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

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

Jan von Delft,¹Florian Marquardt,¹ R. A. Smith,² and Vinay Ambegaokar³

1Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität München, 80333 München, Germany

2School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, England

3Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850, USA 共Received 17 June 2007; published 30 November 2007兲

This is the second in a series of two papers共Papers I and II兲 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 ansatzC˜共t兲=C˜⁰共t兲e^−F共t兲, where the decay functionF共t兲 determines the decoherence rate. We show that in order to obtain a divergence-free expression for the decay function F共t兲, it is sufficient to calculate C˜1共t兲, 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.

DOI:10.1103/PhysRevB.76.195332 PACS number共s兲: 72.15.Rn, 03.65.Yz

I. INTRODUCTION

This is the second in a series of two papers共Papers I and II兲, in which we revisit the problem of decoherence in weak localization, using both an influence-functional approach 共Paper I兲 and a Bethe-Salpeter equation for the cooperon 共Paper II兲 to calculate the magnetoconductivity. The basic challenge is to calculate the interference between two time- reversed trajectories of an electron traveling diffusively in a Fermi sea and coupled to a noisy quantum environment, while taking proper account of the Pauli principle. In Paper I,¹we discussed how this could be done using an influence- functional approach by dressing the spectrum of the noise field by “Pauli factors”关see Eq.共I.66兲; throughout, “I” will indicate formulas from Paper I兴. Moreover, within the influence-functional scheme, we concluded that a divergence-free calculation of the decoherence rate can be obtained by expressing the cooperon in the position-time representation as

C˜共0,t兲 ⯝C˜⁰共0,t兲e^−F共t兲, F共t兲= −C˜¹共0,t兲

C˜⁰共0,t兲, 共1兲 whereC˜1共0 ,t兲 is the first-order term in an expansion of the full cooperon C˜共0 ,t兲 in powers of the interaction. 关In the present paper, this statement will be made more precise:

when reexponentiating, a part of C˜¹共0 ,t兲 has to be omitted that can be determined, in a self-energy-only calculation, to contribute only to the prefactor of the cooperon; see Sec.

II C.兴

These conclusions of Paper I rest entirely on the influence-functional approach and, in the discussion of the Pauli principle, relied on heuristic arguments. Though these are in accord with results derived elsewhere^2–6共as shown in Paper I, Sec. VII兲, it is desirable to compare the approxima- tions used and the results obtained so far against a treatment

relying purely on diagrammatic perturbation theory, the framework within which most of our understanding of disor- dered metals to date has been obtained.

In the present paper, we check and confirm the results mentioned above by diagrammatically setting up a Bethe- Salpeter equation for the cooperon using standard Keldysh diagrammatic perturbation theory 共using conventions sum- marized in Ref. 6兲, which includes self-energy and vertex terms on an equal footing and is free from both infrared and ultraviolet divergences. We then show that this equation can be solved 共approximately, but with exponential accuracy兲 with an ansatz that is precisely of the form of Eq.共1兲, and that the functionF共t兲 so obtained agrees with the form de- rived in Paper I关Eq.共I.65兲兴.

The usual diagrammatic calculation of the cooperon starts from a Dyson equation for a “self-energy-diagrams-only”

version of the cooperon,

¯C

␧,qself共␻兲=¯C

q

0共␻兲关1 +⌺¯

␧,qself共␻兲C¯

␧,qself共␻兲兴. 共2兲 Here, the cooperon self-energy⌺¯

␧,qself

includesonly self-energy diagrams, in which interaction lines connect only forward to forward or backward to backward electron propagators; for these diagrams, the frequency labels along both the forward and the backward propagators are conserved separately, which is why the Dyson equation is a simple algebraic equa- tion forC¯

␧,q

self共␻兲. However, the cooperon self-energy ⌺¯

␧,q self共␻兲 turns out to be infrared divergent in the quasi-two- and quasi- one-dimensional cases. This problem is usually cured by in- serting an infrared cutoff by hand 共as reviewed in Sec. II C below兲. The results so obtained are qualitatively correct but, due to thead hoctreatment of the cutoff, not very accurate quantitatively 关e.g., in the first line of Eq. 共19兲 for F˜

␧self共t兲 below, the exponent is correct, but the prefactor is wrong by roughly a factor of 2 compared to Eq.共I.44兲兴.

1098-0121/2007/76共19兲/195332共17兲 195332-1 ©2007 The American Physical Society

Our goal in the present paper is to obtain, starting from a diagrammatic equation, results free from any cutoffs, infra- red or ultraviolet, that have to be inserted by hand—the theory “should take care of its divergences itself.” This can be achieved if the cooperon self-energy is takento include vertex diagrams, in which interaction lines connect forward and backward electron propagators. Since this brings about frequency transfers between the forward and the backward propagators, their frequency labels are no longer conserved separately. As a consequence, it becomes necessary to study a more complicated version of the cooperon,¯C

q

E共⍀₁,⍀₂兲, la- beled by three frequencies, and governed not by a simple algebraic Dyson equation but by a nonlinear integral equa- tion, which we shall refer to as “Bethe-Salpeter equation.”

Finding an exact solution to the Bethe-Salpeter equation seems to be an intractable problem, which we shall not at- tempt to attack. Instead, we shall transcribe the Bethe- Salpeter equation from the momentum-frequency to the position-time domain, in which it is easier to make an in- formed guess for the expected behavior of the solution. Us- ing the intuition developed in Paper I within the influence- functional approach 关summarized in Eq. 共1兲 of the present paper兴, we shall make an exponential ansatz for C˜␧共r12,t₁,t₂兲, the cooperon in the position-time domain. We shall show that this ansatz solves the Bethe-Salpeter equation with exponential accuracy, in the sense that improving the ansatz would require terms to be added to the exponent that are parametrically smaller 共in powers of 1/g, g being the dimensionless conductance兲than the leading term in the ex- ponent.

II. SETTING UP BETHE-SALPETER EQUATION FOR COOPERON

A. Various expressions for conductivity

The diagrammatic definition of the weak-localization con- tribution to the ac conductivity of a quasi-d-dimensional dis- ordered conductor is given by Fig.1共a兲, which corresponds to the following expression共see Appendix A of Ref. 2, or Appendix C of Ref.6兲:

␦␴d

WL共␻0兲= − ␴d

␲␯ប具C˜

cond␧,␻0典_␧, 共3a兲

C˜

cond

␧,␻0=

冕

^{共d2␻˜兲}

冕

^{共dq兲C¯q␧−␻˜/2共␻0−␻˜,␻0+␻˜兲, 共3b兲}

where

具¯典_␧⬅

冕

^{d␧f共␧−兲}␻⁻0^f共␧+兲

¯ 共4兲

denotes an average over␧, with ␧±=␧±¹₂␻0, and in the dc limit ␻0→0, the weighting function reduces to −f

⬘

共␧兲, the derivative of the Fermi function f共␧兲= 1/关e^␧/T+ 1兴. 共In this paper, temperature is measured in units of frequency, i.e.,T stands forkBT/ប throughout; likewise, although␧will often be referred to as “excitation energy,” it stands for a fre- quency.兲

The full cooperon with general arguments,C¯

q

E共⍀1,⍀2兲, is defined diagrammatically in Fig.1共b兲:Eis the average of the frequencies of the upper and lower electron lines, while⍀1

and⍀2are the outgoing and incoming cooperon frequencies, respectively. In the absence of external time-dependent fields, the average energyEis conserved between incoming and outgoing lines. The cooperon needed for the ac conduc- tivity in Fig. 1共a兲 has incoming upper and lower electron lines with energies ␧+ and ␧−−␻^˜ and outgoing upper and lower electron lines with energies ␧+−␻^{˜ and} ␧−, implying

⍀1=共␻0−␻^˜兲, ⍀2=共␻0+␻^˜兲, and E=␧−¹₂␻^˜, as used in Eq.

共3b兲.

To make contact with the expression for the conductivity in the position-time representation used in Paper I, we re- write Eq.共3b兲as

˜C

cond

␧,␻0⬅

冕

^{dtei␻0tC˜␧+共0,t兲, 共5a兲}

C˜␧+共0,t兲 ⬅C˜␧+

共

^{r12= 0;t1=1}2t,t₂= −¹₂t

兲

^{, 共5b兲}

whereC˜␧+共r₁₂;t₁,t₂兲is a representation of the full cooperon in an energy/position/two-time representation,

FIG. 1. 共a兲Diagram for the weak-localization correction to the ac conductivity, ␦␴^WL共␻0兲 关Eqs. 共3兲兴. 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.共b兲Dia- grammatic definition of full cooperon ¯C

q

E共⍀1,⍀2兲 and schematic depiction of the Bethe-Salpeter equation 共8兲 satisfied by it; E

=¹₂共⍀j ++⍀j

−兲 is the conserved average of the energies of the upper and lower lines, while⍀1and ⍀2are the outgoing and incoming cooperon frequencies共with⍀j=⍀j

+−⍀j

−兲. For the structure of the cooperon self-energy⌺¯fulland details of our diagrammatic conven- tions, see Appendix A and Fig.2.

C˜␧+共r12;t₁,t₂兲=

冕

^{共dq兲共d⍀1兲共d⍀2兲ei关qr12−⍀1t1+⍀2t2兴}

⫻C¯

q

␧+−共1/2兲⍀2共⍀1,⍀2兲 共5c兲

=

冕

^{共dq兲共d2␻˜兲共d␻兲ei关qr12−␻t12+␻˜ t˜12兴}

⫻C¯

q

␧+−共1/2兲共␻˜+␻兲共␻⁻␻^˜,␻⁺␻^˜兲, 共5d兲 wherer₁₂=r₁−r₂, t₁₂=t₁−t₂, t˜₁₂=t₁+t₂, ␻⁼¹₂共⍀2+⍀1兲, and

␻

˜=¹₂共⍀2−⍀1兲. The兰dttime integral in Eq.共5a兲sets␻⁼␻0

and hence␧+−¹₂共␻^˜+␻兲=␧−¹₂␻^{˜ in Eq.}共5c兲, as needed for Eq.

共3b兲. Inserting Eq.共5a兲into Eq.共3a兲, we find

␦␴dWL共␻0兲= − ␴d

␲␯ប

冕

0

⬁

dte^i␻0t具C˜␧+共0,t兲典_␧. 共6兲 Thus, the dc limit␦␴d

WL共0兲is seen to be an energy-averaged version of Eq.共I.1兲. Since our goal is to make contact with the results of Paper I, we shall take the dc limit␻0→0 and

␧+→␧throughout below共it is straightforward to reinstate the

␻0dependence explicitly by replacing the parameter␧by␧+

in all cooperons below兲.

We choose to normalize the full cooperons such that in the absence of interactions, they reduce as follows to their noninteracting versions共⍀12=⍀1−⍀2兲:

C

¯q

E共⍀1,⍀2兲——→

no int

2␲␦共⍀12兲C¯

q

0共⍀1兲, 共7a兲

C˜E共r₁₂;t₁,t₂兲——→

no int

C˜⁰共r₁₂,t₁₂兲. 共7b兲 Here,C¯

q

0共⍀兲=共Eq−i⍀兲⁻¹, with Eq=Dq²+␥H, where ␥His a magnetic-field induced decay rate. For later reference, we also defineE_¯q⁰=Dq¯².

Our strategy for determining the decoherence rate will be to find an approximation forC˜E共0 ,t兲of the form共1兲. To this end, we shall set up a Bethe-Salpeter equation for

¯C

q

E共⍀1,⍀2兲, transcribe it to the position-time domain to find a Bethe-Salpeter equation forC˜E共r12;t₁,t₂兲, and then solve the latter using ansatz共1兲.

B. Bethe-Salpeter equation for cooperon

In the presence of interactions, the full cooperon

¯C

q

E共⍀1,⍀2兲satisfies a Bethe-Salpeter equation of the general form

¯C

q

E共⍀1,⍀2兲=¯C

q

0共⍀1兲

冋

^{2␲␦共⍀12兲}

+

冕

^{共d⍀3兲⌺¯q,fullE 共⍀1,⍀3兲C¯qE共⍀3,⍀2兲}

册

^{, 共8兲}

depicted schematically in Fig.1共b兲. The average energyEis conserved because no external fields are present. For the cooperon self-energy ⌺¯

q,full

E 共⍀1,⍀3兲 occurring herein, we

shall adopt the diagrammatic definition first written down in Ref. 7. The corresponding diagrams and equations for ⌺¯

full

are rather unwieldy and hence have been relegated to Appen- dix A关see Fig. 2 and Eqs.共A2a兲–共A2f兲 in Appendix A 1兴.

This very technical appendix can be skipped by casual read- ers; its contents are summarized in the next two paragraphs, and to be able to follow the developments of the main text below, it should suffice to just occasionally consult the final formulas for the self-energies given in Eqs.共A5a兲–共A5f兲.

The cooperon self-energy⌺¯

fullis itself proportional to the cooperon¯C; thus, the Bethe-Salpeter equation共8兲is nonlin- ear in¯C. Solving it in full glory thus seems hardly feasible.

Therefore, we shall henceforth consider only a “linearized”

version thereof, obtained共in Appendix A 2兲by replacing the full cooperon self-energy⌺¯

fullin Eq.共8兲by a bare one,⌺¯

bare. The latter, given explicitly in Eqs.共A5a兲–共A5f兲, is obtained by making the replacement ¯C

q

E共⍀1,⍀3兲→2␲␦共⍀13兲C¯

q 0共⍀1兲 for every occurrence of the full cooperon in⌺¯

full.

A perturbative expansion of the full cooperon¯Cin powers of the interaction can readily be generated by iterating Eq.

共8兲. This is done explicitly to second order in Appendix A 3 关see Eq. 共A6兲兴. The expansion illustrates two important points: First, due to the frequency transfers between the for- ward and backward propagators generated by the vertex dia- grams, the frequency arguments of⌺¯

bareincreasingly become

“entangled” from order to order in perturbation theory, i.e., they occur in increasingly complicated combinations. This makes it exceedingly difficult to directly construct an explicit solution. Secondly, no ultraviolet divergences arise in pertur- bation theory, confirming the heuristic golden rule arguments of Paper I, Sec. V共and contradicting suggestions to the con- trary implicit in Refs.8–10; see Appendix A 3 for a discus- sion of this point兲.

C. Recover Dyson equation by neglecting vertex terms Before attempting to solve the共linearized兲Bethe-Salpeter equation, it is instructive to start for the moment with a rather strong approximation, namely, to simply neglect all vertex terms共they will be reinstated later兲, thereby avoiding the above mentioned “entanglement” of frequencies. This re- duces the Bethe-Salpeter equation to the more familiar Dyson equation共2兲for the “self-energy-only” cooperon¯C^self, and will allow us to review some standard arguments and to recover some familiar and simple results.

In the absence of vertex diagrams, the cooperon self- energy⌺¯

q,bare

E 共⍀1,⍀2兲of Eq.共A5a兲is proportional to␦共⍀12兲, implying the same for the cooperon¯C

q

␧−⍀2/2,self共⍀1,⍀2兲, so that the cooperons needed on the right-hand sides of Eqs.

共5c兲and共5b兲can, respectively, be written as C¯

q

␧−⍀2/2,self共⍀1,⍀2兲 ⬅2␲␦共⍀12兲C¯

␧,q

self共⍀2兲, 共9a兲

C˜␧,self共0,t兲=

冕

^{共dq兲共d␻兲e−i␻t¯C␧,qself共␻兲. 共9b兲}

The “single-frequency” cooperonC¯

␧,q

self共␻兲 introduced in Eq.

共9a兲is the generalization of the free, single-frequency coop-

eronC¯

q

0共␻兲 to the case of a cooperon for pairs of paths with average energy␧in the presence of self-energy-only interac- tions. From Eq.共8兲, it is seen to satisfy the familiar Dyson equation共2兲, with solution

¯C

␧,qself共␻兲= 1 Eq−i␻⁻⌺¯

␧,qself共␻兲, 共10兲 where the effective cooperon self-energy is given by an in- tegral over all momentum and frequency transfers to the en- vironment,

⌺

¯␧self,q共␻兲=1

ប

冕

^{共d␻¯兲共dq¯兲⌺¯q,q␧,self¯,bare共␻,␻¯兲, 共11兲}

with⌺¯

q,q¯,bare

␧,self 共␻^,␻^¯兲given in Eqs.共A5b兲–共A5d兲.

Now, the standard way to extract the decoherence rate from Eq.共10兲 is to expand the self-energy in powers of Eq

−i␻^:

⌺¯

␧,q

self共␻兲= −␥_␧␸,self,q +共Eq−i␻兲⌺¯

␧,q

⬘

^self+ ¯ , 共12a兲

␥_␧,q^␸,self⬅−关⌺¯

␧,qself共␻兲兴E_q=i␻. 共12b兲 The leading “cooperon mass” term can be identified with the decoherence rate, because Eq. 共9b兲yields共after performing the兰共d␻兲integral by contour integration兲

C˜

␧self共0,t兲 ⯝

冕

^{共dq兲e−t共Eq+␥␧␸,q,self兲共1 +⌺¯␧}

^⬘

^{,qself+ ¯兲. 共13兲}

Since the共dq兲integral is dominated by smallq, let us replace qby 0 in ␥_␧␸,self,q and⌺¯

␧,q

⬘

^self, so that they can be pulled out of the integral. This yields

C˜

␧self共0,t兲 ⯝C˜⁰共0,t兲e^{−F˜␧self共t兲}共1 +⌺¯

␧,0

⬘

^self+ ¯兲, 共14a兲

F˜

␧self共t兲=t␥_␧,0^␸,self, 共14b兲 in whichC˜

␧self共0 ,t兲is expressed in a form reminiscent of Eq.

共1兲: a free cooperon, times the exponential of a decay func- tion, times a factor 1 +⌺¯

␧,0

⬘

^self that renormalizes the overall amplitude of the cooperon 共i.e., it corresponds to “wave- function” renormalization, in analogy to the occurrence of a finite quasiparticle weight Z in a Fermi liquid due to the short-time decay that is not resolved further by this approxi- mation兲.

Since we have to setEq=i␻^{in Eq.}共12b兲andq= 0 in Eq.

共14b兲, it is natural to split the self-energy of Eq.共11兲into two parts, ⌺¯

␧,q self共␻兲=⌺¯

␧,q

self,dec共␻兲+⌺¯

␧,q

self,Z共␻兲, chosen such that

⌺¯

␧,q

self,Z共␻兲vanishes whenEq=i␻^{andq= 0.}关This requirement is, in fact, fulfilled by共and was the motivation for兲the sepa- ration of Eq.共A5b兲into two terms, labeled “dec” and “Z.”兴 Thus,␥_␧,0^␸,selfdepends only on⌺¯

␧,qself,dec共␻兲; using Eq.共A5c兲in Eqs.共11兲and共A5f兲, it can be written as follows for not too large magnetic fields¹¹共␥H/T1兲:

␥_␧␸,self,0 = 1

ប²

冕

^{共d␻¯兲共dq¯兲}共E¯_q^2E¯q0

0兲²+␻^{¯2具VV典¯q␻¯}

pp. 共15兲

Here, the effective propagator 具Vˆ Vˆ典_q¯␻¯

pp arising in Eq. 共15兲 turns out to be precisely the Pauli-principle-modified propa- gator of Eq.共I.66b兲which we conjectured by heuristic argu- ments in Paper I, Sec. V D:

1 ប具VV典q¯pp␻¯

= ImL¯

q¯

R共␻^¯兲

再

^coth

^冋

^2T␻¯

^册

^{+12 tanh}

^冋

^{␧2T−␻¯}

^册

−1

2tanh

冋

^␧2T+␻¯

册 冎

^{. 共16兲}

The coth+ tanh combination occurring in 具Vˆ Vˆ典q¯␻¯

pp limits the frequency integral to 兩␻^¯兩ⱗT, as anticipated by the golden rule discussion in Sec. VC of Paper I关see Eq. 共I.62兲兴.

After performing the共dq¯兲integral in Eq.共15兲, the remain- ing共d␻^¯兲integral turns out to have an infrared divergence for quasi-one- or quasi-two-dimensional samples. To be explicit, if we regularize it by hand by inserting a steplike cutoff function ␪共兩␻^¯兩−␻^¯0兲, we obtain for the quasi-d-dimensional case

␥_␧,0^␸,self⯝ pd

2

冕

␻¯₀

⬁ d␻^¯

␻^¯1−d/2

再

^coth

^冋

^2T␻¯

^册

^+12tanh

^冋

^{␧2T−␻¯}

^册

−1

2 tanh

冋

^␧2T+␻¯

册 冎

^{, 共17a兲}

withp₁=

冑

2␥1/␲^,p2= 1/共2␲^g2兲, andp₃= 1/共

冑

2␥3␲²兲, where

␥1=D共e²/ប␴1兲², g₂=ប␴2/e², and ␥3=D共e²/ប␴3兲⁻². For d

= 3, the integral is well behaved in the limit␻^¯0→0, but not ford= 1 , 2. For example, in the quasi-one-dimensional case, the integral evaluates to

␥0,0␸,self=2T

␲

冋

^2␻¯␥01

册

^1/2

再

^{1 +O}

^{冉 冋}

^បT␻¯0

^册

^1/2

^冊 冎

^{, 共17b兲}

which diverges for ␻^¯0→0. This infrared divergence arises because in the present approach, we have neglected vertex terms, which in general ensure that frequency transfers smaller than the inverse propagation time 1/tdo not contrib- ute共see Paper I, Sec. III兲. Thus, we should choose the infra- red cutoff at ␻^¯0⯝1/t 共as noted in Ref. 6兲, obtaining a time-dependent¹² decay rate, ␥_␧,0^{␸,self= 2}

冑

2/␲^t1/2/共␶_␸,1^AAK兲^3/2, where¹³

1

␶_␸,1^AAK=␥_␸,1^AAK=共T

冑

_␥1兲^2/3=

冉

^Teប²␴

^冑

1^D

冊

^{2/3 共18兲}

is the decoherence rate first derived by Altshuter, Aronov, and Khmelnitskii 共AAK兲.² ␥_␧␸,self,0 grows with time, because with increasing time, the cooperon becomes sensitive to more and more modes of the interaction propagator with increasingly smaller frequencies, whose contribution in Eq.

共17a兲scales like␻^¯−3/2.

Alternatively, instead of␻^¯0= 1/t, the choice ␻^¯0=␥_␧,0^␸,selfis often made, since in weak localization the time duration of relevant trajectories is set by the inverse decoherence rate.

Then, Eq. 共17b兲 is solved self-consistently,^3,8 yielding

␥_␧,0^␸,self=共2

冑

2/␲兲^2/3/␶_␸,1^AAK, with␶_␸,1^AAKagain given by Eq.共18兲.

The decay functions ford= 1 corresponding to the above two choices of␻^¯0 in Eq. 共17b兲 are, respectively 关from Eq.

共14b兲兴,

F˜

␧self共t兲=

再

^共2共2

^{冑 冑}

^{2/2/␲␲兲共t/兲3/2␶共t/␸,1AAK␶␸,1AAK兲3/2兲.}

冎

^共19兲

Evidently, both equations describe decay on the same time scale␶_␸,1^AAK. The second choice does not properly reproduce the 3/2 power law in the exponent that we expect from Eq.

共I.44兲 for F_crw^cl 共t兲 关a fact strongly criticized by Golubev and Zaikin 共GZ兲⁹兴. However, the first choice does, up to a nu- merical prefactor, whose precise value cannot be expected to come out correctly here because it depends on the shape of the infrared cutoff function共arbitrarily chosen to be a sharp step function above兲. We thus recover theclassicalresult for the decoherence rate. The reason is essentially that Pauli blocking 共represented by the tanh terms in 具VV典q¯␻¯

pp兲 sup- presses the effects of quantum fluctuations 关represented by the +1 in coth共␻^¯/2T兲= 2n共␻^¯兲+ 1兴 with frequencies larger thanT, as discussed in detail in Sec. V D of Paper I. More- over, we also obtain the important result that the first “quan- tum correction” to this classical result that arises from self- energy terms 关the O共关␻^¯0/T兴^1/2兲 correction in Eq. 共17b兲兴 is smaller by a factorប/

冑

tT. Fort⬃␶_␸^{, this is}Ⰶ1 in the regime of weak localization关see discussion after Eq.共I.7兲兴, in agree- ment with the conclusions of Vavilov and Ambegaokar.¹⁴

III. BETHE-SALPETER EQUATION IN THE POSITION-TIME DOMAIN

The infrared divergences mentioned above are cured as soon as vertex diagrams are included. However, as men- tioned at the end of Sec. II B and detailed in Appendix A 2, frequency entanglement then renders the momentum- frequency version共8兲 of the Bethe-Salpeter equation intrac- table. This suggests that we try a more pragmatic way of finding an approximate expression for the full cooperon: in- spired by the insight from Paper I共Sec. III D兲that in the case of classical noise, a rather accurate description of the coop- eron can be obtained in theposition-time representation by reexponentiating its expansion to first order in the interaction 关Eq.共1兲of this paper兴, we shall try a similar approach here:

we transcribe the Bethe-Salpeter equation to the position- time domain to obtain an equation for the corresponding cooperonC˜␧共r12,t₁,t₂兲 of Eq. 共5c兲, and solve this equation approximately with an exponential ansatz; this ansatz will turn out to yield precisely the reexponentiation of C˜␧共1兲共r12;t₁,t₂兲, the first-order expansion of the full coop- eron, in full analogy to Eq.共1兲.

A. Transcription to time domain, exponential ansatz Let us now consider the Bethe-Salpeter equation共8兲for

¯C

q

␧−共1/2兲⍀2共⍀1,⍀2兲, i.e., with E=␧−¹₂⍀2, as needed in Eq.

共3b兲when␻0= 0. This equation can be transcribed using Eq.

共5c兲 共with␧+there replaced by␧兲to the form 共−Dⵜr₁

2 +⳵t₁+␥H兲C˜␧共r12;t₁,t₂兲

=␦共r12兲␦共t12兲

+

冕

^dr4dt4dt4

^⬘

^{⌺˜full␧,t4⬘共r14;t1,t4兲C˜␧共r42;t44⬘,t24⬘兲, 共20兲}

where the self-energy in the energy/position/time representa- tion is defined by

⌺˜

full

␧,t₄⬘共r14;t₁,t₄兲 ⬅

冕

^{共dq兲共d⍀1兲共d⍀2兲共d⍀4兲}

⫻e^{i关qr14−⍀1t1+⍀4t4−t4⬘共⍀2−⍀4兲兴}

⫻⌺¯

q,full

␧−⍀2/2共⍀1,⍀4兲. 共21兲 Before trying to solve Eq.共20兲, let us get a feeling for the structure of this equation by calculating the zeroth- and first- order terms ofC˜␧共r12;t₁,t₂兲in an expansion in powers of the interaction propagator 共i.e., ⌺˜

bare兲. To this end, we use the fact that

共−Dⵜr

2+⳵t+␥H兲C˜⁰共r,t兲=␦共r兲␦共t兲, 共22兲 iterate Eq.共20兲once, and replace⌺˜

fullby⌺˜

bare关given by Eqs.

共A5a兲–共A5f兲兴on its right-hand side. We find

C˜␧共r₁₂;t₁,t₂兲=C˜⁰共r₁₂,t₁₂兲+C˜共1兲␧共r₁₂;t₁,t₂兲+ ¯ , 共23a兲 whereC˜共1兲␧共r12;t₁,t₂兲has just the structure discussed in Pa- per I, Sec. III C, describing propagation from 共r2,t₂兲 to 共r1,t₁兲, with interaction vertices along the way at points 共r₄,t₄兲and共r₃,t₃兲:

C˜共1兲␧共r12;t₁,t₂兲=

冕

^{dr3dt3dr4dt4C˜0共r13,t13兲⌺˜bare␧ 共r34;t3,t4兲}

⫻C˜⁰共r₄₂,t₄₂兲, 共23b兲

⌺˜

bare␧ 共r34;t₃,t₄兲 ⬅

冕

^dt4

^⬘

^{⌺˜bare␧,t4⬘共r34;t3,t4兲 共23c兲}

=

冕

^{共dq兲共d⍀3兲共d⍀4兲ei关qr34−⍀1t3+⍀4t4兴}

⫻⌺¯

q,bare

␧−⍀4/2共⍀3,⍀4兲. 共23d兲

Let us now construct an approximate solution of the Bethe-Salpeter equation 共20兲by making an exponential an- satz of the following form:

C˜␧共r12;t₁,t₂兲=C˜⁰共r12,t₁₂兲e^{−F˜␧共r12;t1,t2兲}. 共24兲 The decay function F˜␧ is needed only for t₁₂ⱖ0 关since C˜⁰共r12,t₁₂兲 vanishes otherwise兴 and is required to obey the initial conditionF˜␧共0 ,t2,t₂兲= 0 for allt₂. Ansatz共24兲 solves

Eq.共20兲exactly, provided that the decay functionF˜␧satisfies the equation

−C˜⁰共r12,t₁₂兲

再 ^冋

^{⳵t1−Dⵜr21}

− 2Dⵜr₁C˜⁰共r₁₂,t₁₂兲·ⵜr₁

C˜⁰共r₁₂,t₁₂兲

册

^{F˜␧共r12;t1,t2兲}

−D关ⵜr₁F˜␧共r12;t₁,t₂兲兴²

冎

=

冕

^dr4dt4dt4

^⬘

^{⌺˜full␧,t4⬘共r14;t1,t4兲C˜0共r42,t42兲}

⫻e^{−关F˜␧共r42;t44⬘,t24⬘兲−F˜␧共r12;t1,t2兲兴}. 共25兲

B. Evaluation of the decay functionF˜␧„t…

Let us now evaluate the decay functionF˜␧共t兲 explicitly;

after three simplifying approximations, we shall find that it reproduces the functionF_d,crw^pp 共t兲of Eq.共I.65兲.

Our first simplifying approximation is as follows: instead of trying to solve Eq.共25兲in general, we shall be content to determine the decay function F˜␧ only to linear order in the self-energy, in accord with the fact that we “linearize” the latter by replacing the full self-energy by the bare one.共In- cluding nonlinear contributions would add terms that are smaller than those kept by powers of the small parameter 1/g.兲To this end, it suffices to linearize Eq. 共25兲in F˜␧, by dropping the 共ⵜr₁F˜␧兲² term on the left-hand side and the exponential factore^{−关F˜42␧−F˜12␧兴} on the right-hand side, and re- placing ⌺˜

full by ⌺˜

bare. One readily finds that the resulting linearized equation is solved by

F˜␧共r12;t₁,t₂兲= −C˜共1兲␧共r12;t₁,t₂兲

C˜⁰共r12,t₁₂兲 , 共26兲 whereC˜共1兲␧is given by Eq.共23b兲. Thus, the expansion ofC˜␧

关Eq. 共24兲兴 to first order in F˜␧ reproduces Eq. 共23a兲, as it should, and conversely, Eq.共24兲turns out to be nothing but the reexponentiated version of Eq.共23a兲. Our explicit solu- tion of the Bethe-Salpeter equation, to linear-in-⌺˜ accuracy in the exponent, thus very nicely confirms the heuristic analysis presented in Sec. III D of Paper I in favor of reex- ponentiation strategies.

The second approximation is necessitated by the first:

upon comparing with the structure of the self-energy-only solution关Eq.共14a兲兴, and following the discussion before Eq.

共15兲, we recognize that effectively only a part ofC^共1兲␧ may be reexponentiated共note that this remark would be irrelevant if we were able to find theexact F˜␧兲. Therefore, when evalu- ating C˜共1兲␧ explicitly from Eq. 共23b兲, we insert ⌺¯

bare=⌺¯

bare self

+⌺¯

bare

vert 关Eqs. 共A5a兲兴 into Eq. 共23d兲, but for the self-energy

term ⌺¯

bare

self 关Eq. 共A5b兲兴, we retain only the “decoherence”

contribution⌺¯

bare

self,dec关Eq.共A5c兲兴, because⌺¯

bare

self,Z关Eq.共A5d兲兴 contributes only to the renormalization of the overall ampli- tude of the cooperon共and in any case, its time dependence for long times turns out to be weaker than that arising from

⌺

¯bareself,dec, as is checked explicitly in Appendix C 2兲. In other words, we write C˜共1兲,␧=C˜

dec 共1兲,␧+C˜

self,Z

共1兲,␧ and drop the second term. The resulting expression forC˜

dec 共1兲,␧reads

C˜

dec共1兲␧共r12;t₁,t₂兲= 1

ប

冕

^{共dq兲共d␻兲共dq¯兲共d␻¯兲}

⫻e^iqr12e^−i␻t12兵C¯

q 0共␻兲⌺¯

q,q¯,bare

␧,self,dec共␻^,␻^¯兲C¯

q 0共␻兲 +e^{i␻¯ t˜12}¯C

q

0共␻⁻␻^¯兲⌺¯

q,q¯,bare

␧,vert 共␻^,␻^¯兲C¯

q

0共␻⁺␻^¯兲其.

共27兲 关The quickest way to arrive at Eq. 共27兲 is from the second term of Eq. 共8兲, with C¯

q

E共⍀3,⍀2兲→2␲␦共⍀23兲C¯

q

0共⍀2兲 on the right-hand side, and ⌺¯

full→⌺¯

bare, given by Eqs.

共A5a兲–共A5f兲.兴

Our third approximation for evaluating the first-order de- coherence correction to the cooperon 共and thus the decay function兲consists in retaining only its dominating long-time behavior, for Tt₁₂1. In this limit, terms of order␻/T are Ⰶ1 and may be neglected 关they produce subleading contri- butions forF˜␧共t兲, as is checked explicitly in Appendix C 1兴.

This allows us to keep only the␻= 0 component of the ef- fective environmental propagator L¯

E␻,q¯

dec 共␻^¯兲 关Eq. 共A5f兲兴, which is contained in both ⌺¯

q,q¯,bare

␧,self,dec

and ⌺¯

q,q¯,bare

␧,vert 关see Eqs.

共A5c兲and共A5e兲兴. More formally, after substituting the latter two equations for the⌺¯’s occurring in Eq.共27兲and symme- trizing the integrand with respect to␻^¯↔−␻^¯, we Taylor ex- pandL¯

␧␻,q¯

dec 共␻^¯兲in powers of␻and represent␻^{as i}⳵t₁₂under the Fourier integral, thereby bringing Eq.共27兲into the form

C˜

dec共1兲␧共r12;t₁,t₂兲= −

兺

n=0

⬁

⳵t₁₂ n 1

ប

冕

^{共dq¯兲共d␻¯兲}

⫻L¯

␧共n兲,q¯

dec 共␻^¯兲C˜⁰共r12,t₁₂兲P¯

共r₁₂;t₁,t₂兲 crw 共q¯,␻^¯兲.

共28兲 Its ingredients are defined as follows:

L¯

␧共n兲,q¯

dec 共␻^¯兲=共i⳵␻兲ⁿ 2n! 关L¯

␧␻,q¯ dec 共␻^¯兲+L¯

␧␻,q¯

dec 共−␻^¯兲兴_␻=0, 共29兲

P

¯共crwr₁₂;t₁,t₂兲共q¯,␻^¯兲= 2

冕

^{共d␻兲共dq兲e}_C^˜iqr0_共r¹²₁₂^e−i_,t^␻₁₂^t12_兲

⫻关C¯

q 0共␻兲C¯

q−q¯

0 共␻⁻␻^¯兲C¯

q 0共␻兲

−e^{i␻¯ t˜12}¯C

q

0共␻⁻␻^¯兲C¯

q−q0¯共␻兲C¯

q

0共␻⁺␻^¯兲兴 共30a兲