Gauge-Coupled Fermi Surfaces

A Fermi surface of gauge-charged partons Landau-damps a transverse emergent gauge field and produces a singular non-Fermi-liquid patch theory. The familiar $|\omega|^{2/3}$ self-energy is a one-loop result in two dimensions, not a universal exact exponent. Compactness, global constraints, patch-to-observable reconstruction, pairing, disorder, and the chosen control expansion set the claim’s range.

Required background. Compact U(1) gauge fields supplies monopoles and confinement; critical Fermi-surface patch theories supplies patch coordinates and scaling.

Helpful background. Large-N saddles supplies controlled-expansion cautions.

Landau damping in one patch pair

Choose $k_x$ normal and $k_y$ tangent to antipodal patches. A minimal Euclidean action is

$$\begin{aligned} S={}&\sum_{s=\pm}\int_k \psi_s^\dagger(k) \left(i\omega-sv_Fk_x-\frac{k_y^2}{2m_c}\right)\psi_s(k)\\ &+\sum_{s=\pm}g_s\int_{k,q}a_T(q) \psi_s^\dagger(k+q)\psi_s(k)+S_a, \qquad g_+=-g_-. \end{aligned}$$

The opposite signs express coupling to the antipodal patch currents; an overall reversal of the $a_T$ convention reverses both $g_s$. The particle–hole continuum generates the transverse propagator

$$D_T^{-1}(\omega,\mathbf q) \simeq \chi q_y^2+\gamma\frac{|\omega|}{|q_y|},$$

for $|q_x|\ll|q_y|$ in the low-energy patch regime. Balancing terms gives $\omega\sim q_y^3$, $q_x\sim q_y^2$. With the Dyson convention $G^{-1}=G_0^{-1}-\Sigma$, the one-loop fermion self-energy at the surface is

$$\Sigma(i\omega)\sim -i\lambda\,\operatorname{sgn}(\omega)|\omega|^{2/3}, \qquad \lambda>0,$$

so $-\Sigma$ dominates the bare $i\omega$ in the inverse propagator and destroys a Landau quasiparticle pole Lee 2009. The retarded continuation obeys $-\operatorname{Im}\Sigma^R(\omega)>0$ for nonzero $\omega$ in the scaling regime; its real part is fixed by the Kramers–Kronig relation. This causality check must be preserved when conventions or analytic-continuation phases are changed.

Control and global observables

Naive large $N$ does not uniformly suppress all planar diagrams; controlled approaches also deform the boson dispersion or spatial dimension Mross et al. 2010. Exponents at physical $N$ can receive strong corrections. Curvature, multiple patches, $2k_F$ channels, and Ward identities are needed for global susceptibilities.

Parton conductivity is not directly measurable. In a slave-particle metal with electron $c=b^\dagger f$, integrating the internal field gives the Ioffe–Larkin composition in matrix form,

$$\boldsymbol\rho_{\rm phys}=\boldsymbol\rho_b+\boldsymbol\rho_f,$$

under the stated two-fluid assumptions. Thermodynamics, thermal transport, and spin response have different composition rules. A neutral spinon Fermi surface can contribute heat and spin continua while carrying no direct electric charge.

Instabilities and current status

Gauge exchange can enhance or suppress pairing depending on the channel; eventual Z2 pairing may pre-empt the asymptotic non-Fermi-liquid regime. Compact monopoles, disorder, density waves, and the confinement scale must also be compared with the Landau-damping window. A 2024 constrained U(1) treatment of the $t$–$J$ model illustrates an active microscopic application Liang, Yu, and Luo 2024, but it does not turn the patch exponent into a model-independent material prediction.

This frontier assessment was checked through 10 August 2026. The one-loop patch derivation is stable within its hypotheses; physical exponents, pairing scales, and platform identifications remain active questions. See Quantum Matter and Emergence Research for dated updates.

Exercise

Balance the two terms in $D_T^{-1}=\chi q_y^2+\gamma|\omega|/|q_y|$ and find the boson dynamical exponent.

Solution

Setting $q_y^2\sim|\omega|/|q_y|$ gives $|\omega|\sim|q_y|^3$, hence $z=3$ relative to the tangential momentum. Patch curvature then scales $q_x\sim q_y^2$.

References

• Sung-Sik Lee, “Low-Energy Effective Theory of Fermi Surface Coupled with U(1) Gauge Field in 2+1 Dimensions,” Physical Review B 80 (2009) 165102, doi:10.1103/PhysRevB.80.165102.
• Long Liang, Yue Yu, and Xi Luo, “Non-Fermi-Liquid Behavior of the t–J Model in the Strange-Metal Phase: U(1) Gauge Theory Consistent with Local Constraints,” Physical Review B 110 (2024) 075125, doi:10.1103/PhysRevB.110.075125.
• David F. Mross, John McGreevy, Hong Liu, and T. Senthil, “Controlled Expansion for Certain Non-Fermi-Liquid Metals,” Physical Review B 82 (2010) 045121, doi:10.1103/PhysRevB.82.045121.