Fermi-Surface Patch Theory and Low-Energy Scaling

Fermi-surface patch theory organizes low-energy fermions by a point on a smooth surface and a small momentum normal to it. Under the basic Fermi-surface scaling, frequency and normal momentum shrink while the tangential label remains fixed. Generic four-fermion scattering is kinematically suppressed; forward and Cooper configurations remain marginal and must be treated separately.

Required background. Use Fermi-surface kinematics, momentum-shell RG, and beta-function conventions. Helpful background. The Cooper instability develops the pairing branch.

Patch coordinates and scaling

For a patch centered at $\mathbf k_F(\hat n)$,

$$S_0=\int\frac{\mathrm d\omega\,\mathrm dk_\perp\,\mathrm d^{d-1}k_\parallel}{(2\pi)^{d+1}} \psi^\dagger(i\omega-v_Fk_\perp)\psi.$$

Under $\omega\mapsto s\omega$ and $k_\perp\mapsto sk_\perp$ with the patch label $k_\parallel$ unscaled, the measure contributes $s^2$ and the kernel $s$. Thus the momentum-space field scales as $\psi\mapsto s^{-3/2}\psi$, independently of $d$ because the tangential directions label the surface rather than low-energy distance from it.

Curvature cannot be erased globally. For a finite patch, $\xi\simeq v_Fk_\perp+k_\parallel^2/(2m_c)$ requires the tangential width to shrink as $|k_\parallel|\lesssim\sqrt{k_F\Lambda}$ when the normal shell thickness is $\Lambda$. This relation controls patch number and overlap.

Exceptional interaction channels

Momentum conservation makes a generic scattering configuration leave the shrinking shell, so its low-energy phase space is irrelevant. Two families remain:

• forward scattering, where each outgoing momentum stays close to its incoming patch, yielding the Landau interaction function;
• Cooper scattering, where nearly opposite incoming momenta have nearly opposite outgoing momenta.

For each Cooper angular harmonic $g_\ell$, the one-loop flow has the schematic form

$$\frac{\mathrm dg_\ell}{\mathrm d\ell}=-N(0)g_\ell^2.$$

Repulsion flows toward zero, while attraction diverges at a finite logarithmic scale. The divergence marks loss of the normal-state expansion; it does not by itself compute the ordered state. Shankar gives the full kinematic derivation in Shankar 1994, §§ V–VIII, while Polchinski formulates the same low-energy separation as EFT Polchinski 1993, pp. 235–276.

Failure boundaries

Nesting makes finite-wavevector channels unusually singular. A van Hove point has $v_F=0$ and invalidates the smooth-patch linearization. Long-range gauge or Coulomb forces change power counting, and strong damping can destroy the fermion pole before a patch RG reaches its nominal scale. Patch number, curvature, and tangential cutoff must be varied together to prevent double counting.

Exercises

Solve the one-loop Cooper flow for initial $g_0<0$.

Solution

$1/g(\ell)=1/g_0+N(0)\ell$. The coupling diverges at $\ell_c=-1/[N(0)g_0]>0$, corresponding to an instability scale $\Lambda_c=\Lambda_0e^{-\ell_c}$. Its prefactor and ordered continuation require physics beyond this normal-state flow.

References

• Joseph Polchinski, “Effective Field Theory and the Fermi Surface,” in Recent Directions in Particle Theory, World Scientific (1993) 235–276, doi:10.1142/9789814503577_0001, Open PDF.
• Ramamurti Shankar, “Renormalization-Group Approach to Interacting Fermions,” Reviews of Modern Physics 66 (1994) 129–192, §§ V–VIII, doi:10.1103/RevModPhys.66.129.