Decoherence in weak localization. I. Pauli principle in influence functional

This is the first in a series of two papers, in which we revisit the problem of decoherence in weak localization. The basic challenge addressed in our work is to calculate the decoherence of electrons interacting with a quantum-mechanical environment while taking proper account of the Pauli principle. First, we review the usual influence functional approach valid for decoherence of electrons due to classical noise, showing along the way how the quantitative accuracy can be improved by properly averaging over closed (rather than unrestricted) random walks. We then use a heuristic approach to show how the Pauli principle may be incorporated into a path-integral description of decoherence in weak localization. This is accomplished by introducing an effective modification of the quantum noise spectrum, after which the calculation proceeds analogous to the case of classical noise. Using this simple but efficient method, which is consistent with much more laborious diagrammatic calculations, we demonstrate how the Pauli principle serves to suppress the decohering effects of quantum fluctuations of the environment, and essentially confirm the classic result of Altshuler, Aronov, and Khmelnitskii [J. Phys. C 15, 7367 (1982)] for the energy-averaged decoherence rate, which vanishes at zero temperature. Going beyond that, we employ our method to calculate explicitly the leading quantum corrections to the classical decoherence rates and to provide a detailed analysis of the energy dependence of the decoherence rate. The basic idea of our approach is general enough to be applicable to the decoherence of degenerate Fermi systems in contexts other than weak localization as well. Paper II will provide a more rigorous diagrammatic basis for our results by rederiving them from a Bethe-Salpeter equation for the Cooperon.

Figure 1

(Color online) The decay function $F_d\left(t\right)$ in various dimensions $d$. We have employed Eq. (5) for unrestricted random walks, Eq. (70) for the noise spectrum (including Pauli blocking effects), and used $T{\tau }_{\phi }\approx 100$. The asymptotic behavior is given by $F_1\left(t\right)\sim t^{3∕2}$, $F_2\left(t\right)\sim t\ln \left(t\right)$, and $F_3\left(t\right)\sim t$ [see Eqs. (12)]. As indicated by the dashed lines, ${\tau }_{\phi }$ is determined by the condition $F_d\left({\tau }_{\phi }\right)=1$.

Figure 2

(Color online) Random walks from $1\equiv (r_1,-t∕2)$ to $2\equiv (r_2,+t∕2)$, via $3\equiv (r_3,t_3)$ and $4\equiv (r_4,t_4)$: For given times $t$, $t_3$, and $t_4$, the distance between $r_3$ and $r_4$ is overestimated when the closed random walk (left) is substituted by an unrestricted random walk (right). This leads to increased decoherence and an underestimation of the weak-localization correction to the conductivity (see Fig. 3, dotted line).

Figure 3

The magnetoconductivity for a quasi-one-dimensional wire experiencing decoherence by classical white Nyquist noise, as function of ${\tau }_{\phi ,1}^{\mathrm{AAK}}∕{\tau }_H$, comparing AAK’s exact Airy-function result (42) (solid line) against two approximations: the commonly employed average over unrestricted random walks in the path-integral exponent [Eq. (34b)] and the improved version obtained by averaging over closed random walks [Eq. (34a)], resulting in Eq. (44), with $c_1^{\mathrm{urw}}$ or $c_1^{\mathrm{crw}}$, respectively. For ${\tau }_{\phi }∕{\tau }_H\to 0$, the result for unrestricted (closed) random walks equals 0.808 (0.964) of the exact value. For ${\tau }_{\phi ,1}^{\mathrm{AAK}}∕{\tau }_H⪢1$, all three curves become indistinguishable.

Figure 4

(Color online) Sketch of two many-particle states forming a coherent superposition, and two possible scattering processes contributing to ${\Gamma }_e\left(\lambda \right)$ and ${\Gamma }_h\left(\lambda \right)$, by emission and absorption, respectively.

Figure 5

(Color online) The spectrum ${\mathcal{W}}_{\mathrm{pp}}\left(\overline{\omega }\right)$ of Eq. (67) as a function of frequency and electron energy. It has the properties ${\mathcal{W}}_{\mathrm{pp}}\left(0\right)=1$ and ${\mathcal{W}}_{\mathrm{pp}}\to 0$ for $\overline{\omega }⪢\max \{T,\epsilon \}$, and in the limit $T\to 0$, behaves as $\frac{1}{2}\mid \overline{\omega }\mid \theta (\mid \epsilon \mid -\mid \overline{\omega }\mid )$.

Figure 6

(Color online) Comparison of the commonly employed leading approximation for the decoherence time ${\tau }_{\phi ,d}$ (thin lines) and the full decoherence time ${\tilde{\tau }}_{\phi ,d}$ (thick lines), as a function of temperature, for dimensions $d=1$ (full red lines), $d=2$ (black, long dashed), and $d=3$ (blue, short dashed). The temperature $T^{*}$ has been (arbitrarily) chosen to give $T^{*}{\tau }_{\phi ,d}\left(T^{*}\right)=10$ in all cases (this amounts to fixing the values of the material parameters ${\gamma }_1$, $g_2$, and ${\gamma }_3$). The magnitude of the correction is governed by the small parameter ${\left[T{\tau }_{\phi ,d}\right]}^{-1}$, whose $T$ dependence is shown in the inset: In $d=1$ $(d=3)$, the corrections become smaller (and the weak-localization theory is applicable) towards higher (lower) temperatures, where ${\left[T{\tau }_{\phi ,d}\left(T\right)\right]}^{-1}⪡1$. In $d=2$, the correction amounts to a numerical constant factor (shift on the logarithmic scale, hardly visible in this figure). Note, in particular, the slow decay of the correction in the case of $d=1$, where the correction falls off only like $T^{-1∕6}$ [compare Eqs. (74, 75)].

Figure 7

(Color online) (a) $g_1\left(L_t\right)F_{1,\mathrm{crw}}^{\mathrm{pp}}\left(t\right)$ [Eq. (84)] as function of the parameters $\epsilon t$ and $Tt$. (b) The energy- and temperature-dependent decoherence rate ${\tau }_{\phi ,1}$, defined from the condition $F_{1,\mathrm{crw}}^{\mathrm{pp}}\left({\tau }_{\phi ,1}\right)=1$, as a function of $T∕{\gamma }_1$ and $\epsilon ∕{\gamma }_1$.

Figure 8

(Color online) (a) Vertices of “type one” and (b) of “type two” arising in the Keldysh perturbation theory; the accompanying Keldysh Green’s functions are ${\tilde{G}}^K(\epsilon \mp \overline{\omega })$, respectively, producing Pauli factors $\tanh [(\epsilon \mp \overline{\omega })∕2T]$ that dress the associated interaction propagators ${\overline{\mathcal{L}}}^R$ and ${\overline{\mathcal{L}}}^A$ [Eqs. (92a, 92b, 92c, 92d)]. Standard Keldysh rules generate diagrams such as both (c) and (d), but the latter vanishes upon impurity averaging, since it contains $G^A$ on the forward propagator.