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arXiv:cond-mat/0509675 · Journal of Statistical Mechanics Theory and Experiment

Junctions of three quantum wires

Masaki Oshikawa, Claudio Chamon, Ian K. Affleck

Published Feb 1, 2006• 154 citations
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Abstract

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.

DOI
10.1088/1742-5468/2006/02/p02008

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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. This is the simplest problem of a quantum junction between Tomonaga-Luttinger liquids in which Fermi statistics enter in a nontrivial way. We present a direct connection between this problem and the dissipative Hofstadter problem, or quantum Brownian motion in two dimensions in a periodic potential and an external magnetic field, which in turn is connected to open string theory in a background electromagnetic field. We find nontrivial fixed points corresponding to a chiral conductance tensor leading to an asymmetric flow of the current.
We study the effects of electron-electron interactions on the transport properties of a junction of three quantum wires enclosing a magnetic flux. The wires are modeled as single channel spin-12\frac{1}{2}21​ Tomonaga--Luttinger liquids. The system exhibits a rich phase diagram as a function of the electronic interaction strength, which includes a chiral fixed point with an asymmetric current flow highly sensitive to the sign of the flux and another fixed point where pair tunneling dominates, which is similar to the case of spinless electrons. While in the case of spinless electrons the perturbations that correspond to unequal couplings between the three wires are always irrelevant, we find that, when the electron spin is included, there are small regions in the phase diagram where a current flows only between two of the wires and the third wire is decoupled.
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. Copyright (c) EPLA, 2011
We calculate the conductances of a three-terminal junction setup of spinless Luttinger liquid wires threaded by a magnetic flux, allowing for different interaction strength ${g}_{3}\ensuremath{\ne}g$ in the third wire. We employ the fermionic representation in the scattering state picture, allowing for a direct calculation of the linear response conductances, without the need of introducing contact resistances at the connection points to the outer ideal leads. The matrix of conductances is parametrized by three variables. For these we derive coupled renormalization group (RG) equations, by summing up infinite classes of contributions in perturbation theory. The resulting general structure of the RG equations may be employed to describe junctions with an arbitrary number of wires and arbitrary interaction strength in each wire. The fixed point structure of these equations (for the chiral Y junction) is analyzed in detail. For repulsive interaction (g,g3>0g,{g}_{3}>0g,g3​>0) there is only one stable fixed point, corresponding to the complete separation of the wires. For attractive interaction (g<0g<0g<0 and/or g3<0{g}_{3}<0g3​<0) four fixed points are found, the stability of which depends on the interaction strength. We confirm our previous weak-coupling result of lines of fixed points for special values of the interaction parameters reaching into the strong-coupling domain. We find new fixed points not discussed before, even at the symmetric line g=g3g={g}_{3}g=g3​, at variance with the results of M. Oshikawa et al. [J. Stat. Mech. (2006) P02008]. The pair-tunneling phenomenon conjectured by the latter authors is not found by us.
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 point contact tunneling between two leads of a Tomonaga-Luttinger liquid through two degenerate resonant levels in parallel. This is one of the simplest cases of a quantum junction problem where the Fermi statistics of the electrons plays a nontrivial role through the Klein factors appearing in bosonization. Using a mapping to a ``generalized Coulomb model'' studied in the context of the dissipative Hofstadter model, we find that any asymmetry in the tunneling amplitudes from the two leads grows at low temperatures, so that ultimately there is no conductance across the system. For the symmetric case, we identify a nontrivial fixed point of this model; the conductance at that point is generally different from the conductance through a single resonant level.