Impurity scattering in a Luttinger liquid with electron-phonon coupling

Abstract

We study the influence of electron-phonon coupling on electron transport through a Luttinger liquid with an embedded weak scatterer or weak link. We derive the renormalization group (RG) equations which indicate that the directions of RG flows can change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. This results in the rich phase diagram with up to three fixed points: an unstable one with a finite value of conductance and two stable ones, corresponding to an ideal metal or insulator.PACS numbers: 71.10. Pm, 71.10.Hf, 72.10.Fk, 73.20.Mf Interacting electrons in one dimension are known to form a Luttinger liquid (LL) characterized by powerlaw correlation functions [1]. This characteristic feature of the LL has been established via conductance measurements and a scanning tunneling microscopy (STM) both in carbon nanotubes [2] and semiconductor quantum wires [3]. Embedding a potential impurity into the LL leads [4,5] to a universal (i.e. impurity-independent) power-law (in temperature T ) suppression of the transmission amplitude through the LL and suppression of the tunneling density of states (TDoS) near the impurity, with the latter fading away with the distance [6,7].The electron-phonon (el-ph) coupling in addition to the (Coulomb) electron-electron (el-el) repulsion in the LL is known to result in the formation of two polaron branches with different propagation velocities (see, for example, [8]). In the present Letter we show that embedding a single scatterer into such an el-ph liquid results in a rather rich phase diagram: depending on the relative strength of the el-el and el-ph coupling and on the ratio of the Fermi to sound velocities, the system can be an ideal metal (with conductance g = 1 in the units of e 2 /h as in the absence of impurities [9]) or an ideal insulator for any impurity strength, or to be in an intermediate state from which it can flow either to the metallic or insulating limit, depending on the impurity strength.To show this we consider both the weak back-scatterer (WS) and weak link (WL) limit, the latter corresponding to a weak tunneling coupling between the two halves of the LL. In the phononless case the WS and WL amplitudes are known to scale at low energies as ε γ− and ε γ+ , respectively. For the repulsive el-el interaction the exponents γ − ≡ K − 1 < 0 and γ + = K −1 − 1 > 0 where K < 1 is the Luttinger parameter. Therefore, the renormalization group (RG) flows always go in the direction of stronger scattering (weaker transmission) resulting at T = 0 in the insulating phase for any scatterer [4]. The el-ph coupling leads to γ ± → γ ± , with each of the exponents γ ± changing sign at different values of bulk parameters. In the WS limit which has been previously considered [10], the change of sign of γ − was reversing the RG flows indicating that the weak scatterer becomes irrelevant for the sufficiently strong el-ph interaction and the LL thus remains in the metallic state. But the cons...

Local impurity in a multichannel Luttinger liquid

“…The RG dimension ∆ conf is fully governed by the Luttinger matrix K, (10). Thus its role in defining the RG flows is similar to that of the Luttinger parameter K for the single-channel LL.…”

“…This might happen when the channels are totally independent, e.g. they are spatially remote or have different physical nature, like in the electron-phonon LL [9][10][11] . In this case channel 2 remains conducting whereas oneparticle scattering is dominant in channel 1, so that for isolated channels are either in the cc (for K 1 > 1) or the ic (for K 1 < 1) configuration.…”

Duality in multi-channel Luttinger Liquid with local scatterer

“…Solid state systems like Fractional Quantum Hall systems [1], Quantum Spin Quantum Hall or topological insulators [2,3], carbon nanotubes [4] and quantum wires are now routinely described in terms of few (chiral or not) channels with intra-and inter-channel interactions. Similar description is also applied to cold atoms mixtures [5], ballistic quasi-one-dimensional waveguides [6], hollow-core fibers [7] and one-dimensional electron-phonon systems [8,9].…”

“…We have also established duality transformation between different subsets of Luttinger Liquid phases known for single-channel liquid as duality between weak link and weak scatterer limits. [7] and one-dimensional electron-phonon systems [8,9].…”

“…We have also established duality transformation between different subsets of Luttinger Liquid phases known for single-channel liquid as duality between weak link and weak scatterer limits. [7] and one-dimensional electron-phonon systems [8,9].The intra-channel interactions being taking into account lead to formation of Luttinger Liquid [10] for each individual channel. The perturbations that scatter or tunnel particles within the same channel are either irrelevant (scaling dimension is higher than physical dimension which is unity for a local perturbation) or relevant (the scaling dimension is lower than one) in terms of renormalization group analysis [11].…”