Keldysh approach to the renormalization group analysis of the disordered electron liquid

We present a Keldysh nonlinear sigma-model approach to the renormalization group analysis of the disordered electron liquid. We include both the Coulomb interaction and Fermi-liquid type interactions in the singlet and triplet channels into the formalism. Based on this model, we reproduce the coupled renormalization group equations for the diffusion coefficient, the frequency, and interaction constants previously derived with the replica model in the imaginary time technique. With the help of source fields coupling to the particle-number and spin densities, we study the density-density and spin density-spin density correlation functions in the diffusive regime. This allows us to obtain results for the electric conductivity and the spin susceptibility and thereby to rederive the main results of the one-loop renormalization group analysis of the disordered electron liquid in the Keldysh formalism.

Figure 1

Dynamical correlation functions ${\overline{\chi }}_{nn}^{\mathrm{dyn},\mathrm{R}}$ (top) and ${\chi }_{s_k{s}_k}^{\mathrm{dyn},\mathrm{R}}$ (bottom).

Figure 2

The elements of the RG procedure originating from the noninteracting part of the action. Open ends imply $P$. Closed sleeves correspond to $U$ or $\overline{U}$. When separated by an angle, a gradient acts on one of them. A slow frequency ${\varepsilon }_s$ stands in the vertex marked by a dot.

Figure 3

The elements of the RG procedure originating from the interaction part of the action. A shaded square implies one of the interaction amplitudes. A ladder means that the interaction was dressed by ladder diagrams. Such terms are indicated by the subscript “d.”

Figure 4

Source term.

Figure 5

Dressed interactions.

Figure 6

The four different terms contributing to $\Delta S_D$. For the terms ($a$) and ($b$), a gradient expansion is needed.

Figure 7

Diagram for $\langle S_{\mathrm{int},2}\rangle$. This term vanishes as discussed in the text.

Figure 8

This figure illustrates the two choices of ${\varepsilon }_f$ for the average $\langle S_{\mathrm{int},1}\rangle$, where ${\varepsilon }_f$ symbolizes the fast frequency. In fact, both choices are equivalent and in this way one comes from Eqs. (156) to (157).

Figure 9

Terms of the kind displayed in this figure arise when evaluating the average $-\frac{1}{2}\langle \langle S_1^2{S}_{\mathrm{int},1}\rangle \rangle$, compare the first and third terms in Eq. (161). All frequencies involved are bound to be small. This makes the contributions of this type irrelevant.

Figure 10

Diagrammatic representation for $i\langle \langle S_{\varepsilon }{S}_{\mathrm{int},1}\rangle \rangle$. This term contributes to $\Delta z$.

Figure 11

The average $\langle S_{\mathrm{int},1}\rangle$ as relevant for the calculation of $\Delta z$. Expansion in the slow frequency is needed to be performed.

Figure 12

$\langle S_{\mathrm{int},1}\rangle$ as relevant for the renormalization of the interaction amplitudes. In this case, all frequencies involved are slow, the logarithmic correction arises from an integration over fast momenta .

Figure 13

The pairs of diagrams related to ${\left(\Delta S_{\Gamma }\right)}_i$, $i=1–5$. Diagrams labeled as $\left(a\right)$ give rise to the corrections ${\left(\Delta {\Gamma }_1\right)}_i$ and diagrams labeled as ($b$) to the corrections ${\left(\Delta {\Gamma }_2\right)}_i$. Only those contributions remain, for which all interaction amplitudes are of the ${\Gamma }_2$ type. All other contributions, which contain the amplitudes ${\Gamma }_0$ or ${\Gamma }_1$ at least once, cancel between the two diagrams forming a pair. An important consequence is that the amplitude ${\Gamma }_0$ remains unrenormalized.

Figure 14

These diagrams give rise to the corrections to ${\gamma }_{•}^{\rho /\sigma }$. They are organized into two pairs, in close analogy to the corresponding diagrams in Fig. 13 with the same labels.

Figure 15

The four pairs of diagrams relevant for the vertex corrections.