Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Transport in interacting 1d wires with impurity: RG approach and dephasing

Alois Dirnaichner

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

We study transport in interacting one-dimensional systems with a scattering object. In particular we are interested in inelastic scattering processes. Very recent theoretical [3] and experimental [54] research points toward a strong influence of incoherent scattering on the low-temperature transport features of one-dimensional systems with extended scatterers. Furthermore, if inelastic processes are strong a straightforward application of the renormalization group (RG) approach is not possible. In the field of transport in one-dimensional weakly interacting electron systems, this procedure is the state-of-the art method to extend results for weak interaction to the low temperature regime [18,27,43, 47, 52,57, 62, 67,74]. The underlying assumption, namely that the first-order correction in interaction incorporates all effects of the interaction and reproduces itself in higher orders, is widely accepted as a valid “educated guess” disregarding the fact that inelastic processes, if present, arise in subleading orders. Indeed, for the simple case of a single impurity in an infinite wire it is common believe that the approach is valid [6,74]. Having this in mind, systems with a great variety of additional degrees of freedom were treated in the same fashion: Junctions of wires [43] and more complex geometries [18], anti-resonant [47] and resonant [57, 62] scatterers, just to name a few. A careful examination of the interplay between the renormalization and possible inelastic processes or dephasing in these cases is missing. Our strategy is to calculate the second order in interaction for the single impurity situation, thereby establishing a framework with a real-space diagrammatic technique, and to apply this tool to more complex situations, i.e., a finite interacting region and an additional impurity. The diagrammatic technique enables us to identify specific diagrams where the particle number at a given energy is not conserved or the interaction can lead to dephasing in the system. In particular, by modelling the compound scatterer by an energy dependent S-matrix of the Breit-Wigner type, we are able to find a fingerprint of weak dephasing which carries some features of its counterpart in disordered Luttinger liquids [32].

Related papers

Dephasing and Weak Localization in Disordered Luttinger Liquid

I. V. Gornyi, A. D. Mirlin, Dmitry G. Polyakov

We study the transport properties of interacting electrons in a disordered quantum wire within the framework of the Luttinger liquid model. The conductivity at finite temperature is nonzero only because of inelastic electron-electron scattering. We demonstrate that the notion of weak localization is applicable to the strongly correlated one-dimensional electron system. We calculate the relevant dephasing rate, which for spinless electrons is governed by the interplay of electron-electron interaction and disorder, thus vanishing in the clean limit.

arXiv:cond-mat/0407305•Physical Review Letters•Jul 22, 2005•73 citations

Functional Renormalization-Group Analysis of Luttinger Liquids with Impurities

Sabine Andergassen

In one-dimensional quantum wires the interplay of electron correlations and impurities strongly influences the low-energy physics. The diversity of energy scales and the competition of correlations in interacting Fermi systems can be treated very efficiently with the functional renormalization group (fRG), describing the gradual evolution from a microscopic model Hamiltonian to the effective low-energy action as a function of a continuously decreasing energy cutoff. The fRG provides the universal low-energy asymptotics as well as nonuniversal properties, and in particular an answer to the important question at what scale the ultimate asymptotics sets in. We investigate the influence of impurities on spectral and transport properties of fermionic lattice models with short-range interactions. The results capture relevant energy scales and crossover phenomena, in addition to the universal low-energy asymptotics. For weak and intermediate impurity strengths the asymptotic behavior is approached only at rather low energy scales, accessible only for very large systems. For spin-1/2 systems two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.

arXiv:cond-mat/0603341•Jan 1, 2006•4 citations

Functional renormalization group studies of quantum transport through mesoscopic systems

Severin G. Jakobs, Herbert Schoeller

The progress in micro-fabrication technologies makes it possible to address individual, tailored, mesoscopic structures with sizes on a nanometer to micrometer scale. The transport properties of such quasi zero- or one-dimensional systems are considerably influenced by the electron-electron interaction. In theoretical studies this often becomes apparent in the emergence of divergences in a perturbative treatment of the interaction. A successful approach may then be given by renormalization group (RG) methods which are based on the idea of treating energy scales successively from high to low. In the functional RG (fRG) an exact infinite hierarchy of flow equations for multi-particle Green or vertex functions is derived, that allows for systematic approximations on different levels. In numerous studies the fRG has been found to be a powerful tool for the investigation of low dimensional Fermi systems. Usually it is formulated within Matsubara formalism which applies to a system in thermal equilibrium. A central topic of this thesis is the implementation of the fRG within Keldysh formalism, providing the possibility to treat equilibrium and non-equilibrium situations on equal footing. The resulting Keldysh fRG is used in this thesis to explore the transport properties of quantum wires with several barriers and of the single impurity Anderson model. The thesis starts with an introduction to Keldysh formalism and to related concepts. Then the structure of the Green and vertex functions of Keldysh formalism is investigated. This includes studying the consequences of causality, characterizing thermal equilibrium, and formulating corresponding relations for the multi-particle functions. After that, the implementation of the fRG within Keldysh formalism is presented in detail. The flow equations for the vertex functions are derived via their diagrammatic expansion. Two especially useful flow parameters are determined: one of them cuts off the imaginary frequencies that appear as poles of the Fermi functions of the reservoirs. The second flow parameter is an artificially enhanced reservoir hybridization; on one hand it has the advantage to preserve the fluctuation dissipation theorem in equilibrium even in frequency dependent approximations; on the other hand, its use can be more cumbersome. One of the models studied in this thesis is a tight-binding chain with nearest-neighbour interaction. At weak and intermediate interactions it belongs to the class of Luttinger liquids (LLs) whose low energy properties are characterized by power laws. The transport properties of LLs have been studied experimentally for example on carbon nanotubes. However, the presence of transport barriers at the contacts make the interpretation of the experimental data difficult. A theoretical treatment with the fRG permits a rather large flexibility to model the barrier configuration. In this thesis the fRG is first used in its equilibrium Matsubara implementation in order to analyse the temperature dependence of the linear conductance in the presence of two, three or four barriers. It is shown that temperature induces an averaging over quantum phases of propagation, leading to combinations of the power law exponents of the individual barriers. An experimental set-up that allows to verify this behaviour is proposed. In a second step the Keldysh version of the fRG is applied to the problem of transport through a quantum wire at finite bias voltage. For a wire with two high contact barriers it is found that the non-equilibrium particle distribution in the wire is responsible for the appearance of a two-fold power law in the density of states. An additional weakly attached probe lead can be used to make the two different power laws visible in the conductance. The second model which is analysed is the one of the single Anderson impurity, often considered as a paradigm for the investigation of strong local electron correlations. The latter become noticeable in the appearance of the Kondo scale, an energy exponentially small in the interaction strength, that characterizes the low energy physics of the model in the regime of spin fluctuations. On the basis of this model an advanced frequency dependent approximation for the Keldysh fRG is developed in this thesis. Such an approximation can account for subleading effects of the interaction, namely the fact that two-particle interactions can generate decay rates for single-particle states. The derivation of a consistent approximation scheme is presented, that turns out to yield a good description of the model at moderate interaction strength.

•Jan 1, 2010•7 citations

Transport in a spin-incoherent Luttinger liquid

Gregory A. Fiete, Karyn Le Hur, Leon Balents

We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification $g=2{g}_{c}$, where $g$ is the interaction parameter in a spinless Luttinger liquid and ${g}_{c}$ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.

arXiv:cond-mat/0505186•Physical Review B•Sep 13, 2005•33 citations

Regular networks of Luttinger liquids

K. Kazymyrenko, Benoît Douçot

We consider arrays of Luttinger liquids, where each node is described by a unitary scattering matrix. In the limit of small electron-electron interaction, we study the evolution of these scattering matrices as the high-energy single particle states are gradually integrated out. Interestingly, we obtain the same renormalization group equations as those derived by Lal, Rao, and Sen, for a system composed of a single node coupled to several semi-infinite one-dimensional wires. The main difference between the single node geometry and a regular lattice is that in the latter case, the single particle spectrum is organized into periodic energy bands, so that the renormalization procedure has to stop when the last totally occupied band has been eliminated. We therefore predict a strongly renormalized Luttinger liquid behavior for generic filling factors, which should exhibit power-law suppression of the conductivity at low temperatures ${E}_{F}∕({k}_{F}a)⪡{k}_{B}T⪡{E}_{F}$, where $a$ is the lattice spacing and ${k}_{F}a⪢1$. At lower temperatures $[{k}_{B}T⪡{E}_{F}∕({k}_{F}a)]$, a regular network is generally a coherent conductor, but with a much lower Fermi velocity than for a noninteracting electron gas. Some fully insulating ground states are expected only for a discrete set of integer filling factors for the electronic system. A detailed discussion of the scattering matrix flow and its implication for the low energy band structure is given on the example of a square lattice.

arXiv:cond-mat/0407268•Physical Review B•Feb 17, 2005•26 citations

The Functional Renormalization Group for Zero-Dimensional Quantum Systems in and out of Equilibrium

Christoph Karrasch

We study transport properties of quantum impurity systems using the functional renormalization group. The latter is an RG-based diagrammatic tool to treat Coulomb interactions in a fast and flexible way. Prior applications, which employed a simple first-order (Hartree-Fock-like) scheme to truncate the FRG flow equations within the Matsubara formalism, succeeded in accurately describing linear transport of various quantum dot geometries at zero temperature T=0. In a nutshell, advance in this Thesis is three-fold. First, we introduce a frequency-dependent second-order approximation in order to eventually compute finite-energy properties such as the conductance at T>0 (mainly focusing on the single impurity Anderson model). Second, a generalisation of the Hartree-Fock-like approach to Keldysh space allows for addressing the non-equilibrium steady-state dynamics of the interacting resonant level model. Third, we investigate the physics of a quantum dot Josephson junction as well as the charging of a single narrow level using the first-order scheme.

arXiv:1009.3852•Sep 20, 2010•6 citations
We study transport in interacting one-dimensional systems with a scattering object. In particular we are interested in inelastic scattering processes. Very recent theoretical [3] and experimental [54] research points toward a strong influence of incoherent scattering on the low-temperature transport features of one-dimensional systems with extended scatterers. Furthermore, if inelastic processes are strong a straightforward application of the renormalization group (RG) approach is not possible. In the field of transport in one-dimensional weakly interacting electron systems, this procedure is the state-of-the art method to extend results for weak interaction to the low temperature regime [18,27,43, 47, 52,57, 62, 67,74]. The underlying assumption, namely that the first-order correction in interaction incorporates all effects of the interaction and reproduces itself in higher orders, is widely accepted as a valid “educated guess” disregarding the fact that inelastic processes, if present, arise in subleading orders. Indeed, for the simple case of a single impurity in an infinite wire it is common believe that the approach is valid [6,74]. Having this in mind, systems with a great variety of additional degrees of freedom were treated in the same fashion: Junctions of wires [43] and more complex geometries [18], anti-resonant [47] and resonant [57, 62] scatterers, just to name a few. A careful examination of the interplay between the renormalization and possible inelastic processes or dephasing in these cases is missing. Our strategy is to calculate the second order in interaction for the single impurity situation, thereby establishing a framework with a real-space diagrammatic technique, and to apply this tool to more complex situations, i.e., a finite interacting region and an additional impurity. The diagrammatic technique enables us to identify specific diagrams where the particle number at a given energy is not conserved or the interaction can lead to dephasing in the system. In particular, by modelling the compound scatterer by an energy dependent S-matrix of the Breit-Wigner type, we are able to find a fingerprint of weak dephasing which carries some features of its counterpart in disordered Luttinger liquids [32].
We study the transport properties of interacting electrons in a disordered quantum wire within the framework of the Luttinger liquid model. The conductivity at finite temperature is nonzero only because of inelastic electron-electron scattering. We demonstrate that the notion of weak localization is applicable to the strongly correlated one-dimensional electron system. We calculate the relevant dephasing rate, which for spinless electrons is governed by the interplay of electron-electron interaction and disorder, thus vanishing in the clean limit.
In one-dimensional quantum wires the interplay of electron correlations and impurities strongly influences the low-energy physics. The diversity of energy scales and the competition of correlations in interacting Fermi systems can be treated very efficiently with the functional renormalization group (fRG), describing the gradual evolution from a microscopic model Hamiltonian to the effective low-energy action as a function of a continuously decreasing energy cutoff. The fRG provides the universal low-energy asymptotics as well as nonuniversal properties, and in particular an answer to the important question at what scale the ultimate asymptotics sets in. We investigate the influence of impurities on spectral and transport properties of fermionic lattice models with short-range interactions. The results capture relevant energy scales and crossover phenomena, in addition to the universal low-energy asymptotics. For weak and intermediate impurity strengths the asymptotic behavior is approached only at rather low energy scales, accessible only for very large systems. For spin-1/2 systems two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.
The progress in micro-fabrication technologies makes it possible to address individual, tailored, mesoscopic structures with sizes on a nanometer to micrometer scale. The transport properties of such quasi zero- or one-dimensional systems are considerably influenced by the electron-electron interaction. In theoretical studies this often becomes apparent in the emergence of divergences in a perturbative treatment of the interaction. A successful approach may then be given by renormalization group (RG) methods which are based on the idea of treating energy scales successively from high to low. In the functional RG (fRG) an exact infinite hierarchy of flow equations for multi-particle Green or vertex functions is derived, that allows for systematic approximations on different levels. In numerous studies the fRG has been found to be a powerful tool for the investigation of low dimensional Fermi systems. Usually it is formulated within Matsubara formalism which applies to a system in thermal equilibrium. A central topic of this thesis is the implementation of the fRG within Keldysh formalism, providing the possibility to treat equilibrium and non-equilibrium situations on equal footing. The resulting Keldysh fRG is used in this thesis to explore the transport properties of quantum wires with several barriers and of the single impurity Anderson model. The thesis starts with an introduction to Keldysh formalism and to related concepts. Then the structure of the Green and vertex functions of Keldysh formalism is investigated. This includes studying the consequences of causality, characterizing thermal equilibrium, and formulating corresponding relations for the multi-particle functions. After that, the implementation of the fRG within Keldysh formalism is presented in detail. The flow equations for the vertex functions are derived via their diagrammatic expansion. Two especially useful flow parameters are determined: one of them cuts off the imaginary frequencies that appear as poles of the Fermi functions of the reservoirs. The second flow parameter is an artificially enhanced reservoir hybridization; on one hand it has the advantage to preserve the fluctuation dissipation theorem in equilibrium even in frequency dependent approximations; on the other hand, its use can be more cumbersome. One of the models studied in this thesis is a tight-binding chain with nearest-neighbour interaction. At weak and intermediate interactions it belongs to the class of Luttinger liquids (LLs) whose low energy properties are characterized by power laws. The transport properties of LLs have been studied experimentally for example on carbon nanotubes. However, the presence of transport barriers at the contacts make the interpretation of the experimental data difficult. A theoretical treatment with the fRG permits a rather large flexibility to model the barrier configuration. In this thesis the fRG is first used in its equilibrium Matsubara implementation in order to analyse the temperature dependence of the linear conductance in the presence of two, three or four barriers. It is shown that temperature induces an averaging over quantum phases of propagation, leading to combinations of the power law exponents of the individual barriers. An experimental set-up that allows to verify this behaviour is proposed. In a second step the Keldysh version of the fRG is applied to the problem of transport through a quantum wire at finite bias voltage. For a wire with two high contact barriers it is found that the non-equilibrium particle distribution in the wire is responsible for the appearance of a two-fold power law in the density of states. An additional weakly attached probe lead can be used to make the two different power laws visible in the conductance. The second model which is analysed is the one of the single Anderson impurity, often considered as a paradigm for the investigation of strong local electron correlations. The latter become noticeable in the appearance of the Kondo scale, an energy exponentially small in the interaction strength, that characterizes the low energy physics of the model in the regime of spin fluctuations. On the basis of this model an advanced frequency dependent approximation for the Keldysh fRG is developed in this thesis. Such an approximation can account for subleading effects of the interaction, namely the fact that two-particle interactions can generate decay rates for single-particle states. The derivation of a consistent approximation scheme is presented, that turns out to yield a good description of the model at moderate interaction strength.
We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification g=2gcg=2{g}_{c}g=2gc​, where ggg is the interaction parameter in a spinless Luttinger liquid and gc{g}_{c}gc​ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.
We consider arrays of Luttinger liquids, where each node is described by a unitary scattering matrix. In the limit of small electron-electron interaction, we study the evolution of these scattering matrices as the high-energy single particle states are gradually integrated out. Interestingly, we obtain the same renormalization group equations as those derived by Lal, Rao, and Sen, for a system composed of a single node coupled to several semi-infinite one-dimensional wires. The main difference between the single node geometry and a regular lattice is that in the latter case, the single particle spectrum is organized into periodic energy bands, so that the renormalization procedure has to stop when the last totally occupied band has been eliminated. We therefore predict a strongly renormalized Luttinger liquid behavior for generic filling factors, which should exhibit power-law suppression of the conductivity at low temperatures EF∕(kFa)⪡kBT⪡EF{E}_{F}∕({k}_{F}a)⪡{k}_{B}T⪡{E}_{F}EF​∕(kF​a)⪡kB​T⪡EF​, where aaa is the lattice spacing and kFa⪢1{k}_{F}a⪢1kF​a⪢1. At lower temperatures [kBT⪡EF∕(kFa)][{k}_{B}T⪡{E}_{F}∕({k}_{F}a)][kB​T⪡EF​∕(kF​a)], a regular network is generally a coherent conductor, but with a much lower Fermi velocity than for a noninteracting electron gas. Some fully insulating ground states are expected only for a discrete set of integer filling factors for the electronic system. A detailed discussion of the scattering matrix flow and its implication for the low energy band structure is given on the example of a square lattice.
We study transport properties of quantum impurity systems using the functional renormalization group. The latter is an RG-based diagrammatic tool to treat Coulomb interactions in a fast and flexible way. Prior applications, which employed a simple first-order (Hartree-Fock-like) scheme to truncate the FRG flow equations within the Matsubara formalism, succeeded in accurately describing linear transport of various quantum dot geometries at zero temperature T=0. In a nutshell, advance in this Thesis is three-fold. First, we introduce a frequency-dependent second-order approximation in order to eventually compute finite-energy properties such as the conductance at T>0 (mainly focusing on the single impurity Anderson model). Second, a generalisation of the Hartree-Fock-like approach to Keldysh space allows for addressing the non-equilibrium steady-state dynamics of the interacting resonant level model. Third, we investigate the physics of a quantum dot Josephson junction as well as the charging of a single narrow level using the first-order scheme.