Conductance of Tomonaga-Luttinger liquid wires and junctions with resistances

We study the effect that resistive regions have on the conductance of a quantum wire with interacting electrons which is connected to Fermi liquid leads. Using the bosonization formalism and a Rayleigh dissipation function to model the power dissipation, we use both scattering theory and Green's function techniques to derive the DC conductance. The resistive regions are generally found to lead to incoherent transport. For a single wire, we find that the resistance adds in series to the contact resistance of h / e 2 for spinless electrons, and the total resistance is independent of the Luttinger parameter K W of the wire. We numerically solve the bosonic equations to illustrate what happens when a charge density pulse is incident on the wire; the results depend on the parameters of the resistive and interacting regions in interesting ways. For a junction of Tomonaga-Luttinger liquid wires, we use a dissipationless current splitting matrix to model the junction. For a junction of three wires connected to Fermi liquid leads, there are two families of such matrices; we find that the conductance matrix generally depends on K W for one family but is independent of K W for the other family, regardless of the resistances present in the system.We study the effect that resistive regions have on the conductance of a quantum wire with interacting electrons which is connected to Fermi liquid leads. Using the bosonization formalism and a Rayleigh dissipation function to model the power dissipation, we use both scattering theory and Green's function techniques to derive the DC conductance. The resistive regions are generally found to lead to incoherent transport. For a single wire, we find that the resistance adds in series to the contact resistance of h / e 2 for spinless electrons, and the total resistance is independent of the Luttinger parameter K W of the wire. We numerically solve the bosonic equations to illustrate what happens when a charge density pulse is incident on the wire; the results depend on the parameters of the resistive and interacting regions in interesting ways. For a junction of Tomonaga-Luttinger liquid wires, we use a dissipationless current splitting matrix to model the junction. For a junction of three wires connected to Fermi liquid leads, there are two families of such matrices; we find that the conductance matrix generally depends on K W for one family but is independent of K W for the other family, regardless of the resistances present in the system.

arXiv:1011.5058Europhysics Letters (EPL)Mar 1, 201113 citations

Model of resistances in systems of Tomonaga-Luttinger liquid wires

Abhiram Soori, Diptiman Sen

In a recent paper, we combined the technique of bosonization with the concept of a Rayleigh dissipation function to develop a model for resistances in one-dimensional systems of interacting spinless electrons [Europhys. Lett. 93, 57007 (2011)]. We also studied the conductance of a system of three wires by using a current splitting matrix $M$ at the junction. In this paper, we extend our earlier work in several ways. The power dissipated in a three-wire system is calculated as a function of $M$ and the voltages applied in the leads. By combining two junctions of three wires, we examine a system consisting of two parallel resistances. We study the conductance of this system as a function of the $M$ matrices and the two resistances; we find that the total resistance is generally quite different from what one expects for a classical system of parallel resistances. We do a sum over paths to compute the conductance of this system when one of the two resistances is taken to be infinitely large. We study the conductance of a three-wire system of interacting spin-1$/$2 electrons, and show that the charge and spin conductances can generally be different from each other. Finally, we consider a system of two wires that are coupled by a dissipation function, and we show that this leads to a current in one wire when a voltage bias is applied across the other wire.In a recent paper, we combined the technique of bosonization with the concept of a Rayleigh dissipation function to develop a model for resistances in one-dimensional systems of interacting spinless electrons [Europhys. Lett. 93, 57007 (2011)]. We also studied the conductance of a system of three wires by using a current splitting matrix $M$ at the junction. In this paper, we extend our earlier work in several ways. The power dissipated in a three-wire system is calculated as a function of $M$ and the voltages applied in the leads. By combining two junctions of three wires, we examine a system consisting of two parallel resistances. We study the conductance of this system as a function of the $M$ matrices and the two resistances; we find that the total resistance is generally quite different from what one expects for a classical system of parallel resistances. We do a sum over paths to compute the conductance of this system when one of the two resistances is taken to be infinitely large. We study the conductance of a three-wire system of interacting spin-1$/$2 electrons, and show that the charge and spin conductances can generally be different from each other. Finally, we consider a system of two wires that are coupled by a dissipation function, and we show that this leads to a current in one wire when a voltage bias is applied across the other wire.

arXiv:1103.0883Physical Review BJul 25, 20116 citations

Conductance of junctions of multiple interacting quantum wires and long Aharonov-Bohm-Kondo rings

Zheng Shi

In this thesis, we calculate the linear dc conductance of two types of multi-terminal interacting systems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads, and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantum dot in the Kondo regime respectively. In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, the conductance is formally given by the Landauer formula with renormalized single-particle S-matrix elements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish even in the leads, the conductance has an additional contribution dependent on the interaction strength in the leads. We calculate this additional contribution both at the first order in interaction and in the random phase approximation, and heuristically relate the FL conductance to the TLL conductance through a "contact resistance" between an FL lead and a TLL wire. In long ABK rings, where the interaction is due to spin-flip scattering at a quantum dot in the Kondo regime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form, and a connected part that can be approximately eliminated at low temperatures. For a closed long ABK ring, where the electric current is conserved in the ring, the high-temperature conductance has qualitatively different behaviors for temperatures greater than and lower than the characteristic energy scale v_F /L, where v_F is the Fermi velocity and L is the ring circumference. Meanwhile, for an open long ABK ring where electrons may leak into the side leads coupled to ring arms, as long as the ring arms have both small transmission and small reflection, the ring behaves as a two-path Aharonov-Bohm interferometer, and we predict the observation of a π/2 phase shift due to scattering off the Kondo singlet formed at low energies around the impurity spin.In this thesis, we calculate the linear dc conductance of two types of multi-terminal interacting systems: junctions of interacting quantum wires attached to Tomonaga-Luttinger liquid (TLL) leads, and closed and open long Aharonov-Bohm-Kondo (ABK) rings. In both cases, we obtain corrections to the non-interacting Landauer formula, arising from interactions in the TLL leads and the quantum dot in the Kondo regime respectively. In junctions of interacting quantum wires, if the wires are attached to Fermi liquid (FL) leads, the conductance is formally given by the Landauer formula with renormalized single-particle S-matrix elements. If, however, the wires are attached to TLL leads, i.e. the interaction does not vanish even in the leads, the conductance has an additional contribution dependent on the interaction strength in the leads. We calculate this additional contribution both at the first order in interaction and in the random phase approximation, and heuristically relate the FL conductance to the TLL conductance through a "contact resistance" between an FL lead and a TLL wire. In long ABK rings, where the interaction is due to spin-flip scattering at a quantum dot in the Kondo regime, the linear dc conductance consists of two parts: a disconnected part of the Landauer form, and a connected part that can be approximately eliminated at low temperatures. For a closed long ABK ring, where the electric current is conserved in the ring, the high-temperature conductance has qualitatively different behaviors for temperatures greater than and lower than the characteristic energy scale v_F /L, where v_F is the Fermi velocity and L is the ring circumference. Meanwhile, for an open long ABK ring where electrons may leak into the side leads coupled to ring arms, as long as the ring arms have both small transmission and small reflection, the ring behaves as a two-path Aharonov-Bohm interferometer, and we predict the observation of a π/2 phase shift due to scattering off the Kondo singlet formed at low energies around the impurity spin.

Jan 1, 20170 citations

Transient conductance between Fermi and Luttinger liquids

Thomas Kloss, Joseph Weston, Xavier Waintal

Although the intrinsic conductance of an interacting one-dimensional system is renormalized by the electron-electron correlations, it has been known for some time that this renormalization is washed out by the presence of the (non-interacting) electrodes to which the wire is connected. Here, we study the transient conductance of such a wire: a finite voltage bias is suddenly applied across the wire and we measure the current before it has enough time to reach its stationary value. These calculations allow us to extract the Sharvin (contact) resistance of Luttinger and Fermi liquids. In particular, we find that a perfect junction between a Fermi liquid electrode and a Luttinger liquid electrode is characterized by a contact resistance that consists of half the quantum of conductance in series with half the intrinsic resistance of an infinite Luttinger liquid. These results were obtained using two different methods: a dynamical Hartree-Fock approach and a self-consistent Boltzmann approach. Although these methods are formally approximate we find a perfect match with the exact results of Luttinger/Fermi liquid theory.Although the intrinsic conductance of an interacting one-dimensional system is renormalized by the electron-electron correlations, it has been known for some time that this renormalization is washed out by the presence of the (non-interacting) electrodes to which the wire is connected. Here, we study the transient conductance of such a wire: a finite voltage bias is suddenly applied across the wire and we measure the current before it has enough time to reach its stationary value. These calculations allow us to extract the Sharvin (contact) resistance of Luttinger and Fermi liquids. In particular, we find that a perfect junction between a Fermi liquid electrode and a Luttinger liquid electrode is characterized by a contact resistance that consists of half the quantum of conductance in series with half the intrinsic resistance of an infinite Luttinger liquid. These results were obtained using two different methods: a dynamical Hartree-Fock approach and a self-consistent Boltzmann approach. Although these methods are formally approximate we find a perfect match with the exact results of Luttinger/Fermi liquid theory.

Oct 2, 20170 citations

Junctions of three quantum wires

Masaki Oshikawa, Claudio Chamon, Ian K. Affleck

We study a junction of three quantum wires enclosing a magnetic flux. The wires are modelled as single-channel spinless Tomonaga–Luttinger liquids. This is the simplest problem of a quantum junction between Tomonaga–Luttinger liquids in which Fermi statistics enter in a non-trivial way. We study the problem using a mapping onto the dissipative Hofstadter model, describing a single particle moving on a plane in a magnetic field and a periodic potential coupled to a harmonic oscillator bath. Alternatively we study the problem by identifying boundary conditions corresponding to the low energy fixed points. We obtain a rich phase diagram including a chiral fixed point in which the asymmetric current flow is highly sensitive to the sign of the flux and a phase in which electron pair tunnelling dominates. We also study the effects on the conductance tensor of the junction of contacting the three quantum wires to Fermi liquid reservoirs.We study a junction of three quantum wires enclosing a magnetic flux. The wires are modelled as single-channel spinless Tomonaga–Luttinger liquids. This is the simplest problem of a quantum junction between Tomonaga–Luttinger liquids in which Fermi statistics enter in a non-trivial way. We study the problem using a mapping onto the dissipative Hofstadter model, describing a single particle moving on a plane in a magnetic field and a periodic potential coupled to a harmonic oscillator bath. Alternatively we study the problem by identifying boundary conditions corresponding to the low energy fixed points. We obtain a rich phase diagram including a chiral fixed point in which the asymmetric current flow is highly sensitive to the sign of the flux and a phase in which electron pair tunnelling dominates. We also study the effects on the conductance tensor of the junction of contacting the three quantum wires to Fermi liquid reservoirs.

arXiv:cond-mat/0509675Journal of Statistical Mechanics Theory and ExperimentFeb 1, 2006154 citations

Transient and Sharvin resistances of Luttinger liquids

Thomas Kloss, Joseph Weston, Xavier Waintal

Although the intrinsic conductance of an interacting one-dimensional system is renormalized by the electron-electron correlations, it has been known for some time that this renormalization is washed out by the presence of the (noninteracting) electrodes to which the wire is connected. Here, we study the transient conductance of such a wire: a finite voltage bias is suddenly applied across the wire and we measure the current before it has enough time to reach its stationary value. These calculations allow us to extract the Sharvin (contact) resistance of Luttinger and Fermi liquids. In particular, we find that a perfect junction between a Fermi liquid electrode and a Luttinger liquid electrode is characterized by a contact resistance that consists of half the quantum of conductance in series with half the intrinsic resistance of an infinite Luttinger liquid. These results were obtained using two different methods: a dynamical Hartree-Fock approach and a self-consistent Boltzmann approach. Although these methods are formally approximate, we find a perfect match with the exact results of Luttinger/Fermi liquid theory.Although the intrinsic conductance of an interacting one-dimensional system is renormalized by the electron-electron correlations, it has been known for some time that this renormalization is washed out by the presence of the (noninteracting) electrodes to which the wire is connected. Here, we study the transient conductance of such a wire: a finite voltage bias is suddenly applied across the wire and we measure the current before it has enough time to reach its stationary value. These calculations allow us to extract the Sharvin (contact) resistance of Luttinger and Fermi liquids. In particular, we find that a perfect junction between a Fermi liquid electrode and a Luttinger liquid electrode is characterized by a contact resistance that consists of half the quantum of conductance in series with half the intrinsic resistance of an infinite Luttinger liquid. These results were obtained using two different methods: a dynamical Hartree-Fock approach and a self-consistent Boltzmann approach. Although these methods are formally approximate, we find a perfect match with the exact results of Luttinger/Fermi liquid theory.

arXiv:1710.00895Physical review. B./Physical review. BApr 23, 201813 citations

M

at the junction. In this paper, we extend our earlier work in several ways. The power dissipated in a three-wire system is calculated as a function of

M

and the voltages applied in the leads. By combining two junctions of three wires, we examine a system consisting of two parallel resistances. We study the conductance of this system as a function of the

M

matrices and the two resistances; we find that the total resistance is generally quite different from what one expects for a classical system of parallel resistances. We do a sum over paths to compute the conductance of this system when one of the two resistances is taken to be infinitely large. We study the conductance of a three-wire system of interacting spin-1$/$2 electrons, and show that the charge and spin conductances can generally be different from each other. Finally, we consider a system of two wires that are coupled by a dissipation function, and we show that this leads to a current in one wire when a voltage bias is applied across the other wire.