Non-Markovian dephasing of disordered, quasi-one-dimensional fermion systems
NNon-Markovian dephasing of disordered, quasi-one-dimensional fermion systems
Seth M. Davis and Matthew S. Foster
1, 2 Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA (Dated: April 17, 2020)As a potential window into the transition between the ergodic and many-body-localized phases, westudy the dephasing of weakly disordered, quasi-one-dimensional fermion systems due to a diffusive,non-Markovian noise bath. Such a bath is self-generated by the fermions, via inelastic scatteringmediated by short-ranged interactions. The ergodic (many-body-delocalized) phase can be definedby the nonzero dephasing rate, which makes transport incoherent and classical on long lengthscales. We calculate the dephasing of weak localization perturbatively through second order inthe bath coupling, obtaining a short-time expansion. However, no well-defined dephasing ratecan be identified, and the expansion breaks down at long times. This perturbative expansion is not stabilized by including a mean-field Cooperon “mass” (decay rate), signaling a failure of theself-consistent Born approximation. We also consider a many-channel quantum wire where short-ranged, spin-exchange interactions coexist with screened Coulomb interactions. We calculate thedephasing rate, treating the short-ranged interactions perturbatively and the Coulomb interactionexactly. The latter provides a physical infrared regularization that stabilizes perturbation theoryat long times, giving the first controlled calculation of quasi-1D dephasing due to diffusive noise.At first order in the diffusive bath coupling, we find an enhancement of the dephasing rate, but atsecond order we find a rephasing contribution. Our results differ qualitatively from those obtainedvia self-consistent calculations commonly employed in higher dimensions. Our results are relevantin two different contexts. First, in the search for precursors to many-body localization in theergodic phase of an isolated many-fermion system. Second, our results provide a mechanism for theenhancement of dephasing at low temperatures in spin SU(2)-symmetric quantum wires, beyondthe Altshuler-Aronov-Khmelnitsky result. The enhancement is possible due to the amplification ofthe triplet-channel interaction strength, and provides an additional physical mechanism that couldcontribute to the experimentally observed low-temperature saturation of the dephasing time.
Contents
I. Introduction
II. Dephasing of weak localization (review)
III. Dephasing by a diffusive bath:Perturbation theory IV. Dephasing due to combined diffusive andMarkovian baths: Results
V. Dephasing due to combined diffusive andMarkovian baths: Calculation
VI. Discussion and Conclusion
Acknowledgments A. Vertex operator correlators B. Further details on the purely diffusive bathcalculation C. Further details on the coexisting bathcalculation
D. Perturbation theory for Coulombdephasing E. Diagram folding F. Series acceleration G. Fermionic field theory a r X i v : . [ c ond - m a t . d i s - nn ] A p r
4. Coexisting interaction baths 29
References I. INTRODUCTION
Inelastic collisions between electrons tend to destroyquantum phase coherence in a phenomenon called de-phasing . Dephasing is a key physical process underlyingthe transition between the quantum and classical trans-port regimes in many-body fermion systems and thus iscentral to modern efforts in condensed matter and quan-tum information to understand and exploit macroscopicquantum phenomena.It was understood in the 1980s that quantum in-terference effects in electronic systems induced byweak quenched disorder are governed by the dephasingtimescale τ φ [1–3]. This timescale determines the in-frared cutoff for the weak (anti) localization correctionto transport, which diverges in one or two spatial dimen-sions in the absence of dephasing. The dephasing ratecan be measured through the temperature dependenceof the conductance [3, 4].The dephasing rate for inelastic electron scatteringmediated by dynamically screened Coulomb interactionswas calculated exactly by Altshuler, Aronov, and Khmel-nitsky (AAK) [1], who obtained a τ φ ∼ T − / power lawin temperature ( T ) for quasi-one-dimensional (quasi-1D)many-channel wires. Although this result has been well-confirmed experimentally [4], measurements observing ananomalous low-temperature saturation of τ φ sparked adecade of controversy [4–16]. Plausible explanations forthe saturation include additional phase-breaking due toKondo impurities [13, 14]. The role of itinerant electronspin-exchange scattering and its effect on dephasing [17]in these quasi-1D wires was not extensively investigatedat the time.The theoretical challenges imposed by many body lo-calization (MBL) [18–21] invite us to revisit some ofthese questions. The MBL hypothesis proposes that aninteracting, disordered quantum system can undergo anonzero-temperature transition from the semiclassical er-godic (metallic) phase into an insulating state that failsto self-thermalize. In the MBL phase, local operatorsare long-lived and quantum coherence is not destroyedby dephasing [18, 20, 21]. Understanding the nuancesof dephasing in the ergodic phase could uncover precur-sors to MBL, or even yield an analytical tool for study-ing the ergodic-to-MBL transition [22, 23]. Recent workhas raised concerns about the feasibility of accessing thistransition numerically [24–26], which places a renewedurgency upon identifying analytical approaches that arenot limited to small system sizes.In this work, we revisit quantum coherence in quasi-1D fermion transport, and focus specifically on dephas-ing due to short-ranged inelastic scattering. This isrelevant for neutral ultracold atomic fermion systems that could be platforms for MBL realization. Further-more, in spin SU(2)-symmetric quantum wires, inelas-tic electron-electron scattering is mediated by the com-bination of both short-ranged spin-triplet exchange- andCoulomb-interactions [17]. MBL has been primarily in-vestigated for fermion systems with short-ranged interac-tions [18, 19, 27]. Inelastic scattering due to short-rangedinteractions gives rise to a strongly non-Markovian , dif-fusive noise kernel in the ergodic phase [22, 28]. Bycontrast, the exact solution for τ φ obtained by AAK re-lies crucially on the Markovian nature of the noise baththat arises from Coulomb interactions [1]. As we showin this work, the Markovian case is drastically simplerthan the generic case. An additional reason to revisitdephasing due to spin exchange interactions is the well-known but poorly understood enhancement of the tripletchannel interaction in the theory of the zero-temperatureAnderson-Mott metal-insulator transition (MIT) [29–32].An enhancement of spin exchange interactions could leadto an important role of non-Markovianity near a MIT.The problem of dephasing weak localization by a dif-fusive bath is equivalent to solving a strongly-coupled,auxiliary quantum field theory. The upper critical spa-tial dimension of this theory is d = 4, and previous workusing a d = 4 − (cid:15) expansion identified a nontrivial criticalpoint that could signal a failure of dephasing [22]. Thecritical point obtains from vertex corrections that are notcaptured by the standard self-consistent Born approxi-mation (SCBA) [4, 22, 33, 34]. A key goal of this work isto test the veracity of the SCBA and other self-consistentapproximations.Here we present two different calculations for the de-phasing of weak localization in a quasi-1D wire due toa diffusive bath. In both cases, we consider “order one”strength interactions and weak disorder [1–4]. This is incontrast to the strongly disordered, weakly interactinglimit considered by Basko, Aleiner, and Altshuler (BAA)[18]. The weak-disorder limit minimizes the “bad metal”regime of the ergodic phase discussed by BAA, contract-ing the low-temperature window in which the putativeergodic-to-MBL transition could occur [28].First, for an isolated fermion system with short-rangedinteractions, we calculate the dephasing through sec-ond order in the bath coupling, expanding about theun-dephased Cooperon. We obtain a short-virtual-timeexpansion, unplagued by divergences (a feature uniqueto 1D). However, no well-defined dephasing rate can beidentified, and the expansion breaks down at long times.This perturbative expansion is not stabilized by includ-ing a mean-field Cooperon “mass” (decay rate), signalinga failure of the SCBA. Although the expansion breaksdown at long times, it contains interesting features; wefind that the second-order term in the expansion has apositive sign and actually works against dephasing. Wecall such a term rephasing . This calculation demonstratesthat the long-time behavior due to purely diffusive de-phasing cannot be accessed perturbatively.Second, we consider a many-channel quantum wirewhere short-ranged, spin-exchange interactions coexistwith screened Coulomb interactions. We calculate the de-phasing rate, treating the non-Markovian diffusive bathperturbatively and the Markovian Coulomb bath exactlyvia an extension of the AAK technique [1]. The latterprovides a physical infrared regularization that, unlikethe SCBA, stabilizes perturbation theory at long times.The expansion parameter is the dimensionless ratio of thetwo bath coupling strengths. At first order, the diffusivebath enhances the Markovian AAK dephasing rate. Atsecond order, however, we again find a rephasing con-tribution. Taken together with the short-time expan-sion result, this suggests that higher-order terms couldhave important effects in the purely diffusive limit, ca-pable of slowing or even arresting dephasing. This ex-pansion provides the first controlled calculation of thedephasing effects due to a diffusive bath for a quasi-1D system. Our results disagree qualitatively with self-consistent schemes, commonly employed in higher dimen-sional dephasing calculations, which we show give the in-correct dependence on the bath coupling strength. Inparticular, we show that self-consistent calculations in-correctly predict a suppression of the effects of the diffu-sive bath in the strong-coupling limit.Finally, we also describe how the low-temperature en-hancement of the spin-triplet interaction strength [29–31]can translate into an enhancement of the AAK dephasinglaw τ φ ∼ T − / [1], providing a new mechanism for theapparent saturation of phase-breaking in quantum wires[4–16].Our calculation uses nonstandard techniques and ap-plies generally to any set of coexisting Markovian andnon-Markovian noise baths, and so we give a pedagogi-cal presentation. A. Outline
This paper is organized as follows. Sec. II introducesthe basics of dephasing and reviews the AAK solution forthe Markovian case. In Sec. III, we perturbatively studydephasing due to a purely diffusive bath. We presentour results for coexisting diffusive and Coulomb (Marko-vian) baths in Sec. IV, and discuss their relevance tounderstanding the purely diffusive limit. Sec. V pro-vides an overview of the dual-bath dephasing calcula-tion. This calculation exploits the Airy functions used tosolve the quasi-1D Markovian problem exactly [1]. Theseries expansion for the dephasing rate is expressed interms of amplitudes that involve sums of integrals overproducts of Airy functions; these integrals are ultimatelycalculated numerically. Finally, we discuss our results inSec. VI, including their possible implications for MBLphysics and for the apparent saturation of dephasing inquantum wires.Various technical details are relegated to Appendices.Appendix A collects Gaussian correlator results for ver-tex operators that appear throughout this work. Ap- pendix B provides additional details for the pure dif-fusive bath calculation in Sec. III. Appendices C andE present details for the dual-bath calculation summa-rized in Secs. IV and V. Appendix D applies perturba-tive techniques to the well-understood screened Coulomblimit for the sake of comparison. Appendix F explainsa “series acceleration” technique used to efficiently sumthe Airy function amplitudes that arise in the dual-bathcalculation. Finally, Appendix G explains an alterna-tive field theory approach to the Markovian and non-Markovian dephasing problems, which was exploited inthe (4 − (cid:15) ) expansion calculation for the diffusive bath inRef. [22]. Here we highlight mathematical differences be-tween generic dephasing and the Markovian limit. Wealso show how the AAK result can be derived as aninfinite-order diagrammatic resummation. II. DEPHASING OF WEAK LOCALIZATION(REVIEW)
The weak (anti)localization correction to the dc con-ductivity is determined by the Cooperon, a propagatordefined via the stochastic equation of motion [1, 28, 33,34] (cid:26) ∂ η − D ∇ + i (cid:104) φ cl (cid:16) t + η , x (cid:17) − φ cl (cid:16) t − η , x (cid:17)(cid:105)(cid:27) × c tη,η (cid:48) ( x , x (cid:48) ) = D δ ( η − η (cid:48) ) δ ( d ) ( x − x (cid:48) ) . (2.1)In Eq. (2.1), c tη,η (cid:48) ( x , x (cid:48) ) denotes the Cooperon, t and { η, η (cid:48) } respectively denote center-of- and relative-timearguments, { x , x (cid:48) } are position coordinates in d spatialdimensions, D is the classical diffusion constant due toelastic impurity scattering, and φ cl is the “classical” com-ponent of the scalar potential (as opposed to the quantumcomponent, in the Keldysh formalism [28, 35, 36]). φ cl isa Gaussian stochastic field defined by the correlator (cid:104) φ cl ( ω, k ) φ cl ( − ω, − k ) (cid:105) ≡ ∆( ω, k )= coth (cid:18) ω k B T (cid:19) ρ ( ω, k ) , (2.2)where ∆( ω, k ) is the noise kernel (Keldysh propagator);the spectral function for the bath is ρ ( ω, k ). The noisekernel ∆ encodes real inelastic fermion-fermion scatteringprocesses that are responsible for dephasing and is self-generated by thermal fluctuations of the particle density[1, 4, 28]. The Cooperon arises from interference betweenquantum amplitudes for forward- and backward-in-timepropagating paths in a disordered conductor [2–4]. Thescalar potential φ cl couples to both the causal ( t + η/ t − η/
2) paths, where t is the global timevariable and η is a virtual time argument.The weak (anti)localization correction to the conduc-tivity obtains from [2] δσ = (cid:40) ( − e /π (cid:126) ) P , WL , (+2 e /π (cid:126) ) P , WAL , P = ∞ (cid:90) dη c ( η ) , (2.3)where c ( η ) ≡ (cid:104) c tη, − η ( x , x ) (cid:105) , which is independent of t, x due to the bath averaging (cid:104)· · · (cid:105) . Above, P is the vir-tual return probability and W(A)L corresponds to weak(anti)localization, relevant for the case without (with)spin-orbit coupling. Here we focus on the spin SU(2)-symmetric case, corresponding to WL. Importantly, theCooperon in the un-dephased limit (i.e. φ cl = 0) is givenby c ( η ) = ( D/ πDη ) − d/ , so that in one or two dimensions the integral in Eq. (2.3) diverges in the in-frared, signaling Anderson localization in noninteractingsystems. However, for interacting particles, the pres-ence of φ cl generates a finite decay timescale for thebath-averaged Cooperon, ensuring the convergence ofEq. (2.3). This allows for the definition of the dephasingtime τ φ = − lim η →∞ η log (cid:2) c ( η ) (cid:3) . (2.4)The bath-averaged Cooperon can be expressed via aFeynman path integral [1], (cid:104) c tη,η (cid:48) ( x , x (cid:48) ) (cid:105) = D r ( η )= x (cid:90) r ( η (cid:48) )= x (cid:48) D r ( τ ) exp − D η (cid:90) η (cid:48) dτ [˙ r ( τ )] − η (cid:90) η (cid:48) dτ η (cid:90) η (cid:48) dτ ˜∆ (cid:20) τ − τ , r ( τ ) − r ( τ ) (cid:21) − ˜∆ (cid:20) τ + τ , r ( τ ) − r ( τ ) (cid:21) . (2.5)Setting x (cid:48) = x , η (cid:48) = − η and moving into “relative time” and “center-of-time” coordinates, defined by ρ ( τ ) ≡ r ( τ ) − r ( − τ ) , and R ( τ ) ≡
12 [ r ( τ ) + r ( − τ )] , (2.6)we have c ( η ) = D (cid:90) d R ( η )= x (cid:90) R (0)= R D R ( τ ) ρ ( η )= (cid:90) ρ (0)= D ρ ( τ ) exp − D η (cid:90) dτ (cid:104) ˙ R ( τ ) (cid:105) − D η (cid:90) dτ [ ˙ ρ ( τ )] − S I [ R ( τ ) , ρ ( τ )] , (2.7)where the contribution to the action of the noise kernel is given by S I [ R ( τ ) , ρ ( τ )] = (cid:90) dω π (cid:90) d d k (2 π ) d ∆( ω, k ) η (cid:90) dτ a η (cid:90) dτ b (cid:104) e − iω ( τ a − τ b ) / − e − iω ( τ a + τ b ) / (cid:105) × e i k · [ R ( τ a ) − R ( τ b )] sin (cid:20) k · ρ ( τ a )2 (cid:21) sin (cid:20) k · ρ ( τ b )2 (cid:21) . (2.8)In Eq. (2.7), R is the free boundary condition of the center-of-mass coordinate R ( τ ) at τ = 0. By contrast, therelative coordinate ρ ( τ ) satisfies Dirichlet boundary conditions: ρ ( η ) = ρ (0) = 0 [37]. A. Review of Markovian dephasing
There is a massive reduction in the complexity of the problem in the Markovian case of a frequency-independentbath kernel [1]. In this case, ˜∆( t, x ) = δ ( t ) ˜∆ M ( x ), which removes the direct time-dependence of the bath in Eq. (2.5).Explicitly, if ∆( ω, k ) = ∆ M ( k ), Eq. (2.8) simplifies to S I [ R ( τ ) , ρ ( τ )] → S M [ ρ ( τ )] ≡ (cid:90) d d k (2 π ) d ∆ M ( k ) η (cid:90) dτ (cid:26) sin (cid:20) k · ρ ( τ )2 (cid:21)(cid:27) = η (cid:90) dτ (cid:110) ˜∆ M [0] − ˜∆ M [ ρ ( τ )] (cid:111) . (2.9)In the Markovian limit, the action S M has no depen- dence on the field R ( τ ), and the R -path integration isequal to one. The path integral in Eq. (2.7) reduces tothe propagator for a single-particle quantum mechanicsproblem, c M ( η ) = D (cid:104) ρ = 0 | e − ˆ hη | ρ = 0 (cid:105) , (2.10)where we have defined the single-particle central-potential Hamiltonianˆ h ≡ − D ∇ ρ + ˜∆ M [0] − ˜∆ M [ ρ ] . (2.11)This simplification can also be seen in a field theory ap-proach [22]. In that framework, only a set of maximallycrossed rainbow diagrams (a subset of the SCBA) con-tribute [Appendix G 3]. B. Dynamically screened Coulomb interactions
Here we review dephasing due to dynamically screenedCoulomb interactions [1], focusing on many-channel,quasi-1D wires. The noise kernel is∆ M ( ω, k ) = − (cid:18) ω k B T (cid:19) Im (cid:34) V ( k )1 − D (0) R ( ω, k ) V ( k ) (cid:35) (cid:39) (cid:18) k B Tκ (cid:19) Dk , (2.12)where D (0) R ( ω, k ) = − κ Dk Dk − iω (2.13)is the semiclassical, retarded polarization function de-scribing density diffusion in the disordered conductor, κ is the bare compressibility, and V ( k ) is the bare three-dimensional Coulomb potential.The approximation in Eq. (2.12) is twofold. First, wetake | D (0) R ( ω, k ) V ( k ) | large compared to one, due to theplasmonic (logarithmic) enhancement of V ( k ) as k → ω/k B T ,coth ( ω/ k B T ) (cid:39) k B T /ω + O ( ω/k B T ) . (2.14)By cutting out high-frequency processes, we introducea short-range ultraviolet cutoff for the bath. This isjustified because interaction-mediated processes with | ω | larger than k B T contribute only to the conductivity viathe virtual Altshuler-Aronov correction [4]. A formal cal-culation retaining higher frequencies would not expandthe coth and keep the full quantum form of the noisekernel. In order to avoid inconsistency, in the latter caseit is necessary to also retain Pauli-blocking countertermsthat we have dropped here [10, 11]. These terms playeda role in the theoretical controversy concerning the ob-served low-temperature saturation of the dephasing ratein 1D systems [4, 5, 8, 10]. The expansion in Eq. (2.12) re-placing the quantum bath with classical Johnson-Nyquist noise is that of AAK and is physically correct [1, 4, 10].Eq. (2.12) implies that screened Coulomb interactionscan be well-approximated by a Markovian kernel, so thatthe bath-averaged Cooperon obtains from Eqs. (2.10) and(2.11). The effective central potential is˜∆ M (0) − ˜∆ M ( ρ ) = Γ M D | ρ | , (2.15)where we define the coupling constantΓ M ≡ k B Tκ , (2.16)which sets an intrinsic length scale a ≡ (cid:18) Γ M D (cid:19) − / = (cid:18) κ D k B T (cid:19) / . (2.17)It follows from Eqs. (2.11) and (2.15) that the one-dimensional Cooperon is the imaginary-time propagatorfor the single-particle Hamiltonianˆ h = D (cid:18) − d dρ + | ρ | a (cid:19) . (2.18)Diagonalizing this Hamiltonian gives the bound state en-ergies ε n = − α (cid:48) n (cid:0) D/a (cid:1) ,ε n +1 = − α n (cid:0) D/a (cid:1) , (2.19)and the orthonormal eigenfunctions [40] ψ n ( ρ ; a ) = 1 √ a | α (cid:48) n | − / Ai( α (cid:48) n ) Ai (cid:18) | ρ | a + α (cid:48) n (cid:19) , (2.20a) ψ n +1 ( ρ ; a ) = 1 √ a sgn( ρ )Ai (cid:48) ( α n ) Ai (cid:18) | ρ | a + α n (cid:19) , (2.20b)where n ∈ { , , , ... } , and α n and α (cid:48) n respectively denotethe ( n + 1) th (strictly negative) zero of the Airy functionAi( z ) or its derivative Ai (cid:48) ( z ) ≡ ( d/dz ) Ai( z ).The eigenfunctions in Eqs. (2.20a) and (2.20b) al-low the explicit computation of the expectation value inEq. (2.10) for c M ( η ) in one dimension: c M ( η ) = D a ∞ (cid:88) j =0 | α (cid:48) j | exp (cid:18) − ηDa | α (cid:48) j | (cid:19) ≡ D a f (cid:18) ηDa (cid:19) . (2.21)We note that ( ηD/a ) is a dimensionless time variableand that the sum in Eq. (2.21) is only over the even-parityenergies, since the odd-parity wavefunctions vanish at theorigin. This yields the AAK dephasing timescale τ φ (cid:39) (cid:34) D (cid:18) k B Tκ (cid:19) (cid:35) − / . (2.22)Finally, the return probability is P = ∞ (cid:90) dη c M ( η ) = a ∞ (cid:88) n =0 α (cid:48) n ) = 14 (cid:18) κ D k B T (cid:19) / π / Γ (2 / . (2.23)Via Eq. (2.3), this gives the famous AAK result forthe weak localization correction to the conductivity ofa quasi-1D wire, δσ WL ∝ − ( k B T ) − / [1].The summation over Airy derivative function zeros inEq. (2.23) is a known identity [40]. In the sequel, we willneed to be able to numerically evaluate similar sums,which are slowly convergent. To do so efficiently, we willintroduce a “series acceleration” technique [Appendix F]. III. DEPHASING BY A DIFFUSIVE BATH:PERTURBATION THEORY
In this section, we attempt to understand dephasingin a setting intimately related to many-body localization(MBL) [18, 19, 27]. We study the dephasing of the er-godic phase of an MBL candidate system by perturba-tively evaluating Eq. (2.5) in the presence of a diffusivebath.Our calculation models a 1D ultracold Fermi gas withshort-ranged interactions. We focus on the many-channelversion with weak disorder, so that weak localization the-ory applies at intermediate temperatures [1–4]. Hydro-dynamic modes in the ergodic phase of a dirty fermionsystem are generally diffusive [29]. As a result, in the caseof short-range interactions, the noise bath governing thethermalization of the system is also diffusive [17, 28].The diffusive noise kernel is∆ t ( ω, k ) = Γ t (cid:18) D t k D t k + ω (cid:19) . (3.1)For | ω | (cid:46) k B T , this is the approximate semiclassical,diffusive Keldysh propagator for particle density fluctua-tions in the fermion gas with quenched disorder, as arisesdue to short-ranged inelastic particle-particle collisions[28]. The coupling constant isΓ t = 3 γ t k B T (1 − γ t ) χ , (3.2)where γ t is the dimensionless interaction strength(Finkel’stein coupling parameter [28, 29]).In the sequel, we will consider coexisting diffusiveand Markovian baths, with the former (latter) mediated by short-ranged spin exchange (dynamically screenedCoulomb) interactions. In that context, γ t denotesthe spin triplet channel interaction strength (hence the“ t ” subscript), while χ is the bare spin susceptibil-ity. For contact interactions in an ultracold Fermi gas, χ is the compressibility. The diffusion constant D t inEq. (3.1) differs from the bare one entering the Cooperon[Eq. (2.1)], due to an interaction renormalization [28, 29], β ≡ D t D = 11 − γ t . (3.3)Here and throughout this paper, we will use the sym-bol β to refer to this dimensionless ratio (and not theinverse temperature). In the context of itinerant spinexchange interactions in a quantum wire, one typicallyhas an attractive spin-triplet channel coupling strength γ t <
0. The Stoner instability towards ferromagnetismcorresponds to the limit γ t → −∞ [29]. Repulsive inter-actions instead give γ t > γ t → R ( τ ) and relative ρ ( τ ) coordinates. In this sec-tion, we present a purely perturbative calculation for de-phasing due to the diffusive bath in Eq. (3.1). Withoutmoving to the center-of-time and relative coordinates, weevaluate the Cooperon in Eq. (2.5) via the cumulant ex-pansion, c t ( η ) = D r ( η )=0 (cid:90) r ( − η )=0 D r ( τ ) e − D η (cid:82) − η dτ ˙ r ( τ ) − S I [ r ( τ )] (3.4)= c ( η ) exp (cid:20) −(cid:104) S I (cid:105) + 12 (cid:16) (cid:10) S I (cid:11) − (cid:104) S I (cid:105) (cid:17) + . . . (cid:21) , FIG. 1: The coefficient function G ( β ) for the first-order cu-mulant expansion result in Eqs. (3.7) and (3.8), which de-termines the lowest-order superexponential dephasing of theCooperon due to the diffusive bath [Eq. (3.4)]. Here β is theratio of the interacting and bare diffusion constants definedby Eq. (3.3). where the bare Cooperon is c ( η ) = ( D/ πDη ) − / , (3.5) (cid:104)· · · (cid:105) denotes a functional average with respect to thenoiseless action, and the bath-induced interaction is S I [ r ( τ )] = Γ t (cid:90) dk π η (cid:90) − η dτ a η (cid:90) − η dτ b e ik [ r ( τ a ) − r ( τ b )] × (cid:104) e − D t k | τ a − τ b | / − e − D t k | τ a + τ b | / (cid:105) . (3.6)The cumulant expansion in the bath coupling Γ t boilsdown to the computation of the expectation values (cid:104) S nI (cid:105) .At first order, we only need to compute (cid:104) S I (cid:105) . We notethat the functional average over r ( τ ) affects only the ex-ponential factor in the top line of Eq. (3.6). Performingthis average to obtain the vertex operator correlator [Ap-pendix A] as well as the Gaussian integral over k gives (cid:104) S I (cid:105) = (cid:18) Γ t η / √ D (cid:19) G ( β ) , (3.7)where G ( β ) is a dimensionless function of the diffusion constant ratio β [Eq. (3.3)], G ( β ) = (cid:114) π (cid:90) − dτ a (cid:90) τ a dτ b (cid:34) g ( β, τ a , τ b ) − / − g ( β, τ a , τ b ) − / (cid:35) , (3.8)and where the g , functions are defined by g ( β, τ a , τ b ) = ( β + 1)( τ b − τ a ) −
12 ( τ b − τ a ) ,g ( β, τ a , τ b ) = β | τ b + τ a | + ( τ b − τ a ) −
12 ( τ b − τ a ) . (3.9)The amplitude G ( β ) is plotted in Fig. 1. Since G ( β ) ispositive, the action of the bath is to suppress (dephase)the Cooperon with increasing virtual time η . However,Eqs. (3.7) and (3.4) predict a superexponential damp-ening of the Cooperon, and thus do not define a finitedephasing rate.To better interpret the first-order result, we continuethe calculation to second order, requiring the evaluationof the expectation value (cid:10) S I (cid:11) = Γ t (cid:90) dk π (cid:90) dk π × η (cid:90) − η dτ a η (cid:90) − η dτ b η (cid:90) − η dτ a η (cid:90) − η dτ b (cid:104) e − D t k | τ a − τ b | / − e − D t k | τ a + τ b | / (cid:105) (cid:104) e − D t k | τ a − τ b | / − e − D t k | τ a + τ b | / (cid:105) × (cid:68) e ik [ r ( τ a ) − r ( τ b )] e ik [ r ( τ a ) − r ( τ b )] (cid:69) . (3.10)The 4-point vertex function correlator on the last line ofthis equation is evaluated in closed form in Appendix A.We note that Eq. (3.10) is invariant under the sym-metries τ a ↔ τ b , τ a ↔ τ b , and ( ω , k , τ a , τ b ) ↔ ( ω , k , τ a , τ b ). This 8-fold symmetry group leaves 3 dis-tinct topological classes of the 24 distinct time-orderings,and we may thus reduce to the case where τ a < τ b , τ a < τ b , and τ a < τ a . We define the three inequiva-lent time-sectors Ω s to beΩ s ≡ { τ a < τ b < τ a < τ b } s = 1 , { τ a < τ a < τ b < τ b } s = 2 , { τ a < τ a < τ b < τ b } s = 3 . (3.11)These correspond to the three diagrams shown in Fig. 2.The form of the functional average depends upon thetopological class of the time-ordering.Using the vertex operator correlator from Appendix A,the momentum integrals in Eq. (3.10) can be obtained in closed form. The final result can be expressed as follows, (cid:104) S I (cid:105) = Γ t η D (cid:104) G (1)2 ( β ) + G (2)2 ( β ) + G (3)2 ( β ) (cid:105) , (3.12)where G ( s )2 is a dimensionless function corresponding tothe topological sector s . These functions are definedexplicitly in Appendix B, as parametric integrals overrescaled { τ a , τ b , τ a , τ b } variables [Eq. (3.10)].The final result for the diffusive bath-averagedCooperon, computed through second order in the cumu-lant expansion, is given by c t ( η ) c ( η ) = exp − (cid:18) Γ t η / √ D (cid:19) G ( β )+ 12 (cid:18) Γ t η / √ D (cid:19) G ( T )2 ( β ) + O (Γ t ) , (3.13) FIG. 2: Diagrams giving the three topologically distinct con-tributions to the Cooperon due to the diffusive noise bath atsecond order in perturbation theory. These correspond to thetime-ordering sectors { Ω , , } in Eq. (3.11). From top to bot-tom, we have “sector 1” Ω , “sector 2” Ω , and “sector 3”Ω ; we also refer to these amplitudes as “double,” “crossed,”and “nested,” respectively. We show in Appendix A how theform of the correlator depends on the topology of the time-ordering. where G ( T )2 ( β ) ≡ G (1)2 ( β ) + G (2)2 ( β ) + G (3)2 ( β ) − [ G ( β )] . (3.14)The cumulant expansion in Eq. (3.13) is well-defined be-cause the integrals that determine the amplitude func-tions G ( β ) and G ( T )2 ( β ) are free of divergences in onedimension. (In 2D or higher, the cumulant expansion isplagued by UV divergences; these can be regularized byself-consistency, but see below.) Nevertheless, no finitedephasing rate as in Eq. (2.4) can be identified, becausethe expansion is a series in powers of (Γ t (cid:112) η /D ). More-over, this series evidently breaks down for long virtualtimes η (cid:38) ( D/ Γ t ) / , signaling a failure of perturbationtheory. The Cooperon is needed for arbitrarily large η ,in order to compute the weak localization correction inEq. (2.3).The most interesting aspect of the result in Eq. (3.13)is the sign of the second-order correction. As shown inFig. 3, the net coefficient G ( T )2 ( β ) is positive and nonzerofor β >
0. Fig. 4 shows that this is due to a competi-tion between terms coming from the different topologicalsectors, which do not cancel the square of the first-ordercoefficient. The range plotted in Figs. 3 and 4 corre-sponds to interparticle scattering due to attractive in-teractions γ t < G ( T )2 ( β ) remains positiveand nonzero for β >
1, corresponding to repulsive inter-actions. We conclude that at second order, the interac-tion of the Cooperon with the diffusive bath gives a net rephasing contribution, and this quickly overwhelms thefirst-order dephasing result when η (cid:38) ( D/ Γ t ) / . Theultimate fate of the Cooperon at long times requires anonperturbative treatment of the diffusive bath. FIG. 3: The coefficient function G ( T )2 ( β ) for the second-ordercumulant expansion result in Eqs. (3.13) and (3.14). Since G ( T )2 ( β ) >
0, this shows that the second-order cumulant givesa superexponential rephasing contribution to the Cooperon inEq. (3.13), due to the interaction with the diffusive bath. Here β is the ratio of the interacting and bare diffusion constantsdefined by Eq. (3.3). See Fig. 4 for the plot of the individualcomponents of Eq. (3.14) that yield the total coefficient. We note that the self-consistent Born approximation(SCBA) is not sufficient to stabilize these results. Inthe SCBA, one sums the set of all non-crossing diagrams(as defined using an alternative field theory language, seeAppendix G). This yields the self-consistent equation [22] τ − SCBA = 2 (cid:90) dω π (cid:90) d d k (2 π ) d ∆ t ( ω, k ) Dk − iω + τ − SCBA . (3.15)Evaluating this in 1D, using Eq. (3.1) gives τ − SCBA = (cid:18) Γ t D + D t (cid:19) / ∝ ( k B T ) / , (3.16)identical to the temperature dependence obtained byAAK [1] for the Markovian screened Coulomb bath[Eq. (2.22)]. However, adding the “mass” τ − SCBA to the
ABCD
FIG. 4: Plot of the individual contributions to thetotal second-order coefficient function G ( T )2 ( β ) defined byEq. (3.14), plotted in Fig. 3. The second order contributionto the cumulant expansion in Eq. (3.13) gives a net rephas-ing , due to the combination of the competing terms that donot exactly cancel out. The contributions are indicated as A:double diagram, G (1)2 ( β ), B: the square of first order term,[ G ( β )] , C: the crossed diagram, G (2)2 ( β ), and D: the nesteddiagram, G (3)2 ( β ) [see Fig. 2]. bare Cooperon c ( η ) merely appends the decaying expo-nential prefactor exp( − η/τ SCBA ) to the path integral inEq. (3.4). At large virtual times η → ∞ , linear dephasingis overpowered by the second-order terms contributing toEq. (3.13) that are neglected in the SCBA, and still givea nonzero contribution proportional to η .These results can be compared with those of AAKfor the screened Coulomb Markovian bath, reviewedin Sec. II B. In that case there is an exact solution[Eq. (2.21)]. However, one could instead employ a per-turbative calculation similar to the one presented above.Performing the cumulant expansion for the Markovianbath [Appendix D], one finds the same power-law be-havior in η seen above in Eq. (3.13) [10]. The η / -dependence is generic to perturbing around the bareCooperon in 1D, and is not tied to the diffusive char-acter of the bath.For the diffusive bath, we find that every order in thecumulant expansion is governed by a competition be-tween many dephasing and rephasing terms. Our second-order result in Eq. (3.13) demonstrates that rephasingdiagrams may dominate at any given order. By con-trast, the cumulant expansion for the Markovian bath[Appendix D] yields a single term at each order. Thisdifference in complexity can also be seen in the field the-ory description [Appendix G].The path integral Eq. (2.5) gives a strongly coupledfield theory [22] governing the dephasing of a systemwith a diffusive noise bath. The bare cumulant expansionbreaks down after short virtual times, so that a nonper-turbative technique is required to characterize the de-phasing of the system. However, the SCBA is not suffi-cient to stabilize the theory against additional perturba-tive corrections. In the next two sections, we employ anadditional Markovian noise bath (which we treat exactly)as an infrared regularization; this stabilizes the pertur-bation theory for the diffusive, non-Markovian bath atlong virtual times [see Eq. (4.1)].Finally, we note that the calculation presented in thissection can also be carried out in a field theory formalism,and that there is a well-defined mapping between theFeynman diagrams there and the different contributionsseen here in the cumulant expansion. This connection isdescribed in Appendix G. IV. DEPHASING DUE TO COMBINEDDIFFUSIVE AND MARKOVIAN BATHS:RESULTS
We demonstrated in Sec. III that a naive perturba-tive treatment of the diffusive noise bath modulating theCooperon in Eq. (2.5) is insufficient to determine thedephasing time [Eq. (2.4)]. We argued that the stan-dard partial summation of perturbation theory [the self-consistent Born approximation (SCBA)] does not stabi-lize the calculation against neglected perturbative correc-tions. More work is required to understand dephasing in
FIG. 5: The coefficient function C ( β ) for the first-order cu-mulant expansion result in Eq. (4.1), which describes dephas-ing due to inelastic spin-triplet exchange scattering, in thepresence of an additional screened-Coulomb Markovian bath.Here β is the ratio of the interacting and bare diffusion con-stants defined by Eq. (3.3). The coefficient C ( β ) is expressedin terms of a slowly-converging infinite sum in Eq. (5.15). Inorder to reliably approximate the result, here we have usedthe series acceleration technique described in Appendix F. Weplot the coefficient for 0 < β ≤
1, which corresponds to fer-romagnetic exchange interactions [ γ t < C ( β ) >
0, the first-order correction in Eq. (4.1)enhances the dephasing rate of the pure Markovian result in c M ( η ). such a weakly disordered, quasi-1D fermion system withpure short-ranged interactions.To understand the effects of the diffusive noise bath,in this section we study coexisting interactions. In par-ticular, we consider the diffusive kernel [Eq. (3.1)] in par-allel with the Markovian kernel in Eq. (2.15). This sce-nario corresponds to a quasi-1D, many-channel quantumwire with spin SU(2) symmetry, possessing both long-ranged Coulomb and short-ranged, spin exchange inter-actions. These interactions are respectively associated tothe charge and spin density hydrodynamic modes, andeach gives rise to its own noise bath that interacts withthe Cooperon [17, 28, 29].We treat the Markovian bath exactly, extending theAAK solution reviewed in Sec. II B, whilst simultaneouslyemploying the cumulant expansion [Eq. (3.4)] for the dif-fusive bath. We find that the Markovian bath provides aphysical infrared regularization of the terms computed inthe perturbative expansion for the diffusive bath. Unlikethe bare expansion presented in Sec. III or the SCBA,this regularization stabilizes perturbation theory at longvirtual times. This is due to nontrivial cancellations be-tween higher-order terms, detailed in Sec. V, with noanalogue in the bare expansion [Eq. (3.13)].At first order, we find that the diffusive bath enhancesthe dephasing rate, but at second order we again finda positive rephasing contribution. Interestingly, this ex-actly parallels the short-time expansion for the purelydiffusive bath in Eq. (3.13). By contrast to that calcula-tion, the results here obtain in the limit of large virtualtimes η → ∞ . In the latter limit, the dual bath-averaged0Cooperon can be cast in the form c ( η ) c M ( η ) = A ( β ) exp − ηDa (cid:18) Γ t Γ M (cid:19) C ( β ) − (cid:18) Γ t Γ M (cid:19) C ( T )2 ( β )+ O (cid:18) Γ t Γ M (cid:19) , (4.1)where the prefactor A ( β ) and rate coefficients C ( β ) and C ( T )2 ( β ) are dimensionless functions of the diffusion con-stant ratio β , defined by Eq. (3.3). In Eq. (4.1), c M ( η )is the exact result for the Markovian-dephased Cooperonin Eq. (2.21), while the coupling strengths Γ M and Γ t forthe Coulomb Markovian and spin-triplet diffusive bathswere defined by Eqs. (2.16) and (3.2), respectively.The rate coefficient functions C ( β ) and C ( T )2 ( β ) areboth positive, so that the former (latter) enhances (sup-presses) the dephasing relative to the Markovian result c M ( η ). Similar to the second-order coefficient in the bareexpansion for the diffusive bath [Eqs. (3.13) and (3.14)],the second-order rate coefficient obtains from a combi- A B C
FIG. 6: The coefficient function C ( T )2 ( β ) for the second-order cumulant expansion result in Eq. (4.1). This coeffi-cient is expressed as the combination of terms {C j ( β ) } shownin Eq. (4.2). The individual terms in the latter equationarise from the three different time-ordering sectors depicted inFig. 2, minus the square of the first-order result. The terms C ( β ) and C ( β ) appear already at first order [see Fig. 5],while the rest obtain exclusively from the second-order di-agrams. To evaluate the total coefficient in Eq. (4.2), theterms C and C are each approximated using the series ac-celeration technique described in Appendix F. The remainingterms {C , C , C } are each given by a triply-infinite summa-tion over Airy eigenfunction energy levels [Eq. (2.19)]. Herethey are estimated by series truncation after summing thefirst N ε energy levels. We plot C ( T )2 ( β ) for several values of N ε to show convergence. A: N ε = 1, B: N ε = 2, ( N ε = 3unlabeled), C: N ε = 4. We see that C ( T )2 ( β ) >
0, so that thesecond-order contribution to the dephasing rate in Eq. (4.1)is negative . In other words, the second-order contribution is rephasing . This is similar to the second-order correction tothe short-time expansion for the pure diffusive bath calcula-tion [Eq. (3.13) and Fig. 3]. nation of several terms, corresponding to contributionsfrom the three different time-ordering topologies expli-cated in Eqs. (3.11) and Fig. 2, minus the square of thefirst order result. The net result can be expressed by thecombination C ( T )2 ( β ) ≡ C ( β ) − C ( β ) C ( β ) + C ( β ) + C ( β ) , (4.2)where the components {C j ( β ) } are precisely defined inAppendix C 2. Each of the functions {C j ( β ) } can be ex-pressed through one or more infinite summations overthe Airy bound states that solve the Markovian prob-lem [Eqs. (2.19) and (2.20)]; summands are given by in-tegrals over matrix elements involving these eigenfunc-tions. Figs. 5 and 6 show that the sign of these functionsgive net dephasing and rephasing contributions at firstand second order, respectively. We note that the dimen-sionless perturbative parameter in the dual-bath cumu-lant expansion Eq. (4.1) is the ratio of the bath couplingconstants (Γ t / Γ M ), which is independent of temperature(unless further renormalization of the triplet couplingstrength is taken into account—see Sec. VI B), in con-trast to the pure diffusive bath expansion in Eq. (3.13).The fact that the second-order correction in Eq. (4.1)is rephasing is a main result of this section. This issimilar to the pure diffusive bath result in Eq. (3.13),except that Eq. (4.1) is well-defined in the long-virtual-time limit η → ∞ , and gives a valid dephasing time viaEq. (2.4). The advent of rephasing corrections due tothe diffusive bath beyond first order was anticipated bythe RG study in Ref. [22], which located a nontrivialfixed point in a d = 4 − (cid:15) expansion. The fixed pointarises due to vertex corrections neglected in the SCBA[Eq. (3.15)] that suppress the Cooperon-bath couplingstrength. Such corrections are absent in the Markoviancase [Appendix G 3]. A. Comparison with self-consistent calculations
In Sec. III, we argued that the perturbative calcula-tion of dephasing due to the purely diffusive bath is not stabilized by the SCBA [Eqs. (3.15) and (3.16)]. Here wediscuss how the long-virtual-time result for the combinedbaths obtained in Eq. (4.1) compares to self-consistentcalculations.On physical grounds (but see below), low frequencies | ω | < τ − φ are not expected to contribute to dephasing[17, 33, 34]. This motivates the self-consistent truncationof first-order perturbation theory,1 τ φ = 2 (cid:90) dk π ∞ (cid:90) τ − φ dω π Dk [( Dk ) + ω ] ∆( ω, k ) , (4.3)where ∆( ω, k ) is the noise kernel [ cf . the SCBA inEq. (3.15)]. Applying this procedure to the 1D dual-1bath Cooperon studied in this (and the next) section, wefind the result1 τ φ = 2 Dπ / a (cid:20) (cid:18) Γ t Γ M (cid:19) H ( β ) (cid:21) / = 2 Dπ / a (cid:18) Γ t Γ M (cid:19) H ( β ) − (cid:18) Γ t Γ M (cid:19) [ H ( β )] + O (cid:18) Γ t Γ M (cid:19) , (4.4)where H ( β ) ≡ √ β (1 + √ β )(1 + β ) . (4.5)This result shares some features with Eq. (4.1), in partic-ular the appearance of a second-order rephasing contri-bution. However, the numerical prefactor is incorrect forthe pure Coulomb contribution, and more importantlythe coefficient function at n th order [ H ( β )] n is qualita-tively incorrect. The function H ( β ) vanishes as β / inthe β → γ t → −∞ [Eq. (3.3)]. By contrast, the coeffi-cient functions C ( β ) and C ( T )2 ( β ) shown in Figs. 5 and 6asymptote to nonzero values as β →
0. This discrepancyis an order-of-limits issue; the η → ∞ and β → η ≤ /τ φ .On the other hand, the result in Eq. (4.1) with coeffi-cient functions { C ( β ) , C ( T )2 ( β ) } obtains only in the large η -limit. Since β ∝ D t , the effect of the diffusive bathmust vanish at β = 0, see Eq. (3.6). Eq. (4.1) is valid for Dη/a (cid:38) /β . The controlled result in Eq. (4.1) is par-ticularly relevant in the context of the RG enhancementof the spin exchange interaction γ t near a MIT [29–32].Another key point is that the second-order rephasingterm in the self-consistent result obtained in Eq. (4.4) de-pends sensitively on the cutting scheme, due to the stronginfrared divergence in Eq. (4.3). Instead of applying Eq. (4.3) simultaneously to the Markovian and diffusivebaths, we can instead use the AAK result for the dephas-ing time due to the Coulomb bath, ( τ AAKφ ) − = | α (cid:48) | D/a [Eq. (2.21)] to cut the frequency integral for the correc-tion due to the diffusive bath. This gives (cid:18) τ φ (cid:19) diff = 4 π (cid:18) Γ t √ D (cid:19) H ( β ) (cid:113) τ AAKφ = 2 / π | α (cid:48) | / (cid:18) Da (cid:19) (cid:18) Γ t Γ M (cid:19) H ( β ) , (4.6)with no second-order rephasing correction.We conclude that the sensitive dependence on the in-frared makes self-consistency even qualitatively incorrectfor the dephasing of quasi-1D systems due to a diffusivebath, with or without an additional regularizing Marko-vian bath. V. DEPHASING DUE TO COMBINEDDIFFUSIVE AND MARKOVIAN BATHS:CALCULATIONA. General method
In this section we provide an overview of the calcula-tion leading to the dual-bath result in Eq. (4.1). Thisarises due to inelastic electron-electron scattering medi-ated by both screened Coulomb and spin-triplet exchangeinteractions, encoded respectively in the Markovian AAKbath [Eq. (2.12)] and the diffusive (non-Markovian) bath[Eq. (3.1)]. We employ the same cumulant expansion asin Sec. III, expanding perturbatively in the diffusive bathwhilst treating the Markovian bath exactly. The latterrequires that we work in terms of the relative- ρ ( τ ) andcenter-of-time R ( τ ) coordinates [see Eq. (2.8)]. We de-fine c M ( η ) to be the exact bath-averaged Cooperon in thepure Markovian limit, given by Eqs. (2.10) and (2.21).The dual-bath-averaged Cooperon is expressed as thepath integral c ( η ) = D (cid:90) dR R ( η )= x (cid:90) R (0)= R D R ( τ ) ρ ( η )=0 (cid:90) ρ (0)=0 D ρ ( τ ) exp − η (cid:90) dτ (cid:26) D (cid:104) ˙ R ( τ ) (cid:105) + 14 D [ ˙ ρ ( τ )] + Γ M D (cid:12)(cid:12) ρ ( τ ) (cid:12)(cid:12)(cid:27) − S I [ R ( τ ) , ρ ( τ )] , (5.1)where S I is as in Eq. (2.8), with ∆( ω, k ) → ∆ t ( ω, k ) given by Eq. (3.1).We let (cid:104)· · · (cid:105) R and (cid:104)· · · (cid:105) ρ denote the averages with respect to the noiseless R ( τ ) and Markovian-bath-averaged ρ ( τ )actions, respectively. As in Sec. III, the cumulant expansion boils down to the calculation of expectation values of2powers of the perturbing action. In general, to do an n th order calculation, we must evaluate (cid:104) S nI (cid:105) =Γ nt (cid:90) dk π · · · (cid:90) dk n π η (cid:90) dτ a η (cid:90) dτ b · · · η (cid:90) dτ na η (cid:90) dτ nb × (cid:104) e − D t k | τ a − τ b | / − e − D t k ( τ a + τ b ) / (cid:105) × · · · × (cid:104) e − D t k | τ na − τ nb | / − e − D t k ( τ na + τ nb ) / (cid:105) × (cid:28) exp (cid:20) ik (cid:18) R ( τ a ) − R ( τ b ) (cid:19)(cid:21) × · · · × exp (cid:20) ik n (cid:18) R ( τ na ) − R ( τ nb ) (cid:19)(cid:21)(cid:29) R × (cid:28) sin (cid:20) k ρ ( τ a )2 (cid:21) sin (cid:20) k ρ ( τ b )2 (cid:21) × · · · × sin (cid:20) k n ρ ( τ na )2 (cid:21) sin (cid:20) k n ρ ( τ nb )2 (cid:21)(cid:29) ρ , (5.2)where the elementary frequency integrations have already been carried out via (cid:90) dω π (cid:18) D t k D t k + ω (cid:19) (cid:104) e − iω ( τ a − τ b ) / − e − iω ( τ a + τ b ) / (cid:105) = exp (cid:20) − D t k (cid:18) | τ b − τ a | (cid:19)(cid:21) − exp (cid:20) − D t k (cid:18) τ b + τ a (cid:19)(cid:21) ≡ ˜ T ( D t , k, τ a , τ b ) − ˜ T ( D t , k, τ a , τ b ) . (5.3)We have two functional averages to perform: F nR ( k (cid:48) s, τ (cid:48) s ) ≡ (cid:28) exp (cid:20) ik (cid:18) R ( τ a ) − R ( τ b ) (cid:19)(cid:21) × · · · × exp (cid:20) ik n (cid:18) R ( τ na ) − R ( τ nb ) (cid:19)(cid:21)(cid:29) R (5.4a) F nρ ( k (cid:48) s, τ (cid:48) s ) ≡ (cid:28) sin (cid:20) k ρ ( τ a )2 (cid:21) sin (cid:20) k ρ ( τ b )2 (cid:21) × · · · × sin (cid:20) k n ρ ( τ na )2 (cid:21) sin (cid:20) k n ρ ( τ nb )2 (cid:21)(cid:29) ρ . (5.4b)As in Sec. III, the path integral expectation values willhave nontrivial dependencies on the ordering of the timevariables. In general, there are (2 n )! such orderings, cor-responding to the permutation group S n acting on thetime variables. However, the integral is preserved undera subgroup of order ( n ! · n ), generated by the n ex-change operations τ ja ↔ τ jb and the n ! operations thatpermute ( k j , τ ja , τ jb ) → ( k σ ( j ) , τ σ ( j ) a , τ σ ( j ) b ) , for σ ∈ S n .With these symmetries in mind, we can restrict the τ -integration region so that τ a < τ a < · · · < τ na , and τ ja < τ jb for each j ∈ { , ..., n } . We thus only need toconsider (2 n )! / ( n ! · n ) topologically distinct time orderingsectors, which are in direct, one-to-one correspondencewith the topologically distinct diagrams contributing tothe Cooperon at n th order in the field theory description[see Appendix G]. We define these regions in τ -space as { Ω ns } [with 1 ≤ s ≤ (2 n )! / ( n ! · n )], generalizing the sec-ond order decomposition in Eq. (3.11) [see also Fig. 2].This folding of the integration region produces a leadingfactor of ( n ! · n ).The Gaussian functional average over R ( τ ) inEq. (5.4a) follows from Wick’s theorem, since the ver-tex operator products appearing in it are charge-neutral[see Appendix A]. While the ρ ( τ ) expectation value inEq. (5.4b) cannot be evaluated similarly due to the con-fining potential from the Markovian bath in Eq. (5.1),we can instead express it as an expansion in terms ofthe eigenfunctions { ψ j ( ρ ; a ) } in Eqs. (2.20a) and (2.20b).This gives 2 n + 1 distinct summations over the eigenen- ergies [Eq. (2.19)], and introduces the matrix elements S ij ( k ; a ) ≡ (cid:104) ε i | sin (cid:20) k ˆ ρ (cid:21) | ε j (cid:105) = ∞ (cid:90) −∞ dρ ψ i ( ρ ; a ) sin (cid:18) kρ (cid:19) ψ j ( ρ ; a ) . (5.5)Eq. (5.5) vanishes unless one of the eigenfunctions is evenand the other is odd, a parity selection rule. When eval-uating Eq. (5.4b), we need to keep in mind the Dirichletboundary conditions ρ (0) = ρ ( η ) = 0. Since the odd-parity eigenfunctions in Eq. (2.20b) vanish at the ori-gin, when Eq. (5.4b) is evaluated by inserting 2 n + 1resolutions of the identity, the first and last energies inthe Trotterization must have even parity. Moreover, theparity of ε j must correspond to the parity of j , for all j ∈ { , , , . . . , n } . We give the explicit general form ofthe ρ -correlator in Eq. (5.4b) in Appendix C 1.We can simplify our calculation by scaling our integra-tion variables to make them dimensionless,( τ, ρ, k ) → (cid:18) η τ, a ρ, ka (cid:19) , (5.6)where η is the external virtual time argument of theCooperon in Eq. (5.1), and a denotes the characteris-tic dephasing length scale for the Markovian screened-3Coulomb problem [Eq. (2.17)]. This leaves (cid:104) S n (cid:105) = Γ nt (cid:18) Γ M D (cid:19) n/ η n × (cid:2) dimensionless integrals (cid:3) ≡ (cid:18) Γ t Γ M (cid:19) n f n ( z, β ) , (5.7)where f n is a function only of the dimensionless externalvirtual time variable z ≡ ( ηD/a ) , (5.8) and of the diffusion constant ratio β [Eq. (3.3)]. Thescaling procedure shows that the control parameter inthe cumulant expansion is the ratio of the bath couplingstrengths Γ t Γ M = 3 κ γ t χ (1 − γ t ) . (5.9)The scaling also sends the matrix elements to dimensionless functions S ij ( k ; a ) → S ij ( k/a ; a ) ≡ ˜ S ij ( k ) = 1 (cid:112) | α (cid:48) i | α (cid:48) i ) Ai (cid:48) ( α j ) ∞ (cid:90) dρ Ai (cid:0) ρ + α (cid:48) i (cid:1) Ai (cid:0) ρ + α j (cid:1) sin (cid:18) kρ (cid:19) , (5.10)where i, j correspond to even and odd-parity energies, respectively. For a given pair of levels, ˜ S ij ( k ) can be effi-ciently computed numerically as a function of the dimensionless momentum parameter k . This is facilitated by thesuperexponential fall off of the Airy functions with positive argument, Ai( x (cid:29) ∼ x − / exp (cid:2) − (2 / x / (cid:3) .Our strategy is then as follows. After computing the correlators in Eq. (5.4), Eq. (5.2) requires the evaluation of n momentum integrations and 2 n integrations over the τ variables, the latter of which are partitioned into (2 n )! / ( n ! · n )topological sectors { Ω ns } . Note that each frequency integration produces 2 terms [dubbed “ ˜ T ” and “ ˜ T ” in Eq. (5.3)],so that the full expression has 2 n terms. The τ integrations are elementary and we carry them out in closed form,defining the “ T -kernels” to be T sp p ··· p n ( z, β, { k i } , { σ i } ) ≡ z n f ( z ) (cid:90) Ω ns d n τ ˜ T p ( zβ, k , τ a , τ b ) × · · · × ˜ T p n ( zβ, k n , τ na , τ nb ) × exp (cid:40) − z (cid:34) σ n + n (cid:88) i =1 ˜ τ i ( σ i − − σ i ) (cid:35)(cid:41) , (5.11)where p m ∈ { , } for 1 ≤ m ≤ n , f ( z ) is thepure Markovian-dephased Cooperon amplitude definedvia Eq. (2.21), and ˜ τ j gives the time-ordering of the 2 n { τ mα } variables ( α ∈ { a, b } ). The parameters { σ i } aredimensionless eigenenergies for the pure Markovian prob-lem [Eq. (2.19)]; these are Airy prime zeroes σ i ∈ { α (cid:48) j } (for i ∈ { , , . . . , n } ) or Airy zeroes σ i ∈ { α j } (for i ∈ { , , . . . , n − } ).At this point there are n momentum integrations leftover the T -kernels and the matrix elements in Eq. (5.10).We tabulate the matrix elements as functions of k ahead of time and compute the final momentum inte- grations numerically. This gives a numerical function of z [Eq. (5.8)] for each sector and choice of energies { σ i } .The final result from which the η -dependence of the ex-pectation value (cid:104) S nI (cid:105) can be extracted requires summingover (2 n +1) energy arguments over the appropriate Airyor Airy prime zeroes. B. First order
We now specialize to the first order calculation (cid:104) S I (cid:105) and carry out the time integrations to get f ( z, β ) = ∞ (cid:88) i ,i ,i =0 (cid:113) | α (cid:48) i α (cid:48) i | (cid:90) dk π ˜ S i i ( k ) ˜ S i i ( k ) (cid:2) T ( z, β, k , α (cid:48) i , α i , α (cid:48) i ) − T ( z, β, k , α (cid:48) i , α i , α (cid:48) i ) (cid:3) , (5.12)4where f is defined via Eq. (5.7), and where the “ T -kernels” introduced in Eq. (5.11) take the first-order forms T ( z, β, k , σ , σ , σ ) = 1 f ( z ) (cid:18) σ − σ + θk (cid:19) (cid:34)(cid:32) e zσ e − zθk − e zσ σ − σ + θk (cid:33) + (cid:18) e zσ − e zσ σ − σ (cid:19)(cid:35) , (5.13a) T ( z, β, k , σ , σ , σ ) = 1 f ( z ) (cid:18) σ − σ + θk (cid:19) (cid:34)(cid:32) e zσ e − θzk − e zσ e − βzk σ − σ + ( θ − β ) k (cid:33) + (cid:32) e zσ − e zσ e − βzk σ − σ + βk (cid:33)(cid:35) , (5.13b)where θ ≡ / β/ f ( z ) is given by Eq. (2.21). Wemay now finish the evaluation of Eq. (5.12) by numeri-cally integrating in k , using our analytical expressions for T , and tabulated values of the matrix elements ˜ S ij ( k )as functions of k . Each triplet of Airy and Airy-primezero labels { i , i , i } contributes a summand to f ( z, β )in Eq. (5.12).The described numerical procedure produces the cor-rect z -dependence for any set of these three energy la-bels, but it turns out that only a small subset of possi-ble energy combinations are non-decaying in the large- z (virtual time) limit. In particular, the only contribu-tions that grow with z are linear terms that arise whenboth the initial and final energies are at the ground state, i = i = 0. The linear terms arise from the limit( T ) i = i =0 → z | α (cid:48) | α (cid:48) − α i + θk + (const.) , (5.14)which gives the total asymptotic contribution to the ex-First order contributionsTerm i = i = 0 i > i = 0 i > i = 0 Else T zC ( β, i ) − C ( β, i ) C ( β, i , i ) C ( β, i , i ) 0 T C ( β, , i ) − D ( z ; β, i ) C ( β, i , i ) 0 0 TABLE I: We tabulate the asymptotic, large z → ∞ form ofnon-vanishing contributions to the first-order dual-bath cu-mulant amplitude in Eq. (5.12), retaining all contributionsdecaying slower than 1 /z . Here z is the dimensionless virtualtime argument of the Cooperon [Eq. (5.8)]. T and T corre-spond to the two terms in Eq. (5.12), defined by Eq. (5.13).The indices { i , i , i } refer to the three distinct sums overthe energy eigenvalues [Airy or Airy-prime zeroes, Eq. (2.19)]in Eq. (5.12). The { C i , D } functions are defined in Ap-pendix C 2, and are independent of z , with the exception of D , which decays asymptotically as z − / for large z . The ta-ble indicates that the first-order dephasing rate comes solelyfrom the i = i = 0 energy terms in T . pectation value in Eq. (5.12) as f ( z, β ) → z C ( β ) ≡ z ∞ (cid:88) i =0 C ( β, i ) ≡ z ∞ (cid:88) i =0 (cid:90) dk π (cid:104) ˜ S i ( k ) (cid:105) ( α (cid:48) − α i + θk ) . (5.15)We note that the sum in Eq. (5.15) is slowly convergingbut can be numerically estimated by series acceleration[Appendix F].The full asymptotic analysis is summarized in Table I.Contributions decaying slower than 1 /z will be importantfor the second-order calculation, so we carefully retainthem all in Table I. Explicit expressions for the coeffi-cients listed in this table appear in Appendix C 2. Defin- ABCDE
FIG. 7: Comparison of full numerical calculations with theasymptotic, large-virtual-time z → ∞ approximation to thefirst-order dual-bath cumulant amplitude in Eq. (5.12). Herewe define f ( z ) ( T ) i = i =0 to be the contribution to f [Eqs. (5.7)and (5.12)] from T [Eq. (5.13a)], at energy i = i = 0,with i left unspecified. This plot demonstrates that theasymptotic form for this T contribution with Airy primesummand labels pinned at i = i = 0, as specified in Ta-ble I, is correct. Above, i j gives the energy level of the j th energy and z = ηD/a . In this plot, we vary the Airy sum-mand label i from 0 to 4, and we set β = 1 [Eq. (3.3)].A: i = 0, B: i = 1, C: i = 2, D: i = 3, E: i = 4.The solid curves are the asymptotic approximation in Table I, zC (1 , i ) − C (1 , i ). The symbols obtain from the full nu-merical integration of Eq. (5.12), using the exact expressionfor T ( z, , k , α (cid:48) , α i , α (cid:48) ) from Eq. (5.13a). ABCDEF
FIG. 8: Comparison of full numerical calculations with theasymptotic, large-virtual-time z → ∞ approximation to thefirst-order dual-bath cumulant amplitude in Eq. (5.12). Herewe define f ( z ) ( T ) i = i =0 to be the contribution to f [Eqs. (5.7)and (5.12)] from T [Eq. (5.13b)], at energy i = i = 0, with i left unspecified. This plot demonstrates that the asymp-totic form for this T contribution with Airy prime summandlabels pinned at i = i = 0, as specified in Table I, is cor-rect. Above, i j gives the energy level of the j th energy and z = ηD/a . In this plot, we vary the Airy summand label i from 0 to 4, and we set β = 1 [Eq. (3.3)]. A,B: i = 0, C: i = 1, D: i = 2, E: i = 3, F: i = 4. The solid curves are theasymptotic approximation given by C ( β, , i ), [see Table Iand the second line of Eq. (5.18)]. The symbols obtain fromthe full numerical integration of Eq. (5.12), using the exactexpression for T ( z, , k , α (cid:48) , α i , α (cid:48) ) from Eq. (5.13b). Wenote that the convergence speed is determined by the decayof D ( z ; β, i ) ∼ z − / . ing C j ( β ) ≡ (cid:88) i (cid:48) s C j ( β, i (cid:48) s ) , (5.16) D ( z ; β ) ≡ (cid:88) i (cid:48) s D ( z ; β, i (cid:48) s ) , (5.17)and summing over all energies, we have the final asymp-totic formula for the first order expectation value (cid:104) S I (cid:105) (cid:39) (cid:18) Γ t Γ M (cid:19) (cid:20) z C ( β ) − C ( β ) + 2 C ( β ) −C ( β ) + D ( z ; β ) (cid:21) . (5.18)We find that C ( β, i ) and C ( β, i ) are strictly positive,while C ( β, i , i ) and C ( β, i , i ) can alternate in sign[Appendix C 2]. We plot the asymptotic “ T ” and “ T ”contributions to f ( z, β ) [the first and second lines insidethe square brackets of Eq. (5.18), see Table I] in Figs. 7and 8, and compare these to the direct numerical inte-gration of Eq. (5.12) using the full expressions for T , inEq. (5.13). C. Second order overview
We evaluate the second order contribution to the cu-mulant expansion for the dual-bath model along thelines explained above, although there are complicationsnot seen at first order. As before, we obtain the R -correlator in Eq. (5.4a) in closed form. Here, however,the R -correlator breaks into three terms for the threedistinct topological sectors in Eq. (3.11), pictured inFig. 2; explicit expressions appear in Appendix A. Weagain use the Airy-eigenfunction expansion to Trotter-ize the ρ -correlator Eq. (5.4b), which here gives a 5-foldsummation over the energy states. The τ -integrationsin Eq. (5.2) are again elementary, but here each topo-logical sector Ω s has 4 “ T -kernel” [Eq. (5.11)] terms, { T s , T s , T s , T s } . In terms of the T -kernels, thesecond-order amplitude function [see Eq. (5.7)] is f ( z, β ) = ∞ (cid:88) i , ··· ,i =0 (cid:113) | α (cid:48) i α (cid:48) i | (cid:90) dk π (cid:90) dk π (cid:2) T − T − T + T (cid:3) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i i ( k )+ (cid:2) T − T − T + T (cid:3) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i i ( k )+ (cid:2) T − T − T + T (cid:3) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i i ( k ) , (5.19)where T sjk = T sjk ( z, β ; k , k ; α (cid:48) i , α i , α (cid:48) i , α i , α (cid:48) i ) . Using analytical forms for the 12 second-order T -kernels and tabulated values for the matrix elements˜ S ij ( k ) [Eq. (5.10)], we can numerically perform the mo-mentum integrations. This produces the functional de-pendence on the dimensionless virtual time z [Eq. (5.8)]for a given topological sector and set of energy levels { α (cid:48) i , α i , α (cid:48) i , α i , α (cid:48) i } . As in the first order case, we canextract simple expressions for the asymptotic behavior inthe large z → ∞ limit. The important contributions atsecond order are summarized in Table II. The coefficient functions { C i , D i } listed in Table II aredefined explicitly in Appendix C 2. The D and D am-plitudes in Table II are nontrivial functions of z thatgrow via power laws with exponents approximately equalto − / /
2, respectively. However, these terms donot contribute to the final rate due to cancellations. Thetotal surviving asymptotic contributions to the second6Second order contributionsTerm i = i = i = 0 i > i = i = 0 i > i = i = 0 i > i = i = 0 else T z C ( β, i ) C ( β, i ) − z (cid:20) C ( β, i ) C ( β, i )+ C ( β, i ) C ( β, i ) (cid:21) zC ( β, i ) C ( β, i , i ) 2 zC ( β, i , i , i ) 2 zC ( β, i ) C ( β, i , i ) O (1) T zC ( β, i ) C ( β, , i ) − C ( β, i ) D ( z ; β, i ) 2 zC ( β, i ) C ( β, i , i ) O (1) O (1) O (1) T − zC ( β, i ) D ( z ; β, i )+ C ( β, i ) D ( z ; β, i ) O (1) O (1) O (1) O (1) T O (1) O (1) O (1) O (1) O (1) T zC ( β, i , , i ) O (1) 2 zC ( β, i , i , i ) O (1) O (1) T O (1) O (1) O (1) O (1) O (1) T O (1) O (1) O (1) O (1) O (1) T O (1) O (1) O (1) O (1) O (1) T zC ( β, i , , i ) O (1) 2 zC ( β, i , i , i ) O (1) O (1) T O (1) O (1) O (1) O (1) O (1) T O (1) O (1) O (1) O (1) O (1) T O (1) O (1) O (1) O (1) O (1) TABLE II: We tabulate the asymptotic, large z → ∞ form of non-vanishing contributions to the second-order dual-bathcumulant amplitude in Eq. (5.19). Here z is the dimensionless virtual time argument of the Cooperon [Eq. (5.8)]. The termslabeled { T sjk } correspond to the amplitudes arising from the twelve terms in Eq. (5.19). The indices { i , i , i , i , i } refer tothe five distinct sums over the energy eigenvalues [Airy or Airy-prime zeroes, Eq. (2.19)] in Eq. (5.19). The functions { C i } aredefined in Appendix C 2 and are independent of z . Two additional amplitudes D and D depend on z through asymptoticpowers laws with exponents given by − / /
2, respectively; however, the contributions of these two amplitudes cancelout. From the table, we see that the i = i = i = 0 terms from T give quadratic z contributions that exactly cancel thesquare of the first order terms in the cumulant expansion, Table I and Eq. (5.18). Several other nontrivial cancellations takeplace between the various contributions. Finally, we see that additional linear terms arise from the T s terms in the s = { , } topological sectors [see Fig. 2 for the diagrammatic definition of the latter]. order result are then [via Eq. (5.7)] (cid:104) S I (cid:105) s =10 → (cid:18) Γ t Γ M (cid:19) (cid:40) z C + 2 z C (cid:2) C − C (cid:3) +2 z C − z C (cid:2) C − D ( z ) (cid:3)(cid:41) , (5.20a) (cid:104) S I (cid:105) s =20 → (cid:18) Γ t Γ M (cid:19) z C , (5.20b) (cid:104) S I (cid:105) s =30 → (cid:18) Γ t Γ M (cid:19) z C , (5.20c)where the superscript s denotes the topological sector[Fig. 2]. When we subtract the square of the first or-der contribution, we find several nontrivial cancellations.Using Eq. (5.18), the final second order contribution tothe cumulant expansion [as in Eq. (3.4)] is (cid:104) S I (cid:105) − (cid:104) S I (cid:105) = 2 z (cid:18) Γ t Γ M (cid:19) (cid:20) C − C C + C + C (cid:21) + O (1) . (5.21)This is the basis of the results in Eqs. (4.1) and (4.2),quoted in Sec. IV. In Eq. (5.21), the quadratic z termshave canceled exactly. This cancellation is the key dif-ference between this dual diffusive- and Markovian-bath result, and the pure-diffusive bath expansion studied inSec. III [Eq. (3.13)]. By killing off the higher-order η dependence, this cancellation stabilizes the cumulant ex-pansion at long virtual times and determines a well-defined dephasing rate via Eq. (2.4). In Eq. (5.21), theamplitudes C , C , and D cancel out as well. This isnotable, because the energy-level-resolved amplitudes inTable II, C ( β, i , i ), C ( β, i , i ), and D ( z ; β, i ) con-tribute with level { i j } -dependent signs, so that the over-all sign of the total contribution would require numeri-cally precise summation over many Airy energy levels.The asymptotic results for T , summarized in Ta-ble II are compared to the direct numerical integrationof Eq. (5.19) in Fig. 9.The major dephasing contribution at second ordercomes from the −C C term in Eq. (5.21). We can eval-uate this with series acceleration. Summing C , C , and C to the first four excited states gives the plot in Fig. 6,which definitively shows that the net contribution is posi-tive , and thus that the final second order result is rephas-ing , as discussed in Sec. IV.7 ABCD
FIG. 9: Comparison of full numerical calculations with theasymptotic, large-virtual-time z → ∞ approximation to thesecond-order dual-bath cumulant amplitude in Eq. (5.19).This plot compares the contributions obtained by numeri-cally integrating the terms involving T s with s ∈ { , } in Eq. (5.19) to the asymptotic formulae quoted in Ta-ble II, involving the amplitude functions C ( β, i , i , i ) and C ( β, i , i , i ). Here we define f ( z ) ( T s ) i = i =0 to be the contri-bution to f [Eqs. (5.7) and (5.19)] from T s [Eq. (5.11)], atenergy i = i = 0, with i , i , i left unspecified. This plotdemonstrates that the asymptotic form for the T s contribu-tion with Airy prime summand labels pinned at i = i = 0and s = 2 ,
3, as specified in Table II, is correct. Above, i j givesthe energy level of the j th energy and z = ηD/a . In this plot,we set i = i = 0, vary the Airy summand label i from 0to 1, plot both sectors s = 2 and s = 3, and we set β = 1[Eq. (3.3)]. A: i = 0 , s = 3, B: i = 0 , s = 2, C: i = 1 , s = 3,D: i = 1 , s = 2. The solid curves are the asymptotic ap-proximation in Table II, 2 zC , (1 , , i , T , ( z, , k , k , α (cid:48) , α , α (cid:48) i , α , α (cid:48) )[Eq. (5.11)]. VI. DISCUSSION AND CONCLUSIONA. Dephasing and rephasing
We have investigated the dephasing of quasi-1D sys-tems with diffusive noise baths as a possible analyti-cal window into the physics of the many-body local-ization (MBL) transition. The diffusive noise bath isself-generated by short-ranged interactions at interme-diate temperatures in an isolated fermion system withquenched disorder.In Sec. III, we studied a system with a purely dif-fusive noise bath. This describes a quasi-1D systemof ultracold fermions with contact interactions, whichcould be a potential realization for MBL. We calculatedthe Cooperon through second-order perturbation theoryaround the noninteracting result. This procedure givesa well-defined, divergence-free short-time expansion, butthe latter breaks down at long times and fails to yield ameaningful result for the dephasing time.To better understand our results, we also considered inSecs. IV–V a physical regularization of the previous prob-lem, in which the diffusive bath coexists with the Marko- vian noise bath that arises due to screened Coulomb in-teractions. This corresponds to an SU(2) spin-symmetricmany-channel quantum wire with Coulomb and short-range spin-triplet-exchange interactions. Treating theCoulomb bath exactly (via an extension of the AAKtechnique [1]) and the diffusive bath perturbatively, wefound that the presence of the Coulomb interaction sta-bilizes the perturbation theory, giving well-defined cor-rections to the dephasing rate due to the Coulomb in-teraction. Reminiscent of our results in Sec. III, we findthat the second-order term in this expansion is rephas-ing . Rephasing corrections are consistent with RG re-sults showing that vertex corrections can suppress theCooperon-noise coupling strength [22].We demonstrated that commonly used (in higherdimensional dephasing calculations) self-consistent ap-proaches that bootstrap lowest-order perturbation the-ory fail to capture the correct physics of dephasing dueto the diffusive bath, in both the absence and presenceof an additional Markovian bath.A key goal for future work is to obtain a nonpertur-bative understanding for dephasing due to the diffusivebath in isolation. As articulated in Ref. [22], this can becast as a type of self-interacting polymer problem, witha gyration radius that sets the dephasing length in thelong-virtual-time limit.
B. Enhancement of dephasing in spinSU(2)-symmetric quantum wires via itineratespin-exchange interactions
In the theory of the interacting, disordered(“Anderson-Mott” [29]) zero-temperature metal-insulator transition (MIT), it has long been appreciatedthat short-ranged, spin-triplet-exchange interactionsare enhanced [29–31] due to the presence of quencheddisorder. Such enhancements are generically expecteddue to the confluence of wave function criticality neara MIT (multifractality [41–43]) and the smooth scalingof matrix elements with energy (Chalker scaling [44]).The enhancement of the spin interaction strength inspin SU(2)-symmetric (orthogonal class AI) systems[29–31] means that dephasing due to this channelshould also be enhanced, which could play a role inthe physics of weakly disordered, many-channel 1Dquantum wires. This enhancement might provide anadditional mechanism for the apparent “saturation”of the dephasing rate (deviation from the τ φ ∼ T − / AAK prediction for Markovian-Coulomb dephasing [1])observed in experiments on such systems [4–16].The enhancement of the ferromagnetic exchange inter-action γ t < δ | γ t | = g ( γ t ) (cid:18) T T (cid:19) / , (6.1)8where g ( γ t ) ≥ γ t that van-ishes in the γ t → , T ∝ / ( σ ν ), where σ is the classical conduc-tivity and ν is the density of states. In the context ofdephasing, this leads to a T − / enhancement of the noisecoupling strength Γ t [Eq. (3.2)] at intermediate tempera-tures. The first-order correction to the dephasing due tothe diffusive bath in Eq. (4.1) thus contains an additionalboost due to this enhancement, leading to a slowing ofthe dephasing rate relative to the AAK result τ φ ∼ T − / [1]. This subleading correction to the dephasing rate de-cays as T / . At even lower temperatures, Eqs. (4.1)and (6.1) predict a suppression of dephasing relative toAAK, due to the second-order rephasing correction. The full interplay of enhanced or suppressed dephasing (realprocesses) with all virtual quantum corrections could inprinciple be tackled using the dynamical version of theFinkel’stein nonlinear sigma model [28]. Acknowledgments
We thank Doug Natelson, Jan von Delft, and Igor Bur-mistrov for helpful discussions. This research was sup-ported by NSF CAREER Grant No. DMR-1552327, andby the Welch Foundation Grant No. C-1809.
Appendix A: Vertex operator correlators
In this appendix we record results for the various charge-neutral vertex operator correlators that arise throughoutthe paper. These are Gaussian and obtain from Wick’s theorem. For the first-order pure diffusive bath calculation[Eqs. (3.6) and (3.7)], we have (cid:104) e ikr ( τ b ) e − ikr ( τ a ) (cid:105) r = e − Dh r ( η,k,τ a ,τ b ) , h r ( η, k, τ a , τ b ) = k | τ b − τ a | (cid:20) − η | τ b − τ a | (cid:21) . (A1)The second-order generalization is more complicated and depends on the time ordering. We can use symmetry toreduce to the case where τ a < τ b , τ a < τ b , and τ a < τ a [Eq. (3.11) and Fig. 2]. We have (cid:104) e ik r ( τ b ) e − ik r ( τ a ) e ik r ( τ b ) e − ik r ( τ a ) (cid:105) r = e − Dh r ( η,k ,τ a ,τ b ) e − Dh r ( η,k ,τ a ,τ b ) e Dφ r ( η,k ,k ,τ a ,τ b ,τ a ,τ b ) , (A2)where φ r ( η, k , k , τ a , τ b , τ a , τ b ) = k k η ( τ b − τ a )( τ b − τ a ) − k k , τ a < τ b < τ a < τ b ,τ b − τ a , τ a < τ a < τ b < τ b ,τ b − τ a , τ a < τ a < τ b < τ b . (A3)The expectation values over the center-of-time path integral R ( τ ) used in the coexisting bath calculation presentedin Secs. IV and V are simpler due to the averaging over the endpoint R [Eq. (5.1)]. We find that (cid:104) e ikR ( τ b ) e − ikR ( τ a ) (cid:105) R = e − Dh R ( k , τ a ,τ b ) , h R ( k, τ a , τ b ) = k | τ b − τ a | . (A4)At second order, we have (cid:104) e ik R ( τ b ) e − ik R ( τ a ) e ik R ( τ b ) e − ik R ( τ a ) (cid:105) R = e − Dh R ( k ,τ a ,τ b ) e − Dh R ( k ,τ a ,τ b ) e Dφ R ( k ,k ,τ a ,τ b ,τ a ,τ b ) , (A5)with φ R ( k , k , τ a , τ b , τ a , τ b ) = k k , τ a < τ b < τ a < τ b , ( τ b − τ a ) , τ a < τ a < τ b < τ b , ( τ b − τ a ) , τ a < τ a < τ b < τ b . (A6)9 Appendix B: Further details on the purely diffusive bath calculation
We explicitly define the functions G s ( β ) used in Eq. (3.12). The time sectors { Ω , , } are defined via Eq. (3.11),illustrated by the diagrams in Fig. 2. We find that G s ( β ) = 8 π (cid:90) Ω s d τ (cid:20) (cid:112) Θ s ( β, τ ) − (cid:112) Θ s ( β, τ ) − (cid:112) Θ s ( β, τ ) + 1 (cid:112) Θ s ( β, τ ) (cid:21) , (B1)where Θ ij ( β, τ a , τ b , τ a , τ b ) = (cid:26) g i ( β, τ a , τ b ) g j ( β, τ a , τ b ) − (cid:2) ( τ b − τ a )( τ b − τ a ) (cid:3) (cid:27) , (B2)Θ ij ( β, τ a , τ b , τ a , τ b ) = (cid:26) g i ( β, τ a , τ b ) g j ( β, τ a , τ b ) − (cid:2) ( τ b − τ a )( τ b − τ a ) − τ b − τ a ) (cid:3) (cid:27) , (B3)Θ ij ( β, τ a , τ b , τ a , τ b ) = (cid:26) g i ( β, τ a , τ b ) g j ( β, τ a , τ b ) − (cid:2) ( τ b − τ a )( τ b − τ a ) − τ b − τ a ) (cid:3) (cid:27) , (B4)and the functions { g i } are defined by Eq. (3.9). Appendix C: Further details on the coexisting bath calculation1. General calculation
We compute the functional integral over ρ ( τ ) [Eq. (5.4b)] via the Coulomb-Markovian eigenfunctions defined inEqs. (2.20a) and (2.20b). Let ˜ τ j (where j runs over { , ..., n } ) be the time-ordering of all the time variables { τ ia , τ ib } appearing in Eq. (5.4b). (E.g., restricting so that τ ia < τ ib and τ ia < τ ja for i < j , one has that ˜ τ = τ a [see Fig. 2].The rest of the ˜ τ j will depend on the time-sector topology, as discussed in Secs. III and Appendix A). In general, wethen have (after the scaling applied in Eq. (5.6)) F nρ ( k (cid:48) s, τ (cid:48) s ) ≡ (cid:28) sin (cid:20) k ρ ( τ a )2 (cid:21) sin (cid:20) k ρ ( τ b )2 (cid:21) × . . . × sin (cid:20) k n ρ ( τ na )2 (cid:21) sin (cid:20) k n ρ ( τ nb )2 (cid:21)(cid:29) ρ (C1)= 1 f ( z ) ∞ (cid:88) i ,...,i n =0 ˜ S i n i n − [ k r (2 n ) ] × . . . × ˜ S i i [ k r (1) ] (cid:113) α (cid:48) i α (cid:48) i n exp − η ε n + n (cid:88) j =1 ˜ τ j ( ε j − − ε j ) , (C2)where the ε j are given by Eq. (2.19) and f ( z ) is given by Eq. (2.21). Above, the subscript k r ( j ) indicates that one hasto be careful about allocating the momenta to the expectation values. The ordering of the momenta is dependent onthe topology of the time sector being considered. We formalize this by defining a function r : { , ..., n } → { , ..., n } ,as follows: if ˜ τ j = τ ia or τ ib , then r ( j ) = i . The order of the momenta depends on the time-ordering of the τ variables.
2. First and second order
We give the coefficients defined in the first-order asymptotic analysis in Table I. C ( β, i ) = 2 (cid:90) dk π (cid:104) ˜ S i ( k ) (cid:105) ( α (cid:48) − α i + θk ) , (C3) C ( β, i ) = 2 (cid:90) dk π (cid:104) ˜ S i ( k ) (cid:105) ( α (cid:48) − α i + θk ) , (C4) C ( β, i , i ) = 2 (cid:12)(cid:12)(cid:12)(cid:12) α (cid:48) α (cid:48) i (cid:12)(cid:12)(cid:12)(cid:12) / α (cid:48) − α (cid:48) i (cid:90) dk π ˜ S i ( k ) ˜ S i i ( k )( α (cid:48) − α i + θk ) , (C5)0 C ( β, i , i ) = 2 (cid:12)(cid:12)(cid:12)(cid:12) α (cid:48) α (cid:48) i (cid:12)(cid:12)(cid:12)(cid:12) / (cid:90) dk π ˜ S i ( k ) ˜ S i i ( k )( α (cid:48) − α (cid:48) i + βk )( α (cid:48) − α i + θk ) . (C6)Similarly, the coefficients introduced in the second-order asymptotic analysis Table II are C ( β, i , i , i ) = 4 α (cid:48) − α (cid:48) i (cid:34)(cid:90) dk π ˜ S i ( k ) ˜ S i i ( k )( α (cid:48) − α i + θk ) (cid:35) (cid:34)(cid:90) dk π ˜ S i ( k ) ˜ S i i ( k )( α (cid:48) − α i + θk ) (cid:35) , (C7) C ( β, i , i , i ) = 8 (cid:90) dk π (cid:90) dk π ˜ S i ( k ) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i ( k )( α (cid:48) − α i + θk )( α (cid:48) − α i + θk )(2 α (cid:48) − α (cid:48) i + 2 θk + 2 θk + k k ) , (C8) C ( β, i , i , i ) = 8 (cid:90) dk π (cid:90) dk π ˜ S i ( k ) ˜ S i i ( k ) ˜ S i i ( k ) ˜ S i ( k )( α (cid:48) − α i + θk )( α (cid:48) − α i + θk )(2 α (cid:48) − α (cid:48) i + 2 θk + 2 θk + k k ) . (C9)Tables I and II also contain “anomalous” contributions D and D , which arise in the T , T , and T terms whenall “even” energies ( i , i , i ) are pinned at the ground state. These are defined by D ( z ; β, i ) = 2 z / β α (cid:48) − α i (cid:90) dk π k e − βk (cid:20) ˜ S i (cid:18) k √ z (cid:19)(cid:21) , (C10) D ( z ; β, i ) = 4 z / β α (cid:48) − α i (cid:90) dk π k (1 − e − βk ) (cid:20) ˜ S i (cid:18) k √ z (cid:19)(cid:21) . (C11)In each case, we can expand ˜ S ij ( k ) in powers of k/ √ z for large z , since k/ √ z will be small for the dominant portionof the integrand. Because the leading order contribution to ˜ S ij ( k/ √ z ) is linear in k/ √ z , the leading order asymptoticcontributions to D and D are z − / and z / , as seen numerically. We note again that both D and D cancel fromthe final dephasing expressions.
3. Numerical comparison with asymptotics
In this appendix subsection we provide a provide a col-lection of plots [Figs. 10–17] demonstrating the accuracyof the asymptotic expressions given in Tables I and II,similar to Figs. 7–9. In Figs. 10–17, we numerically cal-culate the full contributions from the terms listed in therows of Tables I and II, (i.e. T j , T sjk ), for some specificchoices of the energy levels, as functions of z = ηD/a .This is done via the method explained in Sec. V. Thesenumerical results are then compared to the asymptoticforms listed in Tables I and II. As in Figs. 7–9, we define f ( z ) ( T j ) i (cid:48) s (cid:16) f ( z ) ( T sjk ) i (cid:48) s (cid:17) to be the contribution to f ( f )from term T j (cid:16) T sjk (cid:17) at the energy levels specified by thesubscript “ i (cid:48) s ”. The plots all show quick convergence tothe expected behavior. Appendix D: Perturbation theory for Coulombdephasing
The exact solution for the Cooperon in the screenedCoulomb (Markovian) case is given in Eq. (2.21). Here weinstead treat this case perturbatively, via the cumulantexpansion method employed throughout this paper.The cumulant expansion requires the evaluation of theperturbing action defined by Eqs. (2.9) and (2.15). We
ABCD
FIG. 10: Comparison of full numerical calculations with theasymptotic, large-virtual-time z → ∞ approximation to thefirst-order dual-bath cumulant amplitude in Eq. (5.12). Herewe define f ( z ) ( T ) i =0 ,i =1 to be the contribution to f [Eqs. (5.7)and (5.12)] from T [Eq. (5.13a)], at energy i = 0 , i = 1,with i left unspecified. This plot demonstrates that theasymptotic form for this T contribution with Airy prime sum-mand labels pinned at i = 0 , i = 1, as specified in Table I, iscorrect. Above, i j gives the energy level of the j th energy and z = ηD/a . In this plot, we vary the Airy summand label i from 0 to 4, and we set β = 1 [Eq. (3.3)]. A: i = 0, B: i = 1,C: i = 2, D: i = 3 ,
4. The solid curves are the asymptoticapproximation in Table I and Eq. (5.18). The asymptotic for-mula in this case is given by C , Eq. (C5). The symbols obtainfrom the full numerical integration of Eq. (5.12), using the ex-act expression for T ( z, , k , α (cid:48) , α i , α (cid:48) ) from Eq. (5.13a). ABCDE
FIG. 11: We define f ( z ) ( T ) i =0 ,i =1 to be the contributionto f [Eqs. (5.7) and (5.12)] from T [Eq. (5.13b)], at en-ergy i = 0 , i = 1, with i left unspecified. This plot givesthe asymptotic form for this T contribution with Airy primesummand labels pinned at i = 0 , i = 1, as specified in Ta-ble I. Above, i j gives the energy level of the j th energy and z = ηD/a . In this plot, we vary the Airy summand label i from 0 to 4, and we set β = 1 [Eq. (3.3)]. A: i = 0, B: i = 1, C: i = 2, D: i = 3, E: i = 4. The decay of thesefunctions shows that these contributions asymptotically van-ish, as indicated in Table I. The symbols obtain from the fullnumerical integration of Eq. (5.12), using the exact expressionfor T ( z, , k , α (cid:48) , α i , α (cid:48) ) from Eq. (5.13b). have (cid:104) S nM (cid:105) = Γ nM D n η (cid:90) dτ n ... η (cid:90) dτ (cid:104)| ρ ( τ n ) | ... | ρ ( τ ) |(cid:105) (D1)= I n (cid:18) M √ πD η / (cid:19) n , (D2)where I n is a numerical prefactor. It is given by I n = ( n !) (cid:90) dτ n τ n (cid:90) dτ n − . . . τ (cid:90) dτ ∞ (cid:90) −∞ dρ n . . . ∞ (cid:90) −∞ dρ × | ρ n | . . . | ρ | (cid:112) (1 − τ n )( τ n − τ n − ) . . . ( τ − τ ) τ × exp (cid:20) − ρ n (1 − τ n ) (cid:21) exp (cid:20) − ( ρ n − ρ n − ) ( τ n − τ n − ) (cid:21) × . . . × exp (cid:20) − ( ρ − ρ ) ( τ − τ ) (cid:21) exp (cid:20) − ρ τ (cid:21) . (D3)We compute I n and the resulting moments numerically;the results are collected in Table III. In the cumulantexpansion [Eq. (3.4)], the coefficient of the n th order termis given by the n th cumulant ≡ κ n , formed from the first n moments of the perturbing action. Using the moments inTable III, we explicitly calculate the first four cumulantsand list them in Table IV. We actually tabulate ¯ κ n ≡ κ n /n !, which is the full numerical coefficient for the n th ABCDE
FIG. 12: We define f ( z ) ( T ) i =1 ,i =0 to be the contributionto f [Eqs. (5.7) and (5.12)] from T [Eq. (5.13b)], at energy i = 1 , i = 0, with i left unspecified. This plot demonstratesthat the asymptotic form for this T contribution with Airyprime summand labels pinned at i = 1 , i = 0, as specifiedin Table I, is correct. Above, i j gives the energy level of the j th energy and z = ηD/a . In this plot, we vary the Airysummand label i from 0 to 4, and we set β = 1 [Eq. (3.3)].A: i = 0, B: i = 1, C: i = 2, D: i = 3, E: i = 4.The solid curves are the asymptotic approximation in Table Iand Eq. (5.18). The asymptotic formula in this case is givenby C , Eq. (C6). The symbols obtain from the full numer-ical integration of Eq. (5.12), using the exact expression for T ( z, , k , α (cid:48) , α i , α (cid:48) ) from Eq. (5.13b). order term in the expansion, so that c M ( η ) = c ( η ) exp − ¯ κ (cid:18) M √ πD η / (cid:19) + ¯ κ (cid:18) M √ πD η / (cid:19) + . . . . (D4)Table IV shows that the perturbative cumulant expan-sion for the Markovian gives alternating dephasing andrephasing terms.Numerical results for I n coefficients I I I I . π/ .
183 0 . . TABLE III: Table collecting numerical results for the I n mo-ment coefficients, defined via Eqs. (D1) and (D3). Numerical results for ¯ κ ¯ κ ¯ κ ¯ κ ¯ κ . π/ . . . TABLE IV: Numerical results for the cumulant coefficients¯ κ n ≡ κ n /n !, which determine the dephasing and rephasingterms in the cumulant expansion for the Markovian bath,Eq. (D4). ABCDE
FIG. 13: We define f ( z ) i = i =1 to be the contribution to f [Eqs. (5.7) and (5.12)] [from both T and T , Eqs. (5.13a)and (5.13b)], at energy i = i = 1, with i left unspecified.This plot gives the asymptotic form for these contributionswith Airy prime summand labels pinned at i = i = 1,as specified in Table I. Above, i j gives the energy level ofthe j th energy and z = ηD/a . In this plot, we vary theAiry summand label i from 0 to 4, and we set β = 1[Eq. (3.3)]. A: i = 0, B: i = 1, C: i = 2, D: i = 3,E: i = 4. Here we see that these contributions asymptot-ically vanish, as indicated in Table I (“Else”). The sym-bols obtain from the full numerical integration of Eq. (5.12),using the exact expressions for T ( z, , k , α (cid:48) , α i , α (cid:48) ) and T ( z, , k , α (cid:48) , α i , α (cid:48) ) from Eqs. (5.13a) and (5.13b). AB FIG. 14: We define f ( z ) ( T ) i = i = i =0 to be the contribution to f [Eqs. (5.7) and (5.19)] from T at energy i = i = i = 0,with i and i left unspecified. This plot demonstrates thatthe asymptotic form for this T contribution with Airy primesummand labels pinned at i = i = i = 0, as speci-fied in Table II, is correct. In this plot, we vary the Airysummand labels i , i from 0 to 1. A: ( i , i ) = (0 , i , i ) = (1 , , (0 , z = 2, while the full numerical re-sults start at z = 0. The full numerical results obtain fromthe integration of Eq. (5.19), using the exact expression for T ( z, , k , k , α (cid:48) , α i , α (cid:48) , α i , α (cid:48) ). Appendix E: Diagram folding
To treat the Markovian noise kernel exactly, we need tofold the time integrations from the region ( − η, η ) to (0 , η )[Eqs. (2.6) and (2.7)]. In general, this procedure foldsthe different diagram topologies associated with time-ordering into one another. This should be consideredif one wants to study the effects of a specific class ofdiagrams, or to make contact with the field theoretic for- ABCDE
FIG. 15: We define f ( z ) ( T )( i ,i ,i ) (cid:54) =(0 , , to be thecontributions to f [Eqs. (5.7) and (5.19)] from T atenergies away from i = i = i = 0. (No i j explic-itly specified.) This plot demonstrates, for several setsof energies, that the asymptotic form for T given inTable II is correct. In this plot, we vary the Airy sum-mand labels i j i , i , i , i , i ) = (1 , , , , i , i , i , i , i ) = (1 , , , , i , i , i , i , i ) =(1 , , , , i , i , i , i , i ) = (0 , , , , i , i , i , i , i ) = (0 , , , , z = 2, while the fullnumerical results start at z = 0. The full numerical resultsobtain from the integration of Eq. (5.19), using the exactexpression for T ( z, , k , k , α (cid:48) i , α i , α (cid:48) i , α i , α (cid:48) i ). ABCDE
FIG. 16: We define f ( z ) ( T ) i = i = i =0 to be the contribution to f [Eqs. (5.7) and (5.19)] from T at energy i = i = i = 0,with i and i left unspecified. This plot demonstrates thatthe asymptotic form for this T contribution with Airy primesummand labels pinned at i = i = i = 0, as speci-fied in Table II, is correct. In this plot, we vary the Airysummand labels i , i from 0 to 1. A: ( i , i ) = (0 , i , i ) = (0 ,
0) (exact), C: ( i , i ) = (1 , i , i ) = (1 ,
0) (exact), E: ( i , i ) = (0 , z = 2, while the full numeri-cal results start at z = 0. The full numerical results obtainfrom the integration of Eq. (5.19), using the exact expressionfor T ( z, , k , k , α (cid:48) , α i , α (cid:48) , α i , α (cid:48) ). The exact and asymp-totic results here differ slightly due to the anomalous D term,which is dropped from the asymptotic expression used here. mulation of the dephasing problem [22], [Appendix G].At first order there is no issue, since there is only asingle diagram topology. At second order, however, wefind a nontrivial mixing of the distinct topologies (“dou-ble,” “crossed,” and “nested”), defined by Eq. (3.11) and3 ABCDE
FIG. 17: We define f ( z ) ( T )( i ,i ,i ) (cid:54) =(0 , , to be thecontributions to f [Eqs. (5.7) and (5.19)] from T atenergies away from i = i = i = 0. (No i j explic-itly specified.) This plot demonstrates, for several setsof energies, that the asymptotic form for T given inTable II is correct. In this plot, we vary the Airy sum-mand labels i j i , i , i , i , i ) = (1 , , , , i , i , i , i , i ) = (1 , , , , i , i , i , i , i ) =(0 , , , , i , i , i , i , i ) = (1 , , , , i , i , i , i , i ) = (0 , , , , z = 2, while the fullnumerical results start at z = 0. The full numerical resultsobtain from the integration of Eq. (5.19), using the exactexpression for T ( z, , k , k , α (cid:48) i , α i , α (cid:48) i , α i , α (cid:48) i ). shown in Fig. 2.To “unfold” an n th -order diagram, any subset of the2 n time variables can be flipped to the negative side ofthe time interval. For each topological sector, we have 2 n distinct preimages under the η -folding map to consider.Table V maps out this inverse-folding for the second-order calculation.From Table V, we see that the (folded) double diagram(sector 1) is really a 50-50 combination of the (unfolded)double and nested diagrams. The (folded) nested andcrossed diagrams (sectors 2 and 3) are both 25-25-50 com-binations of the (unfolded) double, nested, and crosseddiagrams, respectively. However, the terms contributingnon-trivial asymptotics are marked in Table V in bold,and we see that most (but not all) of the contributionscome from diagrams that are originally double. Inter-estingly, the “nontrivial” contributions in the coexistingbath calculation [Secs. IV,V] that give rise to C and C [Table V, Eqs. (4.1) and (4.2)] are actually split equallybetween trivial (double) and nontrivial (crossed, nested)diagrams in the unfolded framework. Appendix F: Series acceleration
We summarize a series acceleration technique used toestimate the dephasing coefficients, which are defined byslowly converging sums. For an absolutely convergentsum (cid:80) ∞ n =0 a n and constants p > C >
0, we have in Diagram unfoldingFolded diagrams Double Nested CrossedFlipped τ ’s T -type Unfolding results {} T Double Nested Crossed { } T Double
Nested Crossed { } T Double
Crossed Double { } T Nested Crossed Nested { } T Nested Double Crossed { } T Double Double Double { } T Nested Crossed Nested { } T Nested Crossed Crossed { } T Nested Crossed Crossed { } T Nested Crossed Nested { } T Double Double Double { } T Nested Crossed Crossed { } T Nested Crossed Nested { } T Double
Double Double { } T Double
Nested Crossed { } T Double Nested Crossed
TABLE V: Table tracing the inverse folding of diagrams.The bold entries are the terms contributing meaningfully todephasing or rephasing in the asymptotic limit, see Table II.We are interested in taking a “folded” diagram of a givenshape and T -kernel type, and diagnosing its diagram shapein the “unfolded” path integral. The right half of the top rowlists the three possible topologies for the folded diagrams. Thefar left column lists which time variables are to be flipped tothe negative side of the time interval ( − η, η ) and the sec-ond column lists the type of contribution (“ T -kernel”) of thefolded diagram. ABCD
FIG. 18: Depiction of the series acceleration techniquedemonstrated on the Airy summation Eq. (F3). A: The trueanswer given by The RHS of Eq. (F3). B: Approximationsby series truncation of the LHS of Eq. (F3) after n terms.We note that the series is slowly converging, and nontrivialerror exists after 1000 terms have been summed. C: Knownanalytical limit of the p -series summation, Cζ ( p ). D: Partialsums of the best p -series approximation in the decompositionEq. (F4). This plot shows that the slowly-converging natureof the original sum can be approximated well by a p -serieswith similar convergence properties. AB FIG. 19: This plot compares convergence speeds for the di-rect summation of Eq. (F3) and the accelerated sum. A:direct summation—LHS of Eq. (F1) applied to Eq. (F3). B:Accelerated sum—RHS of Eq. (F1) applied to Eq. (F3). Wesee that the accelerated sum converges several orders of mag-nitude faster than the original sum. general that ∞ (cid:88) n =0 a n = Cζ ( p ) + ∞ (cid:88) n =0 ( a n − Cn − p ) , (F1)where ζ is the Riemann Zeta function. We call the sumover Cn − p a “ p -series”. In the case that a n → Cn − p rapidly, Eq. (F1) can be used to efficiently estimate thesum: ∞ (cid:88) n =0 a n (cid:39) Cζ ( p ) + N (cid:88) n =0 ( a n − Cn − p ) , (F2)for some sufficiently large N . This procedure worksby packaging the slow convergence of a n into the zeta-function.To illustrate the method, we estimate the sum fromEq. (2.23), (cid:88) n α (cid:48) n ) = 2 π / Γ(2 / , (F3)which was used to derive the exact conductivity cor-rection for the screened Coulomb noise bath. Sincethe zeros of Ai (cid:48) ( x ) are asymptotically given by α n (cid:39)− (3 π/ / n / , the series in Eq. (F3) tends to a p -serieswith 1( α (cid:48) n ) (cid:39) (cid:18) π (cid:19) / n − / . (F4)In Fig. 18, we plot the convergence of the left-hand-sideof Eq. (F3) to the right-hand-side of Eq. (F3) along-side the convergence of (cid:80) ∞ n =0 Cn − p to Cζ ( p ) , with C =(3 π/ − / and n = 4 /
3. We see that both series areslowly converging, but that the convergence rate is ex-tremely similar. In Fig. 19, we compare the convergencespeeds of the original and boosted summations, given by AB FIG. 20: The acceleration technique discussed in this sectionrequires the summands of the series in question to be well-approximated by a p -series. This can be checked empiricallyby fitting a line to the summands in a log-log scale. Thisplot uses this method to demonstrate that the C ( β ) series,defined in Eq. (C3) is well-approximated by a p -series. A: C ( β = 1 , i = n ), B: Cn − p , where C and p are extractedfrom a linear fit. truncating the left- and right-hand sides of Eq. (F1), re-spectively. We see that the accelerated sum convergesseveral orders of magnitude faster than the original sum.We can use the series acceleration technique to ap-proximate the sums C and C [Eqs. (5.18), (5.16), (C3)and (C4)] to all energies, giving us the full first ordercorrection to the dephasing rate. This gives Figs. 5and 6. Fig. 20 shows that the C summation is well-approximated by a p -series, and Fig. 21 compares theconvergences of the original and accelerated series. ABCD
FIG. 21: Depiction of the series acceleration of C ( β ), definedvia Eq. (C3), at β = 1. A: Partial sums of the acceleratedseries. We see that these converge rapidly. B: Partial sumsof the original series, which is slowly converging. C: Knownanalytical limit of the p -series summation, Cζ ( p ). D: Partialsums of the best p -series approximation in the decompositionEq. (F1). We see that C (1) ≈ . Appendix G: Fermionic field theory1. Field theory for Cooperon
We reviewed in Sec. II A how the single-particle pathintegral and relative-time coordinates provide a power-ful tool for the non-perturbative treatment of Markoviannoise kernels [1]. The fluctuation-averaged Cooperonstudied in this paper can also be calculated in terms of areplicated fermionic field theory framework [22, 45]. Thegenerating function of the theory is Z = (cid:90) D ¯Ψ D Ψ D φ cl e − S Ψ [ ¯Ψ , Ψ] − S φ [ φ cl ] − S c [ ¯Ψ , Ψ ,φ cl ] , (G1)with the action components S Ψ [ ¯Ψ , Ψ] = (cid:90) k ,ω ¯Ψ a ( ω, k ) (cid:20) D k − iω (cid:21) Ψ a ( ω, k ) , (G2) S φ [ φ cl ] = 12 Γ (cid:90) k ,ω φ cl ( ω, k ) φ cl ( − ω, − k )∆( ω, k ) , (G3) S c [ ¯Ψ , Ψ , φ cl ] = i √ Γ (cid:90) k ,ω (cid:90) q , Ω φ cl (Ω , q ) × ¯Ψ a ( ω + Ω2 , k + q ) − ¯Ψ a ( ω − Ω2 , k + q ) Ψ a ( ω, k ) . (G4)Above, ∆( ω, k ) is the noise kernel for the theory, D isthe classical diffusion constant due to elastic scattering[Eq. (2.1)], and Γ is the coupling to the bath [as inEqs. (2.16) and (3.2) in the main text]. We choose toembed the Cooperon using the replicated fermion fieldΨ a , where a ∈ { , , . . . , n } and we take n → Z = 1; this is natural in the full dynamicalsigma model [28, 35, 36]. We use replicas here only tolighten the notation.)The full fluctuation-averaged Cooperon is obtained inthe replica limit as the correlation function (cid:104) c tω (cid:48) ,ω ( k ) (cid:105) φ cl = D c ( ω (cid:48) , ω, k ) (G5) ≡ D (cid:104) Ψ a ( ω (cid:48) , k ) ¯Ψ a ( ω, k ) (cid:105) Z , (G6)where (cid:104)· · · (cid:105) Z denotes a functional average over thefull partition function Z , and the “reduced Cooperon”,˜ c R ( ω, k ), follows from averaging over relative frequency FIG. 22: The Feynman rules for the field theory of thefluctuation-averaged Cooperon, defined in Eq. (G1). Dia-grams (A) and (B) represent the bare propagators for theΨ a and φ cl fields, respectively. Diagrams (C) and (D) de-pict the two types of interaction vertices coupling the fields.The vertices in (C) and (D) are the “causal” and “anticausal”vertices, respectively. Ω, ˜ c R ( ω, k ) = 12 (cid:90) Ω ˜ c (cid:18) ω − Ω2 , ω + Ω2 , k (cid:19) , (G7) c ( η ) = D (cid:90) ω, k e − iηω ˜ c R ( ω, k ) . (G8)The Feynman rules for the theory are given in Fig. 22.The corresponding bare propagators for the theory aregiven by (cid:104) Ψ a ( ω, k ) ¯Ψ b ( ω, k ) (cid:105) = δ ab (cid:20) D k − iω (cid:21) − ≡ δ ab ˜ c ( ω, k ) , (G9a) (cid:104) φ cl ( ω, k ) φ cl ( − ω, − k ) (cid:105) = ∆( ω, k ) / Γ . (G9b)All diagrams with closed fermion loops vanish in thereplica limit, and so the only contributing diagrams tothe full Cooperon contain a single fermion line dressedwith noise propagators. This restriction on the diagramtopology leaves (2 n )! / ( n !2 n ) topologically distinct dia-grams at n th order, with a generic diagram depicted inFig. 23.Fig. 22 demonstrates the interaction vertices couplingbetween the Cooperon Ψ a and noise φ cl fields. The twodistinct vertices arise from the first [diagram C] andsecond [diagram D] terms in the noise bath action inEq. (G4). We will refer to these as causal and anti- FIG. 23: The topology of a generic Feynman diagram con-tributing to the Cooperon, before averaging over the bath.The replica limit removes all diagrams with closed Fermionloops, so all relevant diagrams contain a single Fermion linedressed with some number of noise phonons. FIG. 24: Labeling convention that allows us to forget aboutthe causal and anti-causal vertices shown in Fig. 22(C,D) andinstead consider only “type I” (colored blue) and “type II”(colored red) noise phonons (as defined in the main text). Wechoose the frequency of the noise phonon to be ω j ( − ω j ) if theleftmost vertex is causal (anticausal). Type I phonons (A,B)give frequency-diagonal contributions, while type II phonons(C,D) give frequency-off-diagonal contributions. causal vertices, respectively, and they contribute factorsof ± i √ Γ / − Γ / / n th order diagram willhave 2 n colorings of its noise phonons as type I or II,so that each diagram topology generally has competitionbetween exponentially many opposite-sign contributions.
2. Diffusive bath and connection with cumulantexpansion
Re-expanding the cumulant expansion, Eq. (3.4), di-rectly in Γ gives us c ( η ) = c ( η ) (cid:20) − (cid:104) S (cid:105) + 12 (cid:104) S (cid:105) + . . . (cid:21) . (G10)We can also directly expand the Cooperon in powers ofΓ in the replicated fermionic field theory: c ( η ) = c ( η ) + c ( η ) + c ( η ) + . . . (G11) Formally equating terms of the two power series, we findthat (cid:104) S n (cid:105) = ( − n n ! c n ( η ) c ( η ) . (G12)We can thus compute terms in the cumulant expansion—originally framed as expectations in a path integral—directly in the field theory.In particular, calculating the first order type I dia-grams depicted in Fig. 24 for the diffusive noise kernel( ≡ D diff.1 , type I ), we find D diff.1 , type I = − (cid:18) D √ πDη (cid:19) (cid:18) Γ t η / √ D (cid:19) ˜ G ( β ) (G13)˜ G ( β ) ≡ (cid:114) π (cid:90) − dτ a (cid:90) τ a dτ b g ( β, τ a , τ b ) − / , (G14)in line with the results of Eqs. (3.7) and (3.8). Evaluatingthe first order type II diagrams in Fig. 24 then givesthe other contribution to Eq. (3.8). We note that inthe field theory framework, the parametric integrationdefining G ( β ) arises from a Feynman parameter.
3. Markovian Coulomb bath and connection toAAK a. General remarks and divergence regularization
In the case of a Markovian noise kernel, we have seenthat a coordinate change in the path integral formalism[Eq. (2.6)] allows for a massive reduction in complex-ity, recasting the fluctuation-averaged Cooperon as theGreen’s function of a single-particle quantum mechanicsproblem, Eq. (2.11). In the Markovian limit of the fieldtheory description, we find that all but a special classof diagrams vanish exactly, and that summing the re-maining diagrams to all orders recovers the AAK reduc-tion formula for the propagator. Thus, Eqs. (2.10) and(2.11) can be alternatively derived as an infinite-orderfield-theoretic summation.In the Markovian limit, we find IR divergences in themomentum integration due to the divergence of the noisekernel at zero momentum. This is regularized by a per-fect cancellation of all IR divergences between all the di-agrams at a given order. This is easily seen at first orderand can be shown at arbitrary order. This cancellationis fundamentally related to the cancellation of IR diver-gences in the Fourier transform that gave us Eq. (2.15):˜∆ M (0) − ˜∆ M ( ρ ) = 2Γ M D (cid:90) k k (1 − e i k · ρ ) . (G15)Note that individually, ˜∆ M (0) and ˜∆ M ( ρ ) are IR diver-gent in one or two spatial dimensions, but their differ-7 FIG. 25: Examples of non-vanishing diagrams that follow therules outlined in Sec. G 3 b. In all the above diagrams, type Inoise phonons are colored blue and type II noise phonons arecolored red. As required to be non-vanishing, the type I noisephonons pass over no vertices, and the type II noise phononsare in a nested rainbow configuration. ence is finite in 1D (and UV divergent in higher dimen-sions). We will see that the UV contributions of the typeI phonons are responsible for the “ ˜∆ M (0)” term, whilethe type II phonons are responsible for the “ ˜∆ M ( ρ )” termin Eq. (2.11).In Eqs. (G18)–(G21) (below), we will sum all the di-agrams in the perturbative expansion via a two-stepframework. We first sum the type I phonons intoa dressed propagator and then translate the type IIphonon contributions into a self-consistent integral equa-tion. This calculation will require us to treat the type Iand type II phonon lines on separate footing, obscuringthe fact that the IR divergences between the various dia-grams cancel out order-by-order in perturbation theory.We must therefore cancel out the IR divergent portionsof the diagrams in the beginning, before the type I re-summation. In the type I resummation and later in theself-consistent equation, we then drop the IR-divergentportion of the momenta integrations. We will use theintegral superscript “( − IR )” to indicate that it is neces-sary to remove by hand the infrared divergences in themomenta integrations. We also define˜∆ ( − IR ) M ( ρ ) ≡ ( − IR ) (cid:90) k ∆ M ( k ) e i k · ρ . (G16)We point out that in 1D, ˜∆ ( − IR ) M (0) = 0 , while in higherdimensions, ˜∆ ( − IR ) M (0) is dependent on a UV cutoff. b. Diagram non-vanishing requirements Performing the frequency integrations analytically, wefind that most of the Feynman diagrams one can drawvanish in the Markovian case. We can codify this into tworules that a diagram contributing to the Green’s function must satisfy. The rules are as follows:1. There can be no vertices under a type I phonon.If a type I phonon leaves the fermion line, it mustreturn at the very next vertex.2. If a diagram contains any type II phonons, theymust all be nested in a non-crossing rainbow con-figuration.To prove the vanishing rules, one can perform the fre-quency contour integrations in the complex plane andshow that if either rule is violated, the diagram can belabeled so that there is at least one frequency integrationfor which all the poles lie in one half of the plane. Clos-ing the frequency integration contour in the opposite halfof the plane shows that the integral vanishes. Diagramsillustrating the vanishing rules are given in Figs. 25 and26.The non-vanishing rules for this theory require alter-ation of the usual paradigms of field theory. For example,the self-energy cannot be thought of in usual terms. Adiagram with nested type II rainbows, as in Fig. 25(A)above, does not vanish and is 1-particle irreducible. How-ever, it cannot be resummed into a self-energy, becausethe diagram consisting of two sequential copies of it, as inFig. 26(B), vanishes due to violation of rule 2. Thus, nodiagrams containing type II phonons can be resummedinto a self-energy, for they only appear exactly once in theexpansion of the Green’s function. On the other hand, we can sum the type I diagrams to all orders into a “pseudoself-energy” —we take this up in the next subsection. c. Type I phonon resummation
The type I diagrams can be resummed to all ordersinto a “pseudo self-energy”. This is simple and can be
FIG. 26: Two examples of vanishing diagrams, which fail tofollow the rules outlined in Sec. G 3 b. In all the above di-agrams, type I noise phonons are colored blue and type IInoise phonons are colored red. Diagram (A) fails to followthe first rule, since each type I phonon passes over a vertexon the fermion line. Diagram (B) fails to follow the secondrule, since its two type II phonons are not in the requirednested rainbow configuration. We note that the fact that di-agram (A) in Fig. 25 cannot be resummed as a self-energy forthe Cooperon can be understood by noting that Diagram (B)in this figure vanishes. FIG. 27: The diagrammatic infinite-order summation of thetype I noise phonons via a Dyson’s equation with a “pseudoself-energy”. In (A), we define a “type I dressed propagator”(blue fermion line) to be a the sum of all Cooperon diagramsdressed only by type I phonons. In (B) we re-write the sum-mation self-consistently as a Dyson equation. In this case, therole of the self-energy is played by the diagram with a singletype I phonon, which we evaluate directly in Eq. (G18). done exactly since the type I phonons appear in isola-tion and cannot cross; the only diagram that enters intothe pseudo self-energy is the first order type I diagram( ≡ D Coul.1 , type I ) [Fig. 24 (A, B)]. We can then define a type Idressed propagator as shown diagrammatically in Fig. 27.We can perform this summation via the self-consistentequation given by the diagrams in Fig. 27, letting ˜ c I de-note the “type I dressed propagator”. We find˜ c I (cid:18) ω − Ω2 , ω + Ω2 , k (cid:19) = 4 πδ (Ω) Dk − iω + ˜∆ M (0) , (G17)because D Coul.1 , type I = − ( − IR ) (cid:90) ν, q M ( q ) D ( k − q ) − iω − iν = − ( − IR ) (cid:90) q ∆ M ( q )= −
12 ˜∆ ( − IR ) M (0) . (G18) d. Full propagator and connection to AAK solution With the type I propagators summed to infinite ordervia the pseudo self-energy, we can put the perturbativeseries for the full Green’s function into a simpler form.The remaining diagrams to consider are the non-crossingrainbow diagrams with type I dressed fermion propaga-tors and type II noise propagators, as shown in Fig. 28.In this framework there is a single diagram left at eachorder (in the type II phonon) in the series defining theCooperon. The average and internal frequency integra-tions can be performed analytically, giving the n th order contribution to the reduced Cooperon as˜ c ( n ) R ( ω, k ) = 12 ( − IR ) (cid:90) l ∆ M ( l ) . . . ( − IR ) (cid:90) l n ∆ M ( l n ) × Dk + ˜∆ ( − IR ) M (0) − iω × · · ·× D ( k − . . . − l n ) + ˜∆ ( − IR ) M (0) − iω . (G19)We can treat the type II phonon diagrams to all ordersby deriving a self-consistent integral equation for the fullpropagator, as shown diagrammatically in Fig. 28. Thediagrammatic result corresponds to the self-consistentequation for the full Cooperon.˜ c (cid:18) ω − Ω2 , ω + Ω2 , k (cid:19) = 4 πδ (Ω) Dk + ˜∆ ( − IR ) M (0) − iω + (cid:104) Dk + ˜∆ ( − IR ) M (0) − iω + i Ω (cid:105) × (cid:104) Dk + ˜∆ ( − IR ) M (0) − iω − i Ω (cid:105) × ( − IR ) (cid:90) l ∆ M ( l ) (cid:90) ν ˜ c (cid:18) ω − ν , ω + ν , k − l (cid:19) . (G20)Integrating over Ω we find the reduced equation˜ c R ( ω, k ) = 1 Dk + ˜∆ ( − IR ) M (0) − iω + 1 Dk + ˜∆ ( − IR ) M (0) − iω × ( − IR ) (cid:90) l ∆ M ( l ) ˜ c R ( ω, k − l ) . (G21)Defining c R ( η, ρ ) ≡ D (cid:90) ω, k e − iωη e i k · ρ ˜ c R ( ω, k ) , (G22)the position space formulation of Eq. (G21) is (cid:104) ∂ η − D ∇ ρ + ˜∆ M (0) − ˜∆ M ( ρ ) (cid:105) c R ( η, ρ ) = D δ ( η ) δ ( ρ ) . (G23)9 FIG. 28: The diagrammatic infinite-order summation of thetype II noise phonons via a self-consistent equation. In (A),we express the fully dressed propagator (purple fermion line)as the “type I dressed propagator” (blue fermion line) dressedby the summation of all maximally-nested rainbow configu-rations of type II noise phonons. In (B) we re-write the ex-pansion in (A) as a self-consistent equation. The structure ofthe self-consistent equation is reminiscent of that of the self-consistent born approximation [Eq. (3.15)], but here the LHSof the equation is the fully dressed propagator, rather thanthe self-energy. This self-consistent equation sums a propersubset of the usual SCBA diagrams. The diagrammaticsare translated into a Fredholm integral equation [Eqs. (G20),(G21), and (G23)] that turns out to be equivalent to the AAKresult from the main text, Eq. (2.11).
This states that c R is the imaginary-time propagator forthe single-particle quantum mechanics Hamiltonian ˆ h , re-covering the AAK reduction in Eq. (2.11). We see outthat the “ ˜∆ M (0),” term arises from the type-I “pseudo self-energy” while the “ ˜∆ M ( ρ )” term arises from the self-consistent treatment of the type II phonon.
4. Coexisting interaction baths
Finally, we briefly note that the theory for the coex-isting diffusive and screened Coulomb baths can also betreated in the field theory language. In this case, onedefines two distinct species of noise phonon, one for eachnoise bath. (Each noise bath will have both type I andtype II phonons.) As before, the replica limit enforcesthe topological constraints explained by Fig. 23; theCooperon is given by all diagrams with a single fermionline dressed by any combination of the four distinct noisephonons. It turns out that the special vanishing rulesdiscussed in Sec. G 3 b still apply in this more generalscenario, though only to the phonons generated by theMarkovian Coulomb bath. The perturbative calculationscarried out in the main text thus correspond to an ex-act (though asymptotic) partially-infinite-order summa-tion over the diagrams with arbitrarily many Coulombphonons (restricted by the vanishing rules), but up totwo diffusive phonons. We note that the AAK transfor-mation of variables Eq. (2.6) in the single-particle pathintegral formalism allows us to get this result directly interms of the Airy eigenfunction summations exploited inSec. V. [1] B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitsky,Effects of electron-electron collisions with small energytransfers on quantum localisation, J. Phys. C. , 7367(1982).[2] P. A. Lee and T. V. Ramakrishnan, Disordered ElectronSystems, Rev. Mod. Phys. , 287 (1985).[3] B. L. Altshuler and A. G. Aronov, Electron-electron in-teraction in disordered conductors, in Electron-ElectronInteractions in Disordered Systems , edited by M. Pollakand A. L. Efros (North-Holland, Amsterdam, 1985).[4] I. L. Aleiner, B. L. Altshuler, and M. E. Gershenson, In-teraction effects and phase relaxation in disordered sys-tems, Waves Random Media , 201 (1999).[5] P. Mohanty, E. M. Q. Jariwala and R. A. Webb, Intrinsicdecoherence in mesoscopic systems, Phys. Rev. Lett. , 195331,(2007).[11] F. Marquardt, J. von Delft, R. A. Smith, and V. Am-begaokar, Decoherence in weak localization. II. Bethe-Salpeter calculation of the cooperon, Phys. Rev. B. ,195332, (2007).[12] G. Zarand, L. Borda, J. von Delft, and N. Andrei, Theoryof inelastic scattering from magnetic impurities, Phys.Rev. Lett. [17] B. N. Narozhny, G. Zala, and I. L. Aleiner, Interac-tion corrections at intermediate temperatures: Dephas-ing time, Phys. Rev. B , 180202(R) (2002).[18] D. M. Basko, I. L. Aleiner, and B. L. Altshuler,Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann.Phys. (Amsterdam) , 1126 (2006); On the prob-lem of many-body localization, in Problems of Con-densed Matter Physics , edited by A. L. Ivanov and S. G.Tikhodeev (Oxford University Press, Oxford, England,2007); arXiv:cond-mat/0602510.[19] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Interact-ing Electrons in Disordered Wires: Anderson Localiza-tion and Low- T Transport, Phys. Rev. Lett. , 206603(2005).[20] R. Nandkishore and D. A. Huse, Many-body localiza-tion and thermalization in quantum statistical mechan-ics, Annu. Rev. Condens. Matter Phys. , 1538 (2015).[21] S. Gopalakrishnan and S. A. Parameswaran, Dynamicsand Transport at the Threshold of Many-Body Localiza-tion, arXiv:1908.10435.[22] Y. Liao and M. S. Foster, Dephasing Catastrophe in 4 − (cid:15) Dimensions: A Possible Instability of the Ergodic (Many-Body-Delocalized) Phase, Phys. Rev. Lett. , 236601(2018).[23] Jae-Ho Han and Ki-Seok Kim, Boltzmann transport the-ory for many-body localization Phys. Rev. B , 214206(2018).[24] J. ˇSuntajs, J. Bonˇca, T. Prosen, and L. Vidmar,Quantum chaos challenges many-body localization,arXiv:1905.06345.[25] D. A. Abanin, J. H. Bardarson, G. De Tomasi, S.Gopalakrishnan, V. Khemani, S. A. Parameswaran, F.Pollmann, A. C. Potter, M. Serbyn, and R. Vasseur, Dis-tinguishing localization from chaos: challenges in finite-size systems, arXiv:1911.04501.[26] R. K. Panda, A. Scardicchio, M. Schulz, S. R. Taylor,M. ˇZnidariˇc Can we study the many-body localizationtransition?, Europhys. Lett. , 67003 (2020).[27] N. Y. Yao, C. R. Laumann, S. Gopalakrishnan, M. Knap,M. M¨uller, E. A. Demler, and M. D. Lukin, Many-bodylocalization in dipolar systems, Phys. Rev. Lett. ,243002 (2014).[28] Y. Liao, A. Levchenko, and M. S. Foster, Response the-ory of the ergodic many-body delocalized phase: KeldyshFinkel’stein sigma models and the 10-fold way, Ann.Phys. (Amsterdam) , 97 (2017).[29] D. Belitz and T. R. Kirkpatrick, The Anderson-Motttransition, Rev. Mod. Phys. , 261 (1994).[30] A. Punnoose and A. M. Finkel’stein, Metal-InsulatorTransition in Disordered Two-Dimensional Electron Sys-tems, Science , 289 (2005). [31] A. M. Finkel’stein, Disordered Electron Liquid with In-teractions, in
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