Effect of gauge-field interaction on fermion transport in two dimensions: Hartree conductivity correction and dephasing

We consider the quantum corrections to the conductivity of fermions interacting via a Chern–Simons gauge field and concentrate on the Hartree-type contributions. The first-order Hartree approximation is only valid in the limit of weak coupling $\lambda ⪡g^{-1/2}$ to the gauge field ($g⪢1$ is the dimensionless conductance) and results in an antilocalizing conductivity correction $\sim {\lambda }^{2}g{\text{ln}}^{2}T$. In the case of strong coupling, an infinite summation of higher-order terms is necessary, which includes both the virtual (renormalization of the frequency) and real (dephasing) processes. At intermediate temperatures, $T_0⪡T⪡g{T}_0$, where $T_0\sim 1/g^2\tau$ and $\tau$ is the elastic scattering time, the $T$ dependence of the conductivity is determined by the Hartree correction, $\delta {\sigma }^H\left(T\right)-\delta {\sigma }^H\left(g{T}_0\right)\propto g^{1/2}-{\left(T/T_0\right)}^{1/2}\left[1+\text{ln}{\left(g{T}_0/T\right)}^{1/2}\right]$, so that $\sigma \left(T\right)$ increases with lowering $T$. At low temperatures, $T⪡T_0$, the temperature-dependent part of the Hartree correction assumes a logarithmic form with a coefficient of order unity, $\delta {\sigma }^H\propto \text{ln}\left(1/T\right)$. As a result, the negative exchange contribution $\delta {\sigma }^{\text{ex}}\propto -\text{ln}g\text{ln}\left(1/T\right)$ becomes dominant, which yields localization in the limit of $T\to 0$. We further discuss dephasing at strong coupling and show that the dephasing rates are of the order of $T$, owing to the interplay of inelastic scattering and renormalization. On the other hand, the dephasing length is anomalously short, $L_{\phi }⪡L_T$, where $L_T$ is the thermal length. For the case of composite fermions with long-range Coulomb interaction, the gauge-field propagator is less singular. The resulting Hartree correction has the usual sign and temperature dependence, $\delta {\sigma }^H\propto \text{ln}g\text{ln}\left(1/T\right)$, and for realistic $g$ is overcompensated by the negative exchange contribution due to the gauge-boson and scalar parts of the interaction. In this case, the dephasing length $L_{\phi }$ is of the order of $L_T$ for not too low temperatures and exceeds $L_T$ for $T\lesssim g{T}_0$.

Figure 1

Diagrams contributing to the Hartree part of the conductivity correction. The dashed lines represent impurities; the dotted-dashed line denotes the bare gauge-field propagator $U_{\alpha \beta }$ given by Eq. (2.1). Additionally, there is the possibility of diffusons crossing the gauge-field line, as shown in Fig. 2 and discussed in the text.

Figure 2

Diagrams contributing to the effective interaction $\tilde{U}\left(q\right)$. The dotted-dashed line denotes the bare gauge-field propagator $U_{\alpha \beta }$. The two possibilities with the diffuson crossing the interaction line provide a natural low-$k$ cutoff.

Figure 3

Using the effective interaction $\tilde{U}$ as defined in Fig. 2 (thick dotted line); the Hartree diagrams of Fig. 1 can be mapped onto the standard exchange diagrams of Ref. 24.

Figure 4

Upper part: example of a contribution to the current correlator [Eq. (2.9)]. Lower part: the corresponding contribution to the current correlator can be obtained from the TDoS correction (Fig. 5, left) by removing the interaction line and the scalar vertex, while keeping the velocity vertices.

Figure 5

The first-order correction to the TDoS [Eq. (2.16)].

Figure 6

Illustration of the processes contributing to Eqs. (3.18, 3.19). The left and right diagrams give the first and second terms in brackets, respectively.

Figure 7

Illustration of the path-integral transformation, which related weak localization to mesoscopic conductance fluctuations. The detailed presentation of the transformation can be found in Appendix . As indicated in the left part of this figure, the two paths only need to end within one thermal length $L_T$ of each other, changing the short-scale cutoff in Eq. (3.32) (see Ref. 33). For mesoscopic conductance fluctuations, vertex corrections (interaction lines connecting the two copies of the same path) are absent. For the Cooperon (right part of the figure), the vertex corrections are present and add to the self-energy terms.

Figure 8

The interaction-dressed diffuson described by Eq. (4.8). The impurity ladders here denote the “disconnected diffusons” ${\tilde{\mathcal{D}}}_{\infty }$ with only self-energy interaction lines included; the vertex interaction (dashed-dotted) line enters along with the additional diagrams from Fig. 2 which form the effective interaction $\tilde{U}$ (thick dotted line).

Figure 9

The energies of the renormalized diffusons in the Hartree diagrams satisfy $E,E^{\prime }\lesssim T$ and $\omega \gtrsim T$.

Figure 10

Self-energy counterparts to the vertex interaction (Fig. 2), resulting in the diffuson self-energy [Eq. (4.15)]. In addition to the diagrams shown, equivalent possibilities to insert self-energy lines into the advanced Green’s function exist. The detailed calculation can be found in Appendix .

Figure 11

In the presence of self-energy and vertex interactions dressing the diffusons, the effective interaction block $\tilde{U}$ together with the two adjacent dressed diffusons can be more conveniently considered as one diffuson renormalized by the self-energy and vertex interaction lines, and the disconnected part subtracted to ensure the Hartree structure of at least one interaction line connecting the fermionic bubbles.

Figure 12

Schematic plot of the Hartree correction to the conductivity, $\delta {\sigma }^H\left(T\right)$ for $\lambda =1$, in the different ranges of temperatures ($\text{log}T$ scale). The solid and dotted-dashed lines of $\delta {\sigma }^H$ are described by the sum of Eqs. (4.32, 4.33) and by Eq. (4.39), respectively. The exchange contribution (Ref. 17) $\delta {\sigma }^{\text{ex}}$ is shown for comparison. Due to the high dephasing rates of delayed diffusons, the Hartree contribution can be calculated within the diffusive approximation only at temperatures below $T_1$. At low temperatures $T<T_0$, the exchange contribution overcompensates the $T$-dependent part of the Hartree contribution, so that the total correction becomes localizing, $d\sigma /dT>0$.

Figure 13

Schematic plot (log-log scale) of temperature dependence of dephasing length $L_{\phi }$ due to gauge-field interaction for the cases of screened (short-range) interaction [dashed line; Eq. (4.45)] and unscreened Coulomb interaction [dashed-dotted line; Eqs. (5.10, 5.12)]. The conventional dephasing length [Eq. (5.14)] due to the scalar part of interaction (which dominates the lowest-$T$ total dephasing in the unscreened case) is shown by the solid line. For simplicity, the logarithmic factors and $C_{\ast }$ are suppressed.

Figure 14

Schematic plot ($\text{log}T$ scale) of the contributions to the conductivity correction for the unscreened Coulomb interaction: the Hartree contribution $\delta {\sigma }^H$ given by Eq. (5.16), the exchange contribution $\delta {\sigma }^{\text{ex}}$ given by Eq. (19) of Ref. 17, and the standard Altshuler–Aronov contribution $\delta {\sigma }^C$ due to the scalar Coulomb exchange interaction (Ref. 24). For realistically large conductances $\text{ln}g⪡C_{\ast }⪡g$, the exchange term $\delta {\sigma }^{\text{ex}}$ dominates in magnitude over $\delta {\sigma }^H$ in both magnitude and temperature dependence at $T\gtrsim C_{\ast }^2/g^2\tau$, with the dependence in the vicinity of $T\sim C_{\ast }/g\tau$ taking the form $\delta {\sigma }^{\text{ex}}\simeq -2\text{ln}\left(g/C_{\ast }\right)\text{ln}\left(1/T\tau \right)$. The temperature dependence of $\delta {\sigma }^{\text{ex}}$ saturates at low temperatures, so that at very low temperatures, the standard contribution $\delta {\sigma }^C$ becomes dominant. The sum of all contributions is negative for all $T$, in contrast to the situation of composite fermions with short-range interaction.

Figure 15

Generating functionals that lead to the Hartree diagrams. The mirrored versions of diagrams (b) and (c) are also possible.

Figure 16

Examples of diagrams which do not contain the effective interaction $\tilde{U}$. The diagram (a) has an external current vertex adjacent to the diffuson crossing gauge-field line; the diagram (b) has a velocity vertex splitting the diffuson which crosses the gauge-field line.