Effect of Electron-Phonon Coupling on Transmission Through Luttinger

Eﬀect of electron-phonon coupling on transmission through Luttinger liquid hybridized with resonant level

Alexey Galda, Igor V. Yurkevich and Igor V. Lerner School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom (Dated: January 28, 2013) We show that electron-phonon coupling strongly affects transport properties of the Luttinger liquid hybridized with a resonant level. Namely, this coupling significantly modifies the effective energy-dependent width of the resonant level in two different geometries, corresponding to the resonant or antiresonant transmission in the Fermi gas. This leads to a rich phase diagram for a metal-insulator transition induced by the hybridization with the resonant level.

PACS numbers: 73.20.Mf

Low-temperature electronic properties of one- dimensional (1D) systems (like quantum wires or nanotubes) are strongly affected by electron-electron interactions. Electrons in such systems form a Luttinger Liquid (LL) [1]. An arbitrarily weak repulsion in a clean LL leads to power-law decay of various correlation functions with exponents depending on the interaction strength. Such a decay which is a characteristic property of the LL has been been experimentally observed in carbon nanotubes [2] and various quantum wires [3] (see Ref. [4] for a recent review). Inserting a potential impurity or a weak link (e.g., a tunnel barrier) into the LL results in a power-law sup- FIG. 1. The geometries considered: resonant-barrier geome- pression of a local density of states (LDoS) at the im- try (top) and side-attached geometry (bottom), also referred purity site and thus a suppression of the conductance at to in the text as RBG and SAG. low temperatures T [5, 6],changes in characteristics of the Fermi-edge singularity [7] and Friedel oscillations [8], etc. If the barrier interrupting the LL carries a discrete with phonons. localized state resonant with the electron Fermi energy, It is known that the electron-phonon (el-ph) coupling its hybridization with the electronic states in the leads re- in combination with the electron-electron (el-el) repul- sults in a sharp resonant transmission [9]. Similar to the sion results in the formation of two polaron branches with Fermi liquid, it is described by the Breit–Wigner formula different propagation velocities in the clean LL (see, for example, [12, 13]). The formation of polarons modifies but with the resonance width Γ0 replaced by an energy- dependent effective width Γ(ε) vanishing at the Fermi the values of exponents in power laws characteristic for level. Such a resonant transmission can be realized, e.g., the LL. The exponents can change signs as functions of by inserting into a 1D quantum wire a double barrier with the relative strength of the el-el and el-ph coupling and a resonant level or a weakly coupled quantum dot (QD) of the ratio of the Fermi to sound velocities. The effect of this is especially pronounced for the LL with an with sufficiently large level spacing δ (δ T, Γ0) and one level in resonance. We would refer to this geometry as embedded potential scatterer. While a single scatterer resonant-barrier. embedded into the phononless LL makes it going from In a dual geometry, when a QD with such a reso- an ideal metal to an ideal insulator (at T = 0) [5], in nant level is side-attached to the LL, transmission be- the presence of the el-ph coupling such a transition can comes antiresonant: it is reflectance rather than trans- be reversed for a weak scatterer [14] or, in general, can mittance which is described by the Breit-Wigner formula become dependent on the scatterer strength [15]. but with the width Γ(ε) being power-law divergent at the In this Letter we show that the el-ph coupling results Fermi level [10] (with the divergence cut by a tempera- also in a drastic change of electronic transport trough the ture T ). Both geometries are realistic: transmission and LL hybridized with a resonant level both in the resonant- tunnelling measurements in the presence of controlled de- barrier and in the side-attached geometry (see fig. 1). fects have already been performed in both quantum wires We will proceed as follows. First we introduce the [3] and carbon nanotubes for various types of defects [11]. action for the LL with the Coulomb repulsion, el-ph cou- Naturally in any realistic geometry electrons in the LL pling and hybridization with a resonant level. By inte- are inevitably interact not only with each other but also grating out the phonon fields and the fields describing 2 the resonant-level electron, we obtain an effective action Finally, the tunnelling action for both geometries (as- in terms of only the fields describing the conduction elec- suming that the impurity or QD carrying the resonant trons. Then we employ the functional bosonization in level is inserted in the LL or side-attached to it at x = 0) form developed in [16] to describe the polaron formation has the form: in the presence of the resonant level in terms of the mixed Z n ¯ X ¯ o fermion-bosonic action. Finally we use the renormaliza- ST = it id(t)(∂t + ε0) d(t) + t0d(t)ψµ(t) + h.c. , tion group (RG) analysis in form similar to that for the µ phononless LL [5, 10] to calculate transmission through (4) such a polaronic liquid as well as the effect of the el-ph coupling and hybridization on the electronic LDoS in the where ψµ(t) ≡ ψµ(x = 0, t), d(t) is the field correspond- vicinity of the QD. ing to the electron localized at the resonant level with the energy ε0 counted from the Fermi level. In the side- attached geometry (SAG) the index µ simply labels right- EFFECTIVE ACTION and left-movers while in the resonant-barriers geometry (RBG) it refers to the left and right electron subsystems The action consists of three terms, S = SLL + Sel−ph + separated by the barrier. In this case the electron can ST. We assume the usual LL decoupling of the (spin- leave the left subsystem to the QD and enter it from less) electron field into the sum of right- and left-moving the QD only as a right- and left-mover, respectively – electrons, and conversely for the right subsystem so that the labels µ = ±1 mean ip x −ip x ψ(ξ) = ψR(ξ)e F + ψL(ξ)e F , ξ ≡ (x, t), ¯ ¯ ¯ ¯ ψ− = ψ`,L , ψ− = ψ`,R , ψ+ = ψr,R , ψ+ = ψr,L (5) with ψR,L being labelled by η = ±1 below, and unit with ~ = 1 are used. Then the LL part of the action, which in- where L, R refer to the left- and right movers, as be- cludes the electron kinetic energy and the density-density fore, and `, r to the left and right subsystems. We have interaction, has the following form: assumed the tunnelling amplitude t0 to be the same in X Z 1 Z all channels. This is always the case for the SAG, but S = iξ ψ¯ (ξ) i∂ ψ (ξ) − iξ V n2(ξ) . (1) LL η η η 2 0 not necessarily for the RBG if it is implemented as an η=±1 asymmetric double-barrier. In the latter case the resonant properties could be different for t 6= t but we leave Here n ≡ (ψ¯ ψ + ψ¯ ψ ) is the electron density, V 1 2 R R L L 0 it aside here. is the screened Coulomb interaction, iξ stands for ix it The action defined by Eqs. (1), (3) and (4) is quadratic and ∂η ≡ ∂t + ηvF ∂x. The free conduction electron 0 0 in fields d and φ which can thus be integrated out. The Green function, gη(ξ − ξ ), is defined by ∂ηgη(ξ − ξ ) = δ(ξ − ξ0) so that its Fourier transform is given by integration over the phonon fields results only in substituting V0 in the action (1) by the dynamical coupling, −1 gη(ε, q) = [ε + iδ sgn ε − ηvF q] . (2) 2 V (ξ) = V0 + g D0(ξ) . (6) We use here the zero-temperature formalism: the only role played by temperature is providing an alternative Performing the integration over the field d(t) results in low-energy cutoff in the RG equations below. transforming the tunnelling term (4) in the action to the It is not particularly important for what follows how following one: we model the phonon action. We choose a model of 1D Z ˜ X 0 ¯ 0 0 acoustic phonons linearly coupled to the electron density, ST = − it it ψµ(t)Σµ,ν (t − t ) ψν (t ) , assuming the phonon spectrum to be linear with a cutoff µ,ν (7) at the Debye frequency, ωD = cqD . Then the phonon ˆ vF Γ0 action (neglecting possible electron backscattering which Σ(ˆ ε) = , Γˆ0 ≡ Γ0(1ˆ +σ ˆx) , ε − ε0 + iδ sgn ε is justified for T ωD) has the following form: Z where Σ(t − t0) is the Fourier transform of Σ(ε). Γˆ is the h 1 −1 i 0 Sel−ph = iξ − φ(ξ) D φ(ξ) + g φ(ξ) n(ξ) . (3) 2 0 matrix with all elements equal to the tunnelling rate Γ0 ≡ 2 −1 πν0|t0| , where ν0 = (πvF ) is the one-particle DoS of Here φ(ξ) is the phonon field, g the el-ph coupling con- the conduction electrons in the absence of interactions. stant and D0(ξ) the free phonon propagator with the The action given by Eqs. (1) (with V0 = 0) and (7) Fourier transform given by describes the resonant transmission through the Fermi 2 gas hybridized with the resonant level. The hybridiza- ωq D0(ω, q) = , ωq = cq . tion makes the electron Green function to acquire an ω2 − ω2 + iδ 0 0 q off-diagonal part, gµ(ξ − ξ ) → Gµν (ξ, ξ ), describing 3 the resonance-induced backscattering in the SAG or left- transformation which introduces the new bosonic field to-right connection in the RBG. In the mixed position- θ(ξ) ≡ θ(x, t): energy representation it has the following matrix form iθη (ξ) ψη(ξ) → e ψη(ξ) , i∂ηθη(ξ) = ϕ(ξ) . (12) ˆ 0 0 ˆ 0 G(x, x ; ε) =g ˆ(x − x ; ε) + ivF gˆ(x; ε)T(ε)g ˆ(−x ; ε) . (8) The Jacobian of this transformation results [16] in substi- −1 −1 Hereg ˆ is the matrix with diagonal elements given by tuting V + Π for V in eq. (11), where Π = ΠR + ΠL eq. (2), and the Tˆ-matrix has the form is the one-loop electronic polarization operator (exact for the LL), with Πη(ξ) = igη(ξ)gη(−ξ) and gη(ξ) given −iΓˆ Tˆ(ε) = 0 . (9) by eq. (2). Note that one could have started with the i ˆ unscreened Coulomb interaction, as this transformation ε − ε0 + 2 Γ0 sgn ε provides the screening which is, naturally, identical to Although the Green function (8) is formally the same one originally calculated diagrammatically [17]. for both geometries considered, the transmission for the The conduction electron Green function is not invari- RBG is proportional to G12G21 while for the SAG it ant under the gauge transform (12) and so it is dressed is proportional to G11G22. This gives the well-known by θ(ξ). In the absence of the resonance part of the Fermi-gas result with the resonant transmission for the action, eq. (7), the dressed Green function for right- or RBG and the resonant reflection for the SAG (with left-moving electrons is given by R0 = 1 − T0): 0 0 0 iUηη (ξ,ξ ) Gη(ξ − ξ ) = gη(ξ, ξ ) e ,  2 Γ0  2 , RBG  (ε − ε )2 + Γ where iUηη0 ≡ hθηθη0 i is the correlation function of the T (ε) = 0 0 (10) 0 (ε − ε )2 fields θ. To calculate it we notice that eq. (12) for the  0  2 2 , SAG . gauge field θ(ξ) is resolved with the help of the bosonic (ε − ε0) + Γ0 Green function gη which coincides with the free electron We will use the effective action represented by the sum Green function (2): of the terms given by eq. (1) with the substitution (6) Z 0 0 0 and eq. (7) to show that the el-el and el-ph interactions θη(ξ) = iξ gη(ξ − ξ ) ϕ(ξ ) , results in Γ0 → Γ(ε) in the transmission probability (10), with the energy dependence of Γ(ε) being qualitatively Thus we obtain the Fourier transform of U 0 as follows: different from that found in the phononless case [9, 10]. ηη 0 2 2 2 2 ω+ +η vF q V0(ω+ − ωq ) + g ωq U 0 (ω, q) = , ηη ω −η v q 2 2 2 2 2 2 FUNCTIONAL BOSONIZATION + F ω+ − v+q ω+ − v−q (13) where ω ≡ ω + iδ sgn ω and v are velocities of the The first step is the Hubbard–Stratonovich transfor- + ± composite bosonic modes (polarons) given by mation which decouples the n2 term in the action (1) with the substitution (6) and results in the mixed fermionic- q 2 1 2 2 2 2 2 2 2 bosonic action in terms of the auxiliary bosonic field ϕ v = v + c ± (v − c ) + 4αphv c . (14) ± 2 F minimally coupled to ψ: Here we introduced the dimensional el-ph coupling con- 1 Z X Z S = − iξ ϕ V −1 ϕ + i iξ ψ¯ (∂ − ϕ) ψ . (11) stant α ≡ ν g2 while v is the speed of plasmonic exci- eff 2 η η η ph 0 η=±1 tations in the phononless LL,

Here η labels right- and left-moving electrons for both 1/2 −1 v = vF (1 + ν0V0) ≡ vF K , geometries under consideration. We stress that for the RBG there is no interaction between electrons in different where K is the standard Luttinger parameter. We as- halves of the system and in this case the action (11) de- sume that in the absence of phonons the el-el interac- scribes electrons moving in one of the subsystems which tion is always repulsive (V0 > 0) so that K < 1. Note are connected only via the resonant tunnelling. We will that in the limit corresponding to the absence of phonons refer to electrons in different (` and r) subsystems using (c = 0), one has v− = 0 and v+ = v, so that in this indices µ, ν = ±1 which simultaneously label right- and case Uµν reduces to the usual LL plasmonic propagator. left-movers as in eq. (5), keeping η, η0 for referring only This also happens in the absence of the el-ph coupling to the right- and left-movers in both geometries. (αph = 0) when v− = min(c, v) and v+ = max(c, v) and The introduction of the field ϕ in eq. (11) does not these two branches are totally decoupled. We take the constitute the functional bosonization: the latter is in same limit for U when ω > ωD so that c should be put getting rid of the coupling term by the following gauge to 0. 4

Equations (13) and (14) describe well known two- where one would have κ± = 1 without coupling to branch polaronic excitations [12] in the LL in the pres- phonons in which case γ+ → γ˜+ ≡ 1/K −1, the exponent ence of the el-ph coupling. The slow and fast branches describing renormalization of the conductance by a weak have the velocity v∓ obeying the inequalities v− < c, v < scatterer, and γ− → γ˜− ≡ K − 1, the exponent describ- v+. Here we have restricted our considerations to the sta- ing renormalization by a weak link. The el-ph coupling bility region defined by modifies these exponents by the factors κ±(K) given by 2 α ≡ αphK < 1 , (15) " #−1/2 α v where v2 > 0, i.e. we leave out of considerations the κ+ = 1 + √ 2 , β ≡ , (20a) − β + 1 − α c Wentzel–Bardeen instability [18]. ( " #)−1/2 The existence of the two polaron branches may be α interpreted as splitting the LL in the presence of the κ− = (1 − α) 1 + √ . (20b) −1 2 el-ph coupling into the two-component liquid, with the β + 1 − α effective Luttinger parameters K = v /v < K < fast F + It is easy to verify that κ (K) ≤ 1 while κ (K) ≥ 1 ,K = v /v > 1, corresponding to el-el repul- + − slow F − 1, so that γ < γ˜ and γ > γ˜ . Therefore, whereas sion (which becomes stronger with the el-ph coupling) + + − − the phononless exponents are sign-definite,γ ˜ > 0 and and the phonon-mediated attraction. The two-mode na- + γ˜ < 0, this is not necessarily true for the exponents γ ture of the el-ph LL drastically changes a character of − + and γ . We have previously shown [15] that electronic the resonant (or antiresonant) transmission. − transport through the LL with a weak scatter or with a The self-energy part in resonant-level term (7) is weak link, described by the correlation functions with the dressed as a result of the gauge transform (12) by the exponents γ respectively, is strongly influenced by the local field θ(ε) ≡ θ(x = 0, t): ± el-ph coupling due to these exponents changing sign at 0 0 0 −iθµ(t) 0 iθν (t ) different values of parameters α and β. Here we will show Σ(t − t ) → Σµν (t − t ) = e Σ(t − t ) e . (16) that this also strongly affects the resonant transmission It is this dressing which fully governs the resonance-width (reflection) in both geometries where the exponents γ± renormalization, Γ0 → Γ(ε), in the presence of the el-el are changed by the el-ph interaction enter via eq. (18). and el-ph interactions. As in the case of the resonance transport through the phononless LL [9, 10], we shall write a renormalization group (RG) equation for the tunnelling amplitude t . In THE SELF-ENERGY RENORMALIZATION 0 our formalism we should start with renormalizing the self-energy part Σ in the tunnelling action (7) with the The polaron fields θµ(t) entering the self-energy (16) gauge substitution (16). are defined at the origin where the QD (or barriers) car- Having integrated out the fields with x 6= 0, all the in- rying the resonant level is placed. Integrating out all teraction effects enter via the correlation functions of the the fields at x 6= 0 results in the zero-dimensional action bosonic field θ, defined by Eqs. (17) – (20). Therefore, which governs the renormalization of the self-energy in the RG equation for Σ is obtained by a usual integration Eqs. (7) and (16), and thus the renormalization of the over fast components of this field. We do not integrate resonance width in Eqs. (10). In the phononless case this over fast components of the fermionic field ψ(t) and do action is fully equivalent to that used for describing the not rescale the time variable since these two procedures resonant transmission (reflection) through the LL [9, 10]. exactly cancel each other which follows from the absence The only difference due to the el-ph coupling is that the of the renormalization of the self-energy in the noninter- correlation function of the local fields θ(t) is governed by acting case. We assume that the fast Fourier components the two-branch polaron modes, eq. (13). It follows from 0 of the field θµ(t) have frequencies E ≤ |ω| ≤ E , with E this equation that being the running cutoff and E0/E − 1 1. Integrating Z iq πγ them out leads to the following increment for the self- hθ (−ω) θ (ω)i = i U (ω; q) ≡ µν , (17) µ ν 2π µν |ω| energy: where the dimensionless correlation matrix γµν found Z dω γ + γ δΣ (ε) = − µµ νν Σ (ε) from the above integration can be parameterized as µν 2|ω| 2 µν ( E≤|ω|≤E0 δµν γ+ , RBG γµν = (18) 1 − γµν Σµν (ε + ω) . (21) 2 (γ+ − γ−) + γ− δµν , SAG .

Here γ± can be represented as In such an RG scheme [9] the self-energy Σµν acquires a κ+ dependence on the running cutoﬀ E on top of the depen- γ = − 1 , γ = κ K − 1 , (19) + K − − dence on ε. The initial condition for the RG equations 5

is that at the ultraviolet cutoﬀ, E = E0,(E0 ∼ εF is the bandwidth), Σµν is independent of E and has all matrix elements equal to Σ(ε) given by eq. (7). Then as long as E |ε−ε0| one may discard the second term in eq. (21), thus arriving at the following RG equation: dΣ (ε; E) µν = −γ Σ (ε; E) , (22) dl d µν where l ≡ ln E0/E and γd are equal diagonal components of the matrix γµν , eq. (18), given by

γ γ , RBG , (a)Resonant-barrier geometry: Γ ∝ |ε/E0| + with the γ = + (23) off d 1 sign of γ+ depending on the el-ph coupling strength. γ0 ≡ 2 (γ+ + γ−) , SAG . Γdiag = Γoff at ε & ε0 and saturates at ε0 (dashed line) for ε ε . Note that γ+ and γ0 happen to be the edge and bulk DoS . 0 exponents respectively, equal to 1/K−1 and (1−K)2/2K in the phononless case [5] and given by eq. (18) and (20) in the presence of the el-ph interaction. Equation (22) is solved by substituting Σµν (ε; E) in form (7), i.e. with all matrix elements equal, but with Γ0 replaced by Γ(E). This leads to the RG equation for Γ: dΓ(E) = −γ Γ(E) ,E |ε − ε | . (24) dl d 0 With lowering the running cutoff one eventually reaches the region E |ε − ε0| where the second term must be taken into account. The self-energy still has the form of γ (b)Side-attached geometry: Γdiag = Γoff ∝ |ε/E0| 0 for γ eq. (7) but with the substitution ε & ε0;Γdiag saturates and Γoff ∝ |ε/ε0| − for ε . ε0. The sign of γ depends on the el-ph coupling strength, while ˆ ˆ − Γ0 → Γ(E) = Γdiag(E)1ˆ + Γoff (E)ˆσx . (25) γ0 > 0.

For E |ε − ε0| the second term in eq. (21) cancels the FIG. 2. Γdiag(ε) and Γoff (ε) for RBG and SAG (not to scale). first one for µ = ν so that Γdiag saturates at Γ(|ε − ε0|) obtained by solving eq. (24), while Γoff continues to be renormalized according to the following RG equation: by the renormalized ones, Γ(ˆ E =ε), found from Eqs. (24) dΓ (E) and (26). The off-diagonal element of the transmission off = −γ Γ (E) ,E |ε − min{Γ , ε }| , dl ± off 0 0 matrix, Toff , is equal to the transmission or reflection (26) amplitude for, respectively, the RBG or SAG so that 2 with γ+ for the RBG, as in eq. (24), and γ− for the |Toff (ε)| , RBG T (ε) = 2 . (27) SAG. Behavior of renormalized Γ(ε) is shown in fig. 2. 1 − |Toff (ε)| , SAG

Let us stress that the condition of applicability written 2 above does not follow directly from eq. (21) which is per- Here |Toff (ε)| has the following form found from eq. (9) with the substitution (25) at E = ε: turbative in Γ0. However, it follows from considerations non-perturbative in tunnelling (but perturbative in the Γ2 Γ2 − Γ2 interaction strength) [6, 10] that the inequality (26) ac- |T |2 = off , Λ ≡ off diag . off 2 2 4ε tually means the off-resonance condition. The impurity (ε0 − ε + Λ) + Γdiag 0 remains off-resonant if the level width renormalized ac- (28) cording to eq. (24) remains narrow, i.e. Γ(ε ) ε . Only 0 0 For ε |ε |, the diagonal and off-diagonal elements of in this case Γ renormalizes as in eq. (26). Otherwise, & 0 off Γˆ are equal and renormalize in the same way, eq. (24). we should put ε = 0 and describe the resonant situation 0 This leads to the transmission coefficients of the Fermi- entirely in the frame of eq. (24). gas form (10) but with the renormalized tunneling rate,

ε γd TRANSMISSION COEFFICIENT Γ(ε) = Γ0 , (29) E0 The transmission coeﬃcient, T (ε), is obtained by re- which fully describes the case of resonance. When the placing in the Tˆ-matrix (9) the bare tunneling rates, Γˆ0, impurity level is oﬀ-resonance, we substitute ε0 for ε0 −ε 6

γ+ < 0, the eﬀective resonance width diverges with ε → 0 rather than vanishes as in the phononless LL. For the SAG, where γd ≡ γ0, there could be two phase transi- tions: a suﬃciently strong el-ph interaction can decouple the resonant level from conduction electrons, leading to the metallic phase, also for a weak el-el coupling when K is close to 1. This work was supported by the EPSRC Grant T23725/01.

FIG. 3. The RG exponents of the effective resonance width [1] Tomonaga S., Prog. Theor. Phys., 5 (1950) 544; Lut- ∗ − 1 for the RBG, γ+, and the SAG, γ0. Here K ≡ min{1, αph 2 } tinger J. M., J. Math. Phys., 4 (1963) 1154; Haldane is the boundary of the applicability region: for αph > 1 we F. D. M., J. Phys. C, 14 (1981) 2585; von Delft J. stay away from the Wentzel–Bardeen instability [18]. and Schoeller H., Ann. Phys., 7 (1998) 225. [2] Bockrath M. et al., Nature, 397 (1999) 598; Yao Z. et al., ibid, 402 (1999) 273; Ishii H. et al., ibid, 426 (2003) into eq. (28) and take into account that Γoff continues to 540; Lee J. et al., Phys. Rev. Lett., 93 (2004) 166403. renormalize, eq. (26), while Γdiag saturates: [3] Auslaender O. et al., Science, 295 (2002) 825; Slot E. et al., Phys. Rev. Lett., 93 (2004) 176602; Levy E. et ε γ± al., ibid, 97 (2006) 196802; Venkataraman L., Hong Γoff (ε) = Γ(ε0) , Γdiag = Γ(ε0) . (30) Y. S. and Kim P., ibid, 96 (2006) 076601. ε0 [4] Deshpande V. V., Bockrath M., Glazman L. I. and Yacoby A., Nature, 464 (2010) 209. [5] Kane C. L. and Fisher M. P. A., Phys. Rev. Lett., 68 RESONANT CONDUCTANCE (1992) 1220; Phys. Rev. B, 46 (1992) 15233. [6] Matveev K. A., Yue D. and Glazman L. I., Phys. At nonzero but low temperatures T , the two-terminal Rev. Lett., 71 (1993) 3351. conductance g(T ) is proportional to T (ε) with the low- [7] Furusaki A., Phys. Rev. B, 56 (1997) 9352. energy cutoff at ε ∼ T (the Fermi energy corresponds [8] Egger R. and Grabert H., Phys. Rev. Lett., 75 (1995) 3505; White S. R., Affleck I. and Scalapino D. J., to ε = 0). In the off-resonance situation, |Toff (ε)| in Phys. Rev. B, 65 (2002) 165122. eq. (28) vanishes with ε → 0: the resonant level remains [9] Kane C. L. and Fisher M. P. A., Phys. Rev. B, 46 decoupled from conduction electrons even when Γoff (ε) (1992) 7268(R); Polyakov D. G. and Gornyi I. V., diverges when ε → 0. ibid, 68 (2003) 035421; Furusaki A. and Matveev K. A., Phys. Rev. Lett. , 88 (2002) 226404; Nazarov The resonance, ε0 . Γ(ε0), is described by eq. (28) with Λ = 0. This corresponds to the Fermi-gas expres- Y. V. and Glazman L. I., ibid, 91 (2003) 126804. sion, eq. (10), with ε = 0 and Γ substituted by its [10] Lerner I. V., Yudson V. I. and Yurkevich I. V., 0 0 Phys. Rev. Lett., 100 (2008) 256805; Goldstein M. and renormalized value, Γ(ε), eq. (29). The critical exponent Berkovits R., ibid, 104 (2010) 106403. γd, eq. (23), is strongly affected by the el-ph coupling, as [11] Bockrath M. et al., Science, 291 (2001) 283; Ma- illustrated in fig. 3: without phonons γd (i.e. γ+ or γ0) are son N., Biercuk M. J. and Marcus C. M., ibid, 303 monotonically decreasing functions reaching 0 at K = 1. (2004) 655; Terrones M. et al., Phys. Rev. Lett., 89 Crucially, conductance g is either ideal or vanishing at (2002) 075505. [12] Loss D. and Martin T., Phys. Rev. B, 50 (1994) 12160. T = 0, depending on whether γd > 1 or γd < 1. For γ > 1, |T (ε)| in eq. (28) vanishes with ε → 0, as [13] Cazalilla M. A. and Ho A. F., Phys. Rev. Lett., 91 d off (2003) 150403; Mathey L., Wang D.-W., Hofstet- in the case of a strong el-el coupling in the phononless LL ter W., Lukin M. D. and Demler E., ibid, 93 (2004) (K < 1/2 without pair tunnelling [5]). This happens be- 120404. cause Γ(ε) → 0 faster than ε → 0, i.e. the resonant level [14] San-Jose P., Guinea F. and Martin T., Phys. Rev. remains effectively decoupled from conduction electrons. B, 72 (2005) 165427. On the contrary, for γd < 1 we have |Toff (ε)| → 1 for [15] Galda A., Yurkevich I. V. and Lerner I. V., Im- ε → 0 which leads to an ideal resonance for the RBG and purity scattering in Luttinger liquid with electron-phonon antiresonance for the SAG, eq. (27). coupling, arXiv:1008.3270 (2010). [16] Grishin A., Yurkevich I. V. and Lerner I. V., Phys. Thus the effective decoupling of the resonant electron Rev. B, 69 (2004) 165108. level from conduction electrons at γd = 1 leads to a [17] Dzyaloshinskii I. E. and Larkin A. I., Zh. Eksp. Teor. metal-insulator transition. Figure 3 shows that for the Phys., 65 (1973) 411. RBG, where γd ≡ γ+, the el-ph coupling shifts the transi- [18] Wentzel G., Phys. Rev., 83 (1951) 168; Bardeen J., tion towards stronger el-el coupling. Note also that when Rev. Mod. Phys., 23 (1951) 261.