Critical Fermi Surfaces and Patch Theories

Patch theory resolves a small neighborhood of an extended Fermi surface and couples it to a low-momentum critical boson. For a transverse gauge field or Ising-nematic order, two antipodal patches form the minimal closed kinematic unit. Landau damping and curvature produce anisotropic scaling and a $|\omega|^{2/3}$ fermion self-energy in the standard two-dimensional fixed-point treatment.

Required background. Metallic Non-Fermi Liquids and Quasiparticle Breakdown supplies the analytic quasiparticle criterion; Patch Renormalization and Competing Instabilities supplies patch coordinates and interpatch processes. Helpful background. Gauge-Coupled Fermi Surfaces supplies gauge-invariant observables and compactness issues.

Two antipodal patches

Choose patches centered at $\pm k_F\widehat{\mathbf x}$. Let $k_x$ be normal to the Fermi surface and $k_y$ tangent. After absorbing constants,

$$\varepsilon_s(\mathbf k)=s v_Fk_x+\frac{k_y^2}{2m}, \qquad s=\pm .$$

Curvature in $k_y$ must be kept: without it, a patch would contain an unphysical continuum of degenerate tangential modes. A schematic action is

$$\begin{aligned} S={}&\sum_{s=\pm}\int_k \psi_s^\dagger(k) \left(-i\omega+s v_Fk_x+\frac{k_y^2}{2m}\right)\psi_s(k)\\ &+\frac12\int_q D_0^{-1}(q)|\phi(q)|^2 +\sum_s g_s\int_{k,q}\phi(q)\psi_s^\dagger(k+q)\psi_s(k). \end{aligned}$$

For an inversion-even nematic form factor, $g_+=g_-$ in this convention. A transverse gauge field couples to current, so antipodal velocities give $g_+=-g_-$. This sign affects interpatch and pairing channels even though a single-patch self-energy may look identical.

Landau damping and anisotropic scaling

The patch particle–hole continuum generates

$$D^{-1}(\mathbf q,i\Omega_n) \simeq \kappa q_y^2+\gamma\frac{|\Omega_n|}{|q_y|},$$

with $q_x$ subleading under the low-energy scaling

$$q_y\mapsto bq_y,\qquad q_x\mapsto b^2q_x,\qquad \Omega\mapsto b^3\Omega.$$

Both boson terms then scale as $b^2$, while $v_Fk_x$ and $k_y^2/(2m)$ scale together. The one-loop on-shell retarded fermion self-energy obeys

$$-\operatorname{Im}\Sigma^R(\omega) =C|\omega|^{2/3}, \qquad C>0,$$

while the real part is fixed by Kramers–Kronig relations and any microscopic particle–hole asymmetry. The nonpositive retarded imaginary part is required for a nonnegative fermion spectral function. Its magnitude dominates the bare $|\omega|$ and destroys the patch quasiparticle. Lee’s gauge-field formulation makes the anisotropic power counting and singular large-$N$ structure explicit Lee 2009, §§ II–V.

Patch count, global geometry, and observables

A two-patch theory is not the entire Fermi surface. For small momentum transfer, many patch pairs exist; their thermodynamic contributions must be integrated without double counting. The patch width is an intermediate cutoff constrained by $|k_y|\ll k_F$ and by the curvature expansion. Large momentum transfer, BCS pairing between distant patches, and lattice umklapp require additional patch couplings.

The local fermion field may be gauge charged. Its Green function is then gauge dependent, while heat capacity, conductivity, density response, and gauge-invariant composites are physical. For a nematic order parameter, the electron spectral function is directly observable but lattice form factors vary around the Fermi surface.

Patch scaling can violate hyperscaling because the number of low-energy patches grows as the cutoff is lowered. Thermodynamic exponents require this global count; reading them from one patch’s coordinate measure alone is generally wrong.

Control and competing instabilities

The physical $N=2$, $d=2$ problem is strongly coupled. Naive large $N$ is nonuniform because higher-loop graphs can gain powers from singular momentum regions. Controlled deformations include changing the relation between boson dynamics and fermion curvature, co-dimensional or $\epsilon$ expansions, and selected large-flavor limits. Mross and collaborators constructed one such controlled expansion while retaining the non-Fermi-liquid fixed point Mross et al. 2010, §§ II–IV.

The same boson can enhance superconducting pairing. For nematic fluctuations the same-sign antipodal coupling is attractive in broad pairing channels; gauge-current signs produce a different competition. The singular competition between pairing and non-Fermi-liquid behavior in the Ising-nematic problem is analyzed by Metlitski and Sachdev 2010, §§ V–VI. If a pairing scale is higher than the asymptotic non-Fermi-liquid scale, the normal fixed point is a hidden organizing regime rather than an observable zero-temperature phase.

Current evidence boundary

Primary analytic and model literature was checked through 10 August 2026. The $z=3$ patch scaling and $2/3$ one-loop self-energy are robust within the stated two-patch theory, but control, compact gauge dynamics, global transport, and pairing remain model dependent. Ongoing calculations and numerical benchmarks belong in Quantum Matter and Emergence Research.

Exercises

1. Verify that the two terms in $D^{-1}$ have the same patch scaling.

Solution

$q_y^2\mapsto b^2q_y^2$. Also $|\Omega|/|q_y|\mapsto b^3/b=b^2$. The balance therefore gives the bosonic $z=3$ patch scaling.

2. Why must $k_y^2/(2m)$ be kept while the bare $i\omega$ becomes subleading?

Solution

$k_x$ and $k_y^2$ both scale as $b^2$ and jointly resolve distance from the curved Fermi surface. Bare $\omega$ scales as $b^3$, while the generated self-energy scales as $\omega^{2/3}\sim b^2$, so the latter joins the dispersion at leading order.

References

• Lee, S.-S. “Low-Energy Effective Theory of Fermi Surface Coupled with U(1) Gauge Field in $2+1$ Dimensions.” Physical Review B 80 (2009): 165102. DOI.
• Metlitski, M. A., and S. Sachdev. “Quantum Phase Transitions of Metals in Two Spatial Dimensions. I. Ising-Nematic Order.” Physical Review B 82 (2010): 075127. DOI.
• Mross, D. F., J. McGreevy, H. Liu, and T. Senthil. “Controlled Expansion for Certain Non-Fermi-Liquid Metals.” Physical Review B 82 (2010): 045121. DOI.