Decoherence in weak localization. II. Bethe-Salpeter calculation of the cooperon

This is the second in a series of two papers (Papers I and II) on the problem of decoherence in weak localization. In Paper I, we discussed how the Pauli principle could be incorporated into an influence functional approach for calculating the cooperon propagator and the magnetoconductivity. In the present paper, we check and confirm the results so obtained by diagrammatically setting up a Bethe-Salpeter equation for the cooperon, which includes self-energy and vertex terms on an equal footing and is free from both infrared and ultraviolet divergences. We then approximately solve this Bethe-Salpeter equation by the ansatz $\tilde{C}\left(t\right)={\tilde{C}}^0\left(t\right)e^{-F\left(t\right)}$, where the decay function $F\left(t\right)$ determines the decoherence rate. We show that in order to obtain a divergence-free expression for the decay function $F\left(t\right)$, it is sufficient to calculate ${\tilde{C}}^1\left(t\right)$, the cooperon in the position-time representation to first order in the interaction. Paper II is independent of Paper I and can be read without detailed knowledge of the latter.

Figure 1

(a) Diagram for the weak-localization correction to the ac conductivity, $\delta {\sigma }^{\mathrm{WL}}\left({\omega }_0\right)$ [Eqs. (3)]. In contrast to the so-called “interaction corrections” to the conductivity, each current vertex is attached to both a retarded and an advanced electron line. (b) Diagrammatic definition of full cooperon ${\overline{\mathcal{C}}}_q^{\mathcal{E}}({\Omega }_1,{\Omega }_2)$ and schematic depiction of the Bethe-Salpeter equation (8) satisfied by it; $\mathcal{E}=\frac{1}{2}({\Omega }_j^{+}+{\Omega }_j^{-})$ is the conserved average of the energies of the upper and lower lines, while ${\Omega }_1$ and ${\Omega }_2$ are the outgoing and incoming cooperon frequencies (with ${\Omega }_j={\Omega }_j^{+}-{\Omega }_j^{-}$). For the structure of the cooperon self-energy ${\overline{\Sigma }}^{\text{full}}$ and details of our diagrammatic conventions, see Appendix and Fig. 2.

Figure 2

(a) Diagrammatic depiction of the Bethe-Salpeter equation [Eq. (A1)] for the cooperon ${\overline{C}}_q^{\mathcal{E}}({\Omega }_1,{\Omega }_2)$. [(b)–(d)] The contributions to the full self-energy ${\overline{\Sigma }}_{q,\text{full}}^{\mathcal{E}}({\Omega }_1,{\Omega }_3)$ of Eq. (A2a), namely, (b) ${\overline{\Sigma }}_{q,\overline{q},\text{full}}^{\mathcal{E},I∕R,\text{self}}({\Omega }_1,{\Omega }_3,\overline{\omega })$, (c) ${\overline{\Sigma }}_{q,\overline{q},\text{Hikami}}^{\mathcal{E},R}({\Omega }_1,\overline{\omega })$ (which contributes to ${\overline{\Sigma }}_{\text{full}}^{R,\text{self}}$ only), and (d) ${\overline{\Sigma }}_{q,\overline{q},\text{full}}^{\mathcal{E},I∕R,\mathrm{vert}}({\Omega }_1,{\Omega }_3,\overline{\omega })$. The superscripts $I∕R$ on $\overline{\Sigma }$ indicate which type of interaction propagator occurs: ${\overline{\Sigma }}^I$ [generated by $S_I$ in GZ’s approach (Refs. 6, 8] contains ${\overline{\mathcal{L}}}^K$, together with ${\overline{\mathcal{G}}}^R$ and ${\overline{\mathcal{G}}}^A$ on the upper and lower contours, respectively; ${\overline{\Sigma }}^R$ (generated in GZ’s approach by $S_R$) contains a sum of two types of terms. The first type features the combinations ${\overline{\mathcal{L}}}^R$ together with ${\overline{\mathcal{G}}}^K$ and ${\overline{\mathcal{G}}}^A$ on the upper and lower contours, respectively, while the second type features ${\overline{\mathcal{L}}}^A$ together with ${\overline{\mathcal{G}}}^R$ and ${\overline{\mathcal{G}}}^K$ on the upper and lower contours, respectively. The diagrams in (d) for ${\overline{\Sigma }}_{\text{full}}^{I∕R,\mathrm{vert}}$ occur both without and with bracketed labels for the interaction and electron propagators, e.g., $R$ or $\left(A\right)$ on $\mathcal{L}$, to distinguish contributions in which the electron Keldysh Green’s function occurs on the upper or lower electron line, respectively. The vertex to which the electron Keldysh Green’s function is attached is always indicated by a double dot (adopting a convention used in Ref. 6). (e) Diagrammatic conventions; the dressed interaction vertex is denoted by ${\overline{\Gamma }}_{\overline{q}}\left(\overline{\omega }\right)=\left[{\overline{\mathcal{D}}}_{\overline{q}}^0\left(\overline{\omega }\right)\right]∕{\tau }_{\mathrm{el}}$, where ${\overline{\mathcal{D}}}^0$ is the bare diffusion propagator.

Figure 3

(Color online) Schematic illustration of the time evolution in the differential equation for the cooperon: AAK’s scheme (top) evolves both end points symmetrically, while the full Bethe-Salpeter equation (bottom) contains two cooperons, describing propagation in the intervals $(t_2,-t_1)$ and $(-t_1,t_1)$. Curved lines represent interaction propagators.