Nambu–Gor'kov Green Functions and Anomalous Propagators

Nambu–Gor’kov notation packages particle and hole propagation into one matrix Dyson equation. Its off-diagonal entries encode pair conversion and make coherence factors and gap equations economical, but they are basis- and gauge-dependent amplitudes—not directly measured condensates. Physical conclusions must be reconstructed from gauge-invariant spectra or response functions, with the Nambu doubling removed exactly once.

Required background. BCS mean-field theory fixes the uniform saddle. Dyson equations and Ward-consistent vertices supply the matrix self-energy and response constraints.

Matrix propagator in one explicit basis

Use $\Psi_{\mathbf k}=(c_{\mathbf k\uparrow},c^\dagger_{-\mathbf k\downarrow})^T$ and define

$$\mathcal G(\mathbf k,\tau) =-\langle T_\tau\Psi_{\mathbf k}(\tau)\Psi_{\mathbf k}^\dagger(0)\rangle =\begin{pmatrix}G&F\\F^\dagger&-\bar G\end{pmatrix}.$$

Here the lower-right minus sign is part of the definition; other common bases move it into $\bar G$ or the anomalous entries. With

$$\mathcal G^{-1}=i\omega_n\tau_0-\xi_{\mathbf k}\tau_3-\Sigma, \qquad \Sigma=\Sigma_0\tau_0+\Sigma_3\tau_3+ \Phi_1\tau_1-\Phi_2\tau_2,$$

the uniform real-gap mean-field limit is

$$\mathcal G(\mathbf k,i\omega_n) =\frac{i\omega_n\tau_0+\xi_{\mathbf k}\tau_3+\Delta\tau_1} {(i\omega_n)^2-E_{\mathbf k}^2}.$$

Thus

$$G=\frac{u_{\mathbf k}^2}{i\omega_n-E_{\mathbf k}} +\frac{v_{\mathbf k}^2}{i\omega_n+E_{\mathbf k}}, \qquad F=\frac{\Delta}{(i\omega_n)^2-E_{\mathbf k}^2}.$$

The large-frequency check is decisive: $G\sim1/(i\omega_n)$ and $F\sim1/(i\omega_n)^2$. A wrong basis transpose or missing sign often fails this check before it corrupts a spectral calculation. The matrix formulation originates in Gor’kov 1958, pp. 505–508 and the particle–hole representation in Nambu 1960, §§II–III.

Schrieffer 1999, chs. 7–8 provides a detailed translation among the Bogoliubov transformation, anomalous propagators, and the gap equation.

Gauge covariance and redundancy

Under $c(x)\mapsto e^{i\alpha(x)}c(x)$,

$$\Psi(x)\mapsto e^{i\alpha(x)\tau_3}\Psi(x), \qquad \mathcal G(x,y)\mapsto e^{i\alpha(x)\tau_3}\mathcal G(x,y)e^{-i\alpha(y)\tau_3},$$

and $\Delta(x)\mapsto e^{2i\alpha(x)}\Delta(x)$. Consequently $F$ carries charge two and is not gauge invariant. A gauge choice may make a uniform $\Delta$ real, but it cannot turn $F$ into an observable. Gauge-invariant combinations include the excitation poles, local density of states, covariant phase gradient $\boldsymbol\nabla\theta-2e\mathbf A$, and response kernels satisfying the relevant Ward identity.

BdG particle–hole symmetry is also a redundancy of the enlarged basis. If $H_{\mathrm{BdG}}(\mathbf k)\lvert u\rangle=E\lvert u\rangle$, an antiunitary $\mathcal C$ produces a partner at $-E$ and $-\mathbf k$. Thermodynamic traces therefore carry a factor $1/2$ when summed over the full Nambu space. The zero-energy exception needs its own normalization and is treated on the topological boundary-mode page.

Interactions and the gap equation

In an interacting paired state the anomalous self-energy $\Phi(\mathbf k,i\omega_n)$ can depend on momentum and frequency. The exact Dyson equation remains algebraic in Nambu space,

$$\mathcal G^{-1}=\mathcal G_0^{-1}-\Sigma,$$

but a practical closure requires an irreducible pairing vertex. Schematically,

$$\Phi(k)=-T\sum_{k'}\Gamma_{pp}(k,k')F(k').$$

This equation is not controlled merely because it is self-consistent. The approximation for $\Gamma_{pp}$ must share symmetries and, for electromagnetic response, vertex corrections must be consistent with the self-energy. Retarded electron–phonon closure belongs to Eliashberg theory; collective fluctuations belong to the next page.

The diagram marks Nambu notation as a dictionary connecting the saddle to BdG and response, not as an extra physical degree of freedom.

Nambu doubling is a calculational representation. The anomalous line carries gauge charge, and only properly de-doubled spectra and Ward-consistent observables license physical claims. Original schematic, not to scale.

The paired-matter claim test matrix gives the corresponding observable ceilings.

A convention translation test

Suppose another source uses $\widetilde\Psi=(c_{\uparrow},c^\dagger_{\downarrow})^T$ but defines the off-diagonal Hamiltonian as $-\widetilde\Delta\tau_1$. The unitary rephasing $U=\tau_3$ maps the bases: $\widetilde H=UH U^\dagger$ and $\widetilde\Delta=-\Delta$. Poles, density, and response are unchanged. Comparing the printed sign of $F$ without this translation would falsely diagnose disagreement.

Exercise

Check the spectral sum rule. Analytically continue the mean-field $G$ above and show that its spectral function integrates to one.

Solution

$A(\mathbf k,\omega)=-2\operatorname{Im}G^R=2\pi\bigl[u_{\mathbf k}^2\delta(\omega-E_{\mathbf k})+v_{\mathbf k}^2\delta(\omega+E_{\mathbf k})\bigr]$. Therefore $\int\mathrm d\omega\,A/(2\pi)=u^2+v^2=1$. Summing both diagonal Nambu entries would give two because it counts the particle–hole representation twice; that is not a violation of the one-fermion sum rule.

References

• Gor’kov, L. P. (1958). “On the energy spectrum of superconductors.” Soviet Physics JETP 7, 505–508. JETP archive.
• Nambu, Y. (1960). “Quasi-particles and gauge invariance in the theory of superconductivity.” Physical Review 117, 648–663. doi:10.1103/PhysRev.117.648.
• Schrieffer, J. R. (1999). Theory of Superconductivity, revised edition. Westview Press. Publisher record.