Functional RG for Competing Fermi-Surface Instabilities

Functional RG (fRG) compares competing Fermi-surface channels by evolving a regulated effective action and resolving the four-point vertex over momentum, frequency, and internal indices. Its strongest controlled conclusion is a truncation-stable leading susceptibility and stopping scale in a declared weak-coupling regime, not a proof that the corresponding ordered phase exists.

Required background. Use patch scaling and the general functional-RG construction. Helpful background. Functional-method validation supplies regulator and truncation tests.

Regulated vertex flow

Add a quadratic regulator $R_\Lambda$ and define the effective average action $\Gamma_\Lambda$. Its exact flow is

$$\partial_\Lambda\Gamma_\Lambda =\frac12\operatorname{STr}\left[ (\Gamma_\Lambda^{(2)}+R_\Lambda)^{-1}\partial_\Lambda R_\Lambda \right].$$

For fermions the supertrace supplies the loop sign. A common many-patch truncation retains the self-energy $\Sigma_\Lambda$ and four-point vertex $V_\Lambda(k_1,k_2,k_3)$, while neglecting or approximating higher vertices. One then decomposes the vertex into particle–particle, crossed particle–hole, and direct particle–hole channels, projected onto patch or form-factor bases.

The one-loop terms couple the channels: antiferromagnetic fluctuations can enhance a sign-changing Cooper eigenmode, while forward scattering can enhance a nematic response. Because the decomposition is not unique at finite truncation, the full reconstructed vertex and observables, not individual channel labels, are the comparison objects. The systematic formulation and standard approximations are reviewed in Metzner et al. 2012, §§ II–IV.

Stopping and validation

If an eigenvalue reaches the scale at which neglected vertices or symmetry breaking become order one, stop the symmetric-phase flow at $\Lambda_*$. Report:

1. regulator family and initial scale;
2. patch or form-factor and frequency resolution;
3. self-energy and Katanin or higher feedback choices;
4. Ward-identity and crossing residuals;
5. ordering of the leading eigenmodes under refinement; and
6. comparison with a controlled weak-coupling limit or another method.

A robust leading eigenfunction identifies a tendency and its symmetry within the approximation. The ordered gap, transition temperature, and coexistence require a broken-symmetry continuation or independent treatment. A single diverging component can be a basis artifact.

Worked weak-coupling interpretation

For a repulsive square-lattice Hubbard model near, but not exactly at, nesting, particle–hole scattering near $(\pi,\pi)$ may grow first and feed a $d_{x^2-y^2}$ Cooper eigenfunction. The safe statement is conditional: within the specified weak-coupling truncation, these are the leading correlated tendencies above $\Lambda_*$. Changing the next-neighbor hopping, self-energy feedback, or patch resolution can reorder them; this sensitivity is part of the result, not noise to suppress.

Exercises

Why must a stopping-scale comparison use the same regulator and normalization across channels?

Solution

At finite truncation, both the numerical magnitude of a vertex component and the scale at which it reaches a threshold depend on regulator and basis normalization. Only consistently normalized eigenvalues and their refinement behavior can be compared; mixing conventions can manufacture a leading channel.

References

• Walter Metzner, Manfred Salmhofer, Carsten Honerkamp, Volker Meden, and Kurt Schönhammer, “Functional Renormalization Group Approach to Correlated Fermion Systems,” Reviews of Modern Physics 84 (2012) 299–352, §§ II–IV, doi:10.1103/RevModPhys.84.299, Open PDF.