Interaction Corrections and Finkel'stein Scaling

Electron interactions acquire singular low-energy corrections in a diffusive metal because particles revisit the same region for a long time. Exchange and screened Hartree processes suppress or enhance the tunnelling density of states and conductivity according to their singlet/triplet channel amplitudes, while the Finkel’stein sigma model evolves disorder and interaction parameters together. Adding an “interaction correction” to weak localization without a common convention double counts soft modes.

Required background. Weak localization supplies the noninteracting interference correction. The disorder sigma model supplies the diffusive $Q$ field and conductance normalization.

Helpful background. Landau Fermi-liquid theory supplies the spin-symmetric and spin-antisymmetric interaction channels.

Diffusive interaction singularity

In the window

$$T,\lvert\varepsilon\rvert,\lvert\omega\rvert\ll\tau^{-1}, \qquad q\ell\ll1,$$

the particle–hole propagator is diffusive rather than ballistic. A representative exchange contribution to the density of states has the structure

$$\delta\nu(\varepsilon) \propto-\operatorname{Im}\int\frac{d^d q}{(2\pi)^d} \int\frac{d\omega}{2\pi} \frac{V^R(\omega,\mathbf q)} {[Dq^2-i(\omega+\varepsilon)]^2},$$

with the dynamically screened interaction $V^R$. In two dimensions, diffusive momentum integration produces logarithms; an unscreened Coulomb tail can supply a second logarithm. The tunnelling zero-bias anomaly is therefore not a hard gap. Its coefficient and even some transport signs depend on screening, spin degeneracy, and triplet amplitude. Altshuler and Aronov 1979 established the singular density-of-states correction.

The following original diagram makes the shared origin visible. Inspect the branch after the $Q$ field: weak localization and interaction corrections are different insertions into the same diffusive theory.

Interaction and interference corrections share diffusive modes. A consistent calculation fixes density of states, spin convention, screening, conductance units, and infrared cutoff once; the branches cannot be added from incompatible normalizations. The diagram is schematic and not to scale.

From a correction to a coupled field theory

The noninteracting action is augmented by frequency and interaction operators. Schematically,

$$S[Q]=S_\sigma[Q]+S_z[Q]+S_{\Gamma_s}[Q]+S_{\Gamma_t}[Q]+\cdots,$$

where $z$ renormalizes frequency, $\Gamma_s$ is the density singlet amplitude, and $\Gamma_t$ is the spin-triplet amplitude in an $SU(2)$-symmetric problem. Ward identities tie $z$ and the conserved density vertex; long-range Coulomb interaction imposes an additional gauge constraint. The running variables may be expressed as resistance $t\propto1/g$ and dimensionless amplitudes $\gamma_{s,t}=\Gamma_{s,t}/z$.

At weak resistance the renormalization group has the form

$$\frac{dt}{d\ln L}=t^2\,\mathcal B(\gamma_s,\gamma_t)+O(t^3), \qquad \frac{d\gamma_i}{d\ln L}=t\,\mathcal F_i(\gamma_s,\gamma_t)+O(t^2).$$

The functions depend on symmetry, valley number, and interaction range; quoting one beta function without those choices is meaningless. A growing triplet amplitude may signal magnetic strong coupling, while an attractive Cooper channel can signal superconductivity. Neither endpoint is controlled merely because the initial flow was perturbative. Finkel’stein 1983 gives the coupled sigma-model construction, and Belitz and Kirkpatrick 1994, §§III–V reviews its symmetry and control structure.

Observable checks and failure boundaries

A tunnelling density of states, thermodynamic compressibility, and conductivity are distinct observables. The Einstein relation involves the renormalized compressibility and diffusion constant; a tunnelling anomaly cannot be inserted as a transport lifetime. Required checks include:

• the same total or per-spin $\nu$ and $g$ convention in every channel;
• screening appropriate to the gate geometry and dimensional crossover;
• $T\tau\ll1$ for diffusive, rather than ballistic, corrections;
• dephasing and Zeeman/spin–orbit gaps in the triplet sector;
• a stop when $t$, $\lvert\gamma_i\rvert$, or a competing susceptibility becomes order one.

The canonical disorder and glass claim test matrix separates a perturbative anomaly, a coupled RG trajectory, and a phase claim.

Exercise

Return-probability logarithm. In two dimensions the diffusion return probability is $P(0,t)=(4\pi Dt)^{-1}$. Show that an interaction process weighted by $\int_\tau^{1/T}dt\,P(0,t)$ is logarithmic.

Solution

Direct integration gives

$$\int_\tau^{1/T}dt\,P(0,t) =\frac{1}{4\pi D}\ln\frac{1}{T\tau}.$$

The ultraviolet cutoff is the elastic time, where diffusion begins; temperature supplies the infrared cutoff in this simplified estimate. The prefactor and sign still require the exchange/Hartree channel calculation, so the return probability predicts the logarithm but not a universal coefficient.

References

• Boris L. Altshuler and Arkady G. Aronov, “Zero Bias Anomaly in Tunnel Resistance and Electron–Electron Interaction,” Solid State Communications 30 (1979) 115–117. DOI
• Dietrich Belitz and Theodore R. Kirkpatrick, “The Anderson–Mott Transition,” Reviews of Modern Physics 66 (1994) 261–380. DOI
• Alexander M. Finkel’stein, “Influence of Coulomb Interaction on the Properties of Disordered Metals,” Soviet Physics JETP 57 (1983) 97–108. Open PDF