Fermi Liquid Fixed Point Deformations due to Codimension Two Defects

We show that codimension-two defects in Fermi liquids deform the renormalization group flow via a marginally relevant coupling. The mechanism for generating the flow is distinct from the case of the Kondo problem (codimension-three defects) in that the effective particle-hole asymmetry that leads to the log running is due to the spatial anisotropy generated by the defect. The mechanism for the log generation has a simple geometric explanation which shows that hole fluctuations are suppressed as the incoming momentum is taken to be along the direction of the defect. The RG flow time is shown to scale with the length of the defect. We also show that the dislon, the Goldstone mode localized to the defect, couples in a non-derivative fashion to the bulk fermions and becomes relevant above the dislons’ Debye frequency which depends upon the defect tension.

💡 Research Summary

The paper investigates how codimension‑two defects—specifically straight dislocations (or “dislons”)—modify the renormalization‑group (RG) flow of a conventional three‑dimensional Fermi liquid. The authors contrast this situation with the classic Kondo problem, which involves a codimension‑three point impurity and requires a dynamical impurity spin to generate a marginally relevant coupling. In the dislocation case, no internal dynamics are needed; instead, the spatial anisotropy introduced by the line defect creates an effective particle‑hole asymmetry that prevents the usual cancellation between particle and hole contributions in the one‑loop correction.

The key geometric insight is that when an incoming electron’s momentum is aligned with the defect axis (taken as the z‑direction), momentum conservation forbids hole excitations in the intermediate state. Consequently, the one‑loop diagram yields a logarithmic term, leading to a marginally relevant coupling g(θ₁,θ₂) that runs according to dg/dℓ ∝ g². The RG “time” ℓ scales with the physical length of the defect, so an infinitely long dislocation reproduces the familiar logarithmic divergence, while a finite‑length defect cuts off the flow at ℓ∼L/ξ (ξ being a microscopic length such as the electron mean free path).

To treat the problem systematically, the authors construct an effective field theory (EFT) that combines the standard Fermi‑surface patch EFT with a world‑sheet description of the dislocation. The dislocation breaks translational invariance in the two transverse directions and also breaks boost invariance along the line, leaving three Goldstone modes (two transverse “dislons” and one longitudinal mode φ). The world‑sheet action is written as S_dis = ∫ dτ dσ √−g F(B, ĤB), where B and ĤB are invariants built from derivatives of φ. After fixing static gauge (X⁰ = t, X³ = z) the longitudinal mode becomes a scalar field on the line.

A crucial part of the analysis is power counting. The small expansion parameter λ = E/E_F controls the scaling of momenta perpendicular (k_⊥ ∼ λ) and parallel (k_∥ ∼ 1) to the Fermi surface. The fermion field scales as ψ ∼ λ^{-1/2}. Bulk four‑fermion interactions are marginal only in the BCS or forward‑scattering kinematics because the momentum‑conserving delta function effectively contributes a factor λ^{-1}. When coupling the dislocation to the fermions, the authors consider operators of the form S_int = ∫ dt dz X^a(t,z) ∂_a