fit, uncertainty, or density-of-states relabeling is added. [Gia60, figs. 1–2]

Relative to the normal control, the 1.6-K zero-field curve suppresses low-bias current and the author’s relative slope rises to a finite-bias maximum before approaching the normal reference. This is measured support for a superconducting tunneling threshold under the stated junction conditions. It is not, by itself, a pointwise measurement of ${N_{\mathrm s}(E)}$ , a precision weak-coupling gap ratio, or evidence that matrix elements, temperature, lifetime, and circuit response are negligible.

Return now to the lecture’s illustrative weak-coupling parameter set: the following numbers are model outputs, not a fit to Giaever’s Pb junction. That parameter set predicts a threshold scale ${|V|=|\Delta(0)|/e=0.733\,\mathrm{mV}}$ , and at ${|eV|=2|\Delta(0)|}$ it predicts ${G/G_{\mathrm N}=1.155}$ under those ideal assumptions. Finite temperature rounds the threshold; lifetime broadening and an energy-dependent matrix element can change the line shape. A broadened raw conductance curve must therefore not be relabeled as the density of states without stating the deconvolution model.

Thermodynamics interrogates the same ${E_{\mathbf k}}$ without a junction. In the particle-hole-symmetric constant-DOS approximation, evaluate the thermodynamic potential at fixed chemical potential; the density-induced chemical-potential shift is neglected. A convenient equilibrium object is the free-energy difference

$\begin{aligned} F_{\mathrm s}(\Delta,T)-F_{\mathrm n}(T) ={}&\frac{{|\Delta|}^2}{g} +\sum_{\mathbf k}\bigl(|\xi_{\mathbf k}|-E_{\mathbf k}\bigr)\\ &-2k_{\mathrm B}T\sum_{\mathbf k} \ln\!\left[ \frac{1+\mathrm e^{-E_{\mathbf k}/(k_{\mathrm B}T)}} {1+\mathrm e^{-|\xi_{\mathbf k}|/(k_{\mathrm B}T)}} \right], \end{aligned}$

where the momentum sum and spin convention are the same as in the gap equation. The physical branch uses the self-consistent ${\Delta:[0,T_{\mathrm c}]\to\mathbb R_{\geq0}}$ , and

$C_{\mathrm s}(T)-C_{\mathrm n}(T) =-T\frac{\mathrm d^2}{\mathrm dT^2} \left\{F_{\mathrm s}[\Delta(T),T]-F_{\mathrm n}(T)\right\}.$

At ${1\,\mathrm K}$ , the present parameter set has ${\Delta(0)/(k_{\mathrm B}T)=8.51}$ , so the occupation scale at the gap edge is ${f[\Delta(0)]\simeq2.0\times10^{-4}}$ . This is the origin of the activated low-temperature electronic heat capacity in a fully gapped sample.

In the particle-hole-symmetric, weak-coupling constant-DOS model of Example 20.2, derive the leading near- ${T_{\mathrm c}}$ free-energy coefficients, ${\Delta(T)}$ , and the heat-capacity jump. Use single-spin DOS ${N_0}$ per volume. To fix the normalization, take the mean-field grand-potential density difference at fixed chemical potential, with ${s=|\Delta|^2}$ , ${E=\sqrt{\xi^2+s}}$ , and ${g>0}$ : $F(s,T)-F(0,T)=\frac{s}{g} -2N_0\int_0^{E_{\mathrm c}}\!\mathrm d\xi\left[ E-\xi+2k_{\mathrm B}T\ln\frac{1+\mathrm e^{-E/(k_{\mathrm B}T)}}{1+\mathrm e^{-\xi/(k_{\mathrm B}T)}} \right].$ Here ${F}$ denotes this fixed- ${\mu}$ potential, not a silently changed ensemble. The constant-DOS approximation neglects the density-induced shift of ${\mu}$ . Distinguish a derivative at fixed ${s}$ from one along its equilibrium minimum. For the convergent coefficient, use $\frac{\tanh y}{y} =8\sum_{n\ge0}\frac1{\pi^2{(2n+1)}^2+4y^2}.$

First differentiate the supplied potential. Since ${\partial_s E=1/(2E)}$ and ${\partial_E[2k_{\mathrm B}T\ln(1+\mathrm e^{-E/(k_{\mathrm B}T)})]=-2f(E)}$ , $\partial_sF =N_0\left[\frac1{gN_0} -\int_0^{E_{\mathrm c}}\frac{\tanh(E/(2k_{\mathrm B}T))}{E}\,\mathrm d\xi\right].$ This fixes both the gap equation and its thermodynamic prefactor. Knowing only that the bracket vanishes would not determine the energy or heat capacity.

Subtract the linearized gap equation at ${T_{\mathrm c}}$ and expand for ${|T-T_{\mathrm c}|\ll T_{\mathrm c}}$ , ${s\ll{(k_{\mathrm B}T_{\mathrm c})}^2}$ , and ${k_{\mathrm B}T_{\mathrm c}\ll E_{\mathrm c}}$ . The temperature term is ${\ln(T/T_{\mathrm c})}$ . With ${y=\xi/(2k_{\mathrm B}T)}$ , the term linear in ${s}$ is ${sI/[8{(k_{\mathrm B}T)}^2]}$ , where $I=\int_0^\infty\left[ \frac{\tanh y}{y^3}-\frac{\operatorname{sech}^2y}{y^2} \right]\mathrm dy.$ The integrand tends to ${2/3}$ at zero and decays as ${y^{-3}}$ . Differentiate the supplied partial fraction and divide by ${-y}$ : $I=64\sum_{n\ge0}\int_0^\infty \frac{\mathrm dy}{{[\pi^2{(2n+1)}^2+4y^2]}^2} =\frac8{\pi^2}\sum_{n\ge0}\frac1{{(2n+1)}^3} =\frac{7\zeta(3)}{\pi^2}.$ Here ${2y=a\tan\theta}$ gives ${\int_0^\infty\mathrm dy/{(a^2+4y^2)}^2=\pi/(8a^3)}$ ; the positive terms justify exchanging sum and integral. The odd-integer sum is ${(1-2^{-3})\zeta(3)}$ .

Put ${\tau=(T-T_{\mathrm c})/T_{\mathrm c}}$ . To leading near-critical order, $\partial_sF=N_0\left[\tau+ \frac{7\zeta(3)s}{8\pi^2{(k_{\mathrm B}T_{\mathrm c})}^2}+\cdots\right].$ The omitted dimensionless terms include ${\tau^2}$ , ${\tau s/{(k_{\mathrm B}T_{\mathrm c})}^2}$ and ${s^2/{(k_{\mathrm B}T_{\mathrm c})}^4}$ . Integrating with respect to ${s}$ , with zero difference at ${s=0}$ , gives $F(s,T)-F(0,T) =N_0\left[\tau s+ \frac{7\zeta(3)s^2}{16\pi^2{(k_{\mathrm B}T_{\mathrm c})}^2}\right]+\cdots.$ The positive quartic coefficient stabilizes the small paired minimum below ${T_{\mathrm c}}$ : $\Delta(T)\simeq \sqrt{\frac{8\pi^2}{7\zeta(3)}}\,k_{\mathrm B}T_{\mathrm c} \sqrt{1-\frac{T}{T_{\mathrm c}}} \simeq3.06\,k_{\mathrm B}T_{\mathrm c}\sqrt{1-\frac{T}{T_{\mathrm c}}}.$ Substitution before differentiation yields $F_{\mathrm{eq}}(T)-F_{\mathrm n}(T) =-\frac{4\pi^2N_0k_{\mathrm B}^2}{7\zeta(3)}{(T-T_{\mathrm c})}^2+\cdots.$ Thus ${-T\,\mathrm d^2(F_{\mathrm{eq}}-F_{\mathrm n})/\mathrm dT^2}$ gives $\Delta C=\frac{8\pi^2N_0k_{\mathrm B}^2T_{\mathrm c}}{7\zeta(3)},\qquad \gamma_{\mathrm n}=\frac{2\pi^2}{3}N_0k_{\mathrm B}^2,\qquad \frac{\Delta C}{\gamma_{\mathrm n}T_{\mathrm c}}=\frac{12}{7\zeta(3)}\simeq1.43.$ Both heat capacities are densities under this ${N_0}$ convention; the normal coefficient counts both spin species.

The equilibrium energy difference is negative below ${T_{\mathrm c}}$ , and its first temperature derivative vanishes at ${T_{\mathrm c}}$ : branch stability and entropy continuity are distinct checks. The jump is positive and has heat-capacity units; the ratio cancels ${N_0}$ . Differentiating at fixed gap instead would miss the equilibrium curvature. This clean isotropic weak-coupling result is not a universal jump for a nodal or strongly coupled system. The nodal count that follows changes that physical input rather than inheriting this number.

A full gap suppresses low-temperature quasiparticle occupation, producing activated electronic heat capacity. The factor two in ${\gamma_{\mathrm n}}$ includes both spin species while ${N_0}$ remains single-spin. Thus the tunneling threshold, activated low-temperature heat capacity, heat-capacity jump, and closing gap are four consequences of the same self-consistent spectrum. Strong coupling, gap anisotropy, multiple bands, or pair breaking can move the numerical ratio and jump; disagreement then diagnoses missing physics rather than permission to fit each probe with an unrelated gap.

Observation	Quantity from this calculation	Shared prediction	Main idealization
NIS tunneling	${N_{\mathrm s}(E)}$ convolved with ${-f'(E-eV)}$	edge near ${0.733\,\mathrm{mV}}$	flat normal DOS and constant matrix element
Low- ${T}$ heat capacity	thermal occupation of ${E_{\mathbf k}}$	activated scale ${\mathrm e^{-\Delta/(k_{\mathrm B}T)}}$	clean, fully gapped spectrum
Transition anomaly	curvature of ${F_{\mathrm s}[\Delta(T),T]}$	weak-coupling jump ratio ${1.43}$	mean-field equilibrium
Gap closing	finite- ${T}$ gap equation	${T_{\mathrm c}=4.82\,\mathrm K}$	constant ${N_0}$ , one band, sharp cutoff

By contrast, the square-lattice ${d_{x^2-y^2}}$ -wave function $\Delta_{\mathrm d_{x^2-y^2}}(\mathbf k) =\Delta_0[\cos(k_xa)-\cos(k_ya)]$ changes sign under a quarter rotation and vanishes on ${k_x=\pm k_y}$ . Physical nodes occur where those nodal lines cross the Fermi surface. The order-parameter amplitude and minimum excitation gap are therefore not interchangeable in an anisotropic system.

For a declared normal-state dispersion ${\xi:\mathcal K_{\mathrm p}\to\mathbb R}$ and gap function ${\Delta:\mathcal K_{\mathrm p}\to\mathbb C}$ , define the Fermi-surface set ${\mathcal F=\xi^{-1}(\{0\})}$ and gap-zero set ${\mathcal Z_\Delta=\Delta^{-1}(\{0\})}$ . Because $E_{\mathbf k}^2=\xi_{\mathbf k}^2+{|\Delta_{\mathbf k}|}^2,$ a zero-energy quasiparticle occurs exactly at ${\mathbf k\in\mathcal F\cap\mathcal Z_\Delta}$ . A zero of ${\Delta}$ away from ${\mathcal F}$ is not a quasiparticle node.

Nodal counterexample. A sign-changing gap function can have zeros that miss the actual Fermi surface. Then ${\Delta_{\mathbf k}=0}$ somewhere in the Brillouin zone but ${E_{\mathbf k}}$ remains nonzero on every low-energy normal-state momentum. Conversely, a physical zero-energy node requires both ${\xi_{\mathbf k}=0}$ and ${\Delta_{\mathbf k}=0}$ . Geometry, not the word ${d}$ -wave by itself, decides the observable gap.

When a two-dimensional nodal line does cross the Fermi surface, expand near one node using local momenta ${(q_\perp,q_\parallel)}$ :

$E(\mathbf q)\simeq \sqrt{{(\hbar v_{\mathrm F}q_\perp)}^2 +{(\hbar v_\Delta q_\parallel)}^2}.$

Rescaling to ${p_1=\hbar v_{\mathrm F}q_\perp}$ and ${p_2=\hbar v_\Delta q_\parallel}$ turns the region ${E(\mathbf q)<E}$ into a disk of area ${\pi E^2}$ . Differentiation gives, per node and per spin,

$N_{\mathrm{node}}(E) =\frac{|E|}{2\pi\hbar^2v_{\mathrm F}v_\Delta}.$

The low-energy DOS is therefore linear after multiplying by the declared node and spin degeneracies. The isotropic square-root edge disappeared because a point node supplies Dirac phase space rather than a one-dimensional energy turning point. Its integrated state count grows as ${E^2}$ , providing an independent check on the power law.

In Figure 20.5, identify where the normal crossing is avoided, where particle and hole weights are equal, and which plotted gap zeros actually intersect the declared Fermi surface. Predict which panel retains low-energy spectral weight.

Figure 20.5. The exact BdG avoided crossing and two gap functions on one circular Fermi contour separate Nambu redundancy, a hard gap, and the simultaneous-zero condition for physical nodes.

Panel (a) plots ${E/\Delta_0=\pm\sqrt{{(\xi/\Delta_0)}^2+1}}$ against the two dashed normal branches. Panels (b) and (c) then hold the circular Fermi contour fixed while changing only the gap function from ${\Delta_s(\theta)=\Delta_0}$ to ${\Delta_d(\theta)=\Delta_0\cos2\theta}$ . The black points are therefore exactly the four intersections required by the superconducting node criterion, not decorative symmetry markers.

The full-gap panel has a minimum positive energy ${|\Delta_0|}$ at the normal-state Fermi crossing. The nodal panel retains low-energy excitations only at simultaneous Fermi-surface and gap zeros. Coherence-factor color or weight changes character across the avoided crossing but still sums to one; it must not be read as an extra Nambu state.

Verify ${\{\hat\gamma,\hat\gamma^\dagger\}=1}$ for the displayed transformation. Explain ${1-2f(E)}$ physically, and determine whether ${\Delta_0[\cos(k_xa)-\cos(k_ya)]}$ is fully gapped on a Fermi surface crossing the diagonals.

A Bogoliubov quasiparticle is neither an electron nor a second independent Nambu copy. ${\Delta_{\mathbf k}}$ is an order parameter; the observed minimum gap also depends on where the Fermi surface intersects its nodes.

Near one physical two-dimensional node let $E(\mathbf q)=\sqrt{{(\hbar v_{\mathrm F}q_\perp)}^2+{(\hbar v_\Delta q_\parallel)}^2},$ with ${v_{\mathrm F},v_\Delta>0}$ . Count positive quasiparticle states per unit area per declared independent node/spin species below energy ${E}$ , and derive the low-temperature heat-capacity power. Do not add a degeneracy without naming it.

Set ${p_1=\hbar v_{\mathrm F}q_\perp}$ , ${p_2=\hbar v_\Delta q_\parallel}$ . This is the ellipsoid-to-sphere method of the ellipsoidal-pocket count, now in two dimensions and without its electron spin factor. The Jacobian is ${\mathrm d^2q=\mathrm d^2p/(\hbar^2v_{\mathrm F}v_\Delta)}$ , and the allowed disk has area ${\pi E^2}$ . Thus $\mathcal N(E)=\frac{E^2}{4\pi\hbar^2v_{\mathrm F}v_\Delta},\qquad \rho(E)=\frac{\mathrm d\mathcal N}{\mathrm dE} =\frac{E}{2\pi\hbar^2v_{\mathrm F}v_\Delta}\quad(E>0).$ If there are ${g_{\mathrm n}}$ independent positive-energy species under the chosen nonredundant convention, multiply once by ${g_{\mathrm n}}$ . Writing ${A=g_{\mathrm n}/(2\pi\hbar^2v_{\mathrm F}v_\Delta)}$ , the thermal energy and heat capacity per unit area are $U_{\mathrm{th}}=A\int_0^\infty\frac{E^2\,\mathrm dE}{\mathrm e^{E/(k_{\mathrm B}T)}+1} =\frac32A\zeta(3){(k_{\mathrm B}T)}^3, \quad C=\frac92A\zeta(3)k_{\mathrm B}^3T^2.$ The integral follows from the alternating exponential series: $\int_0^\infty\frac{x^2}{\mathrm e^x+1}\,\mathrm dx =2\sum_{n\ge1}\frac{{(-1)}^{n+1}}{n^3}=\frac32\zeta(3).$ Integrating ${\rho}$ returns the disk count, and its units are states per area per energy, two separate checks. The result vanishes at zero energy without a hard threshold, unlike the full gap. It assumes a linear node and temperatures below its cutoff; gap zeros away from the Fermi surface supply no such states. The consistency triangle now has to use the same gap geometry in spectra, amplitudes and thermodynamics.

The consistency triangle of spectrum, amplitudes, and gap equation

A useful way to detect a hidden convention error is to derive the same zero-temperature limit from three sides. Begin with the BdG matrix $h_{\mathrm{BdG}}(\mathbf k)= \begin{pmatrix} \xi_{\mathbf k}&-\Delta\\ -\Delta^*&-\xi_{\mathbf k} \end{pmatrix}.$ Its trace is zero and determinant is ${-\xi_{\mathbf k}^2-{|\Delta|}^2}$ , so its eigenvalues are ${\pm E_{\mathbf k}}$ . This check is independent of any chosen eigenvector phase. For the positive branch, normalized amplitudes may be chosen so that ${|u_{\mathbf k}|}^2=\frac12\left(1+\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right), \qquad {|v_{\mathbf k}|}^2=\frac12\left(1-\frac{\xi_{\mathbf k}}{E_{\mathbf k}}\right).$ Far above the Fermi surface, ${u\to1}$ and ${v\to0}$ ; far below it, the limits reverse. At ${\xi=0}$ , particle and hole weights are equal. Any formula that fails one of these limits has assigned a branch or Nambu component incorrectly.

The product of amplitudes carries the pairing phase. With the conventions used here, $u_{\mathbf k}^*v_{\mathbf k}=\frac{\Delta}{2E_{\mathbf k}}$ so that the displayed Bogoliubov transformation reproduces the anomalous average used in the gap equation. A simultaneous rephasing of the Nambu spinor and amplitudes may alter intermediate signs, but not this agreement among declared conventions. Thermal quasiparticle occupation then reduces the anomalous expectation by $1-2f(E_{\mathbf k}) =\tanh\!\left(\frac{E_{\mathbf k}}{2k_{\mathrm B}T}\right).$ This factor has a direct interpretation: creating either quasiparticle blocks the coherent empty–doubly-occupied superposition in that pair sector. At zero temperature it approaches one; at temperatures large compared with the mode energy it approaches zero.

Substitution into the definition of ${\Delta}$ closes the third side of the triangle. For a constant attractive interaction in a symmetric shell, $1=gN_0\int_0^{\hbar\omega_{\mathrm c}} \frac{\mathrm d\xi}{\sqrt{\xi^2+{|\Delta|}^2}} \tanh\!\left(\frac{\sqrt{\xi^2+{|\Delta|}^2}}{2k_{\mathrm B}T}\right).$ At ${T=0}$ , the integral is an inverse hyperbolic sine. In weak coupling it yields ${|\Delta(0)|\simeq2\hbar\omega_{\mathrm c}\exp[-1/(gN_0)]}$ . The result is nonanalytic at ${g=0}$ , echoing the Cooper logarithm, yet its exponent differs from the single-pair binding energy because the many-pair self-consistency equation has a different kernel.

These three calculations must agree: the spectrum fixes ${E}$ , the normalized eigenvector fixes coherence factors, and those factors reproduce the self-consistency equation. Agreement is stronger evidence than memorizing any one formula. It also localizes errors: a wrong determinant corrupts the spectrum, a failed limiting weight corrupts the eigenvector, and an incorrect thermal sign corrupts the gap equation.

From Nambu mixing to thermal self-consistency

The pairing field reorganizes each oriented two-mode pair block into Nambu coordinates: one annihilation component and one creation component. A canonical Bogoliubov transformation then fixes the excitation energy and particle–hole weights of both physical partner modes. Quasiparticle occupations block the anomalous pair amplitude, and substitution of that amplitude into the definition of ${\Delta_{\mathbf k}}$ closes the thermal gap equation; the free energy decides which self-consistent solution is stable. Finally, the simultaneous conditions ${\xi_{\mathbf k}=0}$ and ${\Delta_{\mathbf k}=0}$ distinguish physical nodes from formal zeros of a basis function. Exact particle-number conservation changes how order is diagnosed, but it does not erase the pair correlations or the excitation spectrum developed here.

The next question is geometric rather than superconducting: how can phases of Bloch eigenstates encode observable information even when there is no pairing order parameter?

Preserve canonical anticommutation under a Bogoliubov transformation, derive the positive BdG energy, explain thermal pair blocking, and locate symmetry-enforced nodes for a declared Fermi surface.

First normalize the coherence factors and recover the two-mode canonical map. Explain why its second Nambu component is a creation operator and why both oriented momentum blocks are retained. Then derive the thermal blocking factor and reconstruct the gap equation without double counting. A change of Nambu ordering or condensate gauge changes components but not the spectrum or a consistently transformed tunneling prediction. Coherence factors govern how the microscopic electron operator overlaps the modes; thermal blocking governs their occupation; and the gap equation governs the mean-field state. These three roles cannot be collapsed into the doubled eigenvalue plot.

Problems

1. Canonical BdG diagonalization

For one ${\mathbf k\in\mathcal K_{\mathrm p}}$ , use $h_{\mathrm{BdG}}= \begin{pmatrix} \xi&-\Delta\\ -\Delta^*&-\xi \end{pmatrix}, \qquad \hat\Psi={(\hat c_{\mathbf k\uparrow}, \hat c_{-\mathbf k\downarrow}^{\dagger})}^{\mathsf T}.$ Write ${\Delta=|\Delta|\mathrm e^{\mathrm i\chi}}$ , choose ${u\ge0}$ , and use ${v=\mathrm e^{\mathrm i\chi} \sqrt{(1-\xi/E)/2}}$ .

Derive the eigenvalues, normalized ${u,v}$ , and ${u^*v=\Delta/(2E)}$ without quoting Theorem 20.1.
Construct ${U=\bigl(\begin{smallmatrix}u&-v\\v^*&u^*\end{smallmatrix}\bigr)}$ , verify ${Uh_{\mathrm{BdG}}U^\dagger=\operatorname{diag}(E,-E)}$ , write the inverse operator transformation, and check one self- and both types of cross-anticommutator.
For ${\xi=3.0\,\mathrm{meV}}$ and ${\Delta=4.0\,\mathrm{meV}\,\mathrm e^{\mathrm i\pi/3}}$ , compute ${E,u,v}$ and ${u^*v}$ . Check the ${\xi\to\pm\infty}$ particle/hole limits.
For generic ${\{\mathbf k,-\mathbf k\}}$ , identify the four distinct spin orbitals and their two disjoint pair blocks. Explain how the two positive-energy partner modes of each block preserve the original Fock-space degree count, and why an adjoint Nambu coordinate is not an additional physical mode.

2. Change temperature or gap symmetry

Consider a two-dimensional circular Fermi surface $\mathbf k_{\mathrm F}(\theta) =k_{\mathrm F}(\cos\theta,\sin\theta), \quad 0\le\theta<2\pi, \quad \xi_{\mathbf k}\simeq\hbar v_{\mathrm F}(k-k_{\mathrm F}),$ with ${k_{\mathrm F}=8.0\,\mathrm{nm}^{-1}}$ and ${v_{\mathrm F}=1.5\times10^5\,\mathrm{m\,s^{-1}}}$ . Compare $\Delta_s(\theta)=\Delta_0, \qquad \Delta_d(\theta)=\Delta_0\cos2\theta, \qquad \Delta_0=2.0\,\mathrm{meV}.$

Apply the superconducting node criterion to list every Fermi-surface node for each gap. Explain why the sign change of ${\Delta_d}$ is not, by itself, the criterion.
Evaluate ${B(T,E)}$ at an antinode with ${E=\Delta_0}$ for ${T=0}$ by limit and for ${k_{\mathrm B}T=0.5\,\mathrm{meV}}$ . Also evaluate it at an ideal zero-energy node for the nonzero temperature and interpret the result.
Near one ${d}$ -wave node, show that ${\hbar v_\Delta=2\Delta_0/k_{\mathrm F}}$ , calculate ${v_\Delta}$ , and derive the low-energy DOS per node and spin. Then include four nodes and two spins and compare its power law with the ${s}$ -wave hard gap.
State which conclusions survive if the circular Fermi surface is replaced by a contour that misses all four lines ${\cos2\theta=0}$ .

3. Consistency triangle

Use the same numerical block as Problem 1(c), now at ${k_{\mathrm B}T=1.0\,\mathrm{meV}}$ , and take a one-level attraction ${V=-g_{\mathrm{lev}}}$ with ${g_{\mathrm{lev}}=2.0\,\mathrm{meV}}$ . A proposed audit reports ${(u^*v)}_{\mathrm{prop}}=0.4\,\mathrm e^{-\mathrm i\pi/3}$ and, for a continuum shell with the Lecture 19 single-spin DOS ${N_0}$ , $1\stackrel{\mathrm{prop}}{=} 2gN_0\int_{-\hbar\omega_{\mathrm c}}^{+\hbar\omega_{\mathrm c}} \frac{\mathrm d\xi}{2\sqrt{\xi^2+{|\Delta|}^2}} \tanh\!\left( \frac{\sqrt{\xi^2+{|\Delta|}^2}}{2k_{\mathrm B}T} \right).$

Recompute ${u,v,u^*v}$ and identify every phase or normalization error in the proposed product.
Calculate ${B(T,E)}$ , the anomalous amplitude ${\langle\hat b\rangle=u^*vB(T,E)}$ , and this level’s contribution ${-V\langle\hat b\rangle}$ to the gap.
Audit the proposed continuum equation. Starting from the sum over ${\mathcal K_{\mathrm p}}$ , write one correct positive- ${\xi}$ integral for a symmetric shell and show the exact overcounting factor in the proposal.
Verify that the corrected spectrum, coherence product, thermal amplitude, and gap contribution transform with one common phase and use one cutoff and DOS convention.

Larger-picture return. These problems separate three jobs that are often collapsed into one formula: the canonical transformation fixes what a probe can create, the thermal occupation fixes how much pair amplitude remains, and self-consistency fixes whether the assumed gap reproduces itself. Later descriptions of collective phase dynamics or unconventional pairing must preserve that distinction.

(advanced): Retardation and the Boundary of the Reduced BCS Model

In a phonon-mediated setting, the attraction is retarded and competes with a screened Coulomb repulsion. The reduced constant-coupling shell represents this frequency dependence by an effective attraction and cutoff; it does not derive either one.

For a minimal frequency ledger, let ${\omega_n=(2n+1)\pi k_{\mathrm B}T/\hbar}$ , ${n\in\mathbb Z}$ , be fermionic Matsubara frequencies. The gap is now an ordinary complex sequence ${\Delta:\mathbb Z\to\mathbb C}$ , ${n\mapsto\Delta_n}$ , rather than only a momentum function. In a linearized, isotropic, ${Z_n=1}$ approximation, a retarded kernel has the schematic but dimensionally closed form $\Delta_n=\pi k_{\mathrm B}T \sum_{m:\,|\hbar\omega_m|<E_{\mathrm F}} \left[ \lambda_{\mathrm{ph}} \frac{\omega_{\mathrm D}^2} {{(\omega_n-\omega_m)}^2+\omega_{\mathrm D}^2} -\mu \right] \frac{\Delta_m}{|\hbar\omega_m|}.$ The phonon attraction is concentrated at frequency transfers below ${\omega_{\mathrm D}}$ , whereas the screened Coulomb term acts up to an electronic scale ${E_{\mathrm F}/\hbar}$ . This different frequency support is the essential information lost when both terms are replaced immediately by one constant.

Eliminating electronic frequencies between ${\omega_{\mathrm D}}$ and ${E_{\mathrm F}/\hbar}$ gives the Morel–Anderson estimate $\mu^*= \frac{\mu} {1+\mu\ln(E_{\mathrm F}/\hbar\omega_{\mathrm D})}.$ For $\frac{E_{\mathrm F}}{\hbar\omega_{\mathrm D}}=100, \qquad \mu=0.30, \qquad \lambda_{\mathrm{ph}}=0.25,$ one obtains $\mu^*=\frac{0.30}{1+0.30\ln100}=0.1260, \qquad \lambda_{\mathrm{eff}} =\lambda_{\mathrm{ph}}-\mu^*=0.1240.$ Without retardation, the naive difference ${\lambda_{\mathrm{ph}}-\mu=-0.05}$ would predict no net attraction. Scale separation instead reduces the low-energy repulsion enough to leave a positive pairing channel. If one inserts ${\lambda_{\mathrm{eff}}}$ into the crude weak-coupling estimate, $\frac{k_{\mathrm B}T_{\mathrm c}}{\hbar\omega_{\mathrm D}} \sim1.134\,\mathrm e^{-1/\lambda_{\mathrm{eff}}} =3.57\times10^{-4},$ about ${1.7\%}$ of the corresponding ${\mu^*=0}$ scale. The numerical value is illustrative; the robust conclusion is the change in the low-energy competition after the two frequency ranges are treated separately.

Model	Gap argument	Interaction information retained	New output
Reduced BCS	${\Delta_{\mathbf k}}$ in a sharp energy shell	one constant ${g}$ and cutoff ${\omega_{\mathrm c}}$	canonical BdG spectrum and static gap equation
Retarded kernel above	${\Delta_n}$ , optionally also momentum dependent	frequency transfer, phonon scale, and electronic Coulomb range	${\mu^*}$ and frequency-dependent pairing
Full Eliashberg theory	${\Delta_n}$ and renormalization ${Z_n}$	material spectral function and self-energy feedback	strong-coupling spectra and corrected ratios

The pseudopotential formula assumes a wide separation of electronic and phonon scales and does not replace a material calculation. Full Eliashberg theory also solves for ${Z_n}$ , retains the detailed electron–phonon spectrum, and continues the frequency-dependent solution to real frequencies. Multiband BdG is a separate Lecture 20 extension candidate and remains deferred; it is not silently approximated by this frequency-only example.

This appendix is deletion-safe: removing it leaves the canonical diagonalization, thermal pair blocking, static self-consistency, problems, and Lecture 21 bridge intact. It cannot be used to repair missing normalization or sign consistency in the core BdG calculation.

References

[Gia60] Giaever, I. (1960). “Energy gap in superconductors measured by electron tunneling.” Physical Review Letters 5 (4), 147–148. https://doi.org/10.1103/PhysRevLett.5.147.

Lecture 20: Bogoliubov Excitations and Thermal Self-Consistency

20.1 Nambu representation and its redundancy

20.2 Canonical Bogoliubov diagonalization

20.3 Checking the Bogoliubov transformation before thermal averaging

20.4 Just-in-time statistical mechanics: thermal blocking

20.5 Anomalous average from quasiparticle occupations

20.6 Gap integral and its thermal limits

20.7 Fully gapped and nodal observables

20.7.1 From the ideal density of states to tunneling and heat capacity

20.8 The consistency triangle of spectrum, amplitudes, and gap equation

20.9 From Nambu mixing to thermal self-consistency

20.10 Problems