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Physical Review B

Coulomb drag effect between Luttinger liquids

P. Schlottmann

Published Jan 23, 2004• 16 citations
Publisher

Abstract

The exact solution of the one-dimensional Hubbard model, the supersymmetric $t\ensuremath{-}J$ model, and the gas of fermions interacting via a $\ensuremath{\delta}$-function potential is used to calculate the drag current between two parallel quantum wires. The critical exponent for the drag current at low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. For a repulsive interaction between the carriers the drag current is opposite to the driving current, while for attractive potentials the two currents are parallel.

DOI
10.1103/physrevb.69.035110

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The exact solution of the one-dimensional Hubbard model, the supersymmetric t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model, and the gas of fermions interacting via a \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potential is used to calculate the drag current between two parallel quantum wires. The critical exponent for the drag current at low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. For a repulsive interaction between the carriers the drag current is opposite to the driving current, while for attractive potentials the two currents are parallel.
The exact solution of the one-dimensional Calogero–Sutherland models with spin-exchange and sinh− 2(r) and r − 2 interactions, and the supersymmetric t–J-model with long-range exchange interaction is used to calculate the drag current between two parallel quantum wires. The critical exponent for the drag current at low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. Since the interaction in the above models is repulsive, the drag current is opposite to the driving current.
Charge carriers moving in a quantum wire may induce, via the Coulomb interaction and momentum conservation, a drag current in another quantum wire located nearby. In addition, tunnelling of carriers between the two parallel wires is considered. For spinless carriers the problem can be mapped onto the one-dimensional Hubbard model. The exact solution of the Hubbard chain and conformal field theory are used to calculate the critical exponents for the temperature dependence of the drag current between the two wires. The exponents are non-universal and depend on the interaction strength, the band filling and the tunnelling amplitude. For a repulsive interaction the drag current is opposite to the driving current, while for an attractive interaction they are parallel.
We calculate the critical exponent of the temperature dependence of the drag current between two parallel quantum wires. The two wires have carriers with equal mass but they may have different electron density. The electrons interact via a $\ensuremath{\delta}$-function spin-exchange potential, which is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or spin-triplet state. This interaction leads to the formation of spin-singlet or spin-triplet bound states of the Cooper type (preformed hard core bosons that do not condensate). Depending on the parameters, the drag current can be parallel or opposite to the driving current. The critical exponent of the drag current is calculated using the Bethe Ansatz solution of the model and conformal field theory.
The exact solution of the one-dimensional Hubbard model (for repulsive and attractive U)U)U) is used to calculate the tunneling rates of single electrons and Cooper pairs between two quantum wires. The critical exponents for the tunneling at small frequencies and low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. The situations of crossing wires, tunneling at a boundary, and resonant tunneling (through a bound state) are considered.
In a one-dimensional electron gas at low enough density, the magnetic (spin) exchange energy JJJ between neighboring electrons is exponentially suppressed relative to the characteristic charge energy, the Fermi energy EF{E}_{F}EF​. At nonzero temperature TTT, the energy hierarchy J⪡T⪡EFJ⪡T⪡{E}_{F}J⪡T⪡EF​ can be reached, and we refer to this as the spin-incoherent Luttinger liquid state. We discuss the Coulomb drag between two parallel quantum wires in the spin-incoherent regime, as well as the crossover to this state from the low-temperature regime by using a model of a fluctuating Wigner solid. As the temperature increases from zero to above JJJ for a fixed electron density, the 2kF2{k}_{F}2kF​ oscillations in the density-density correlations are lost. As a result, the temperature dependence of the Coulomb drag is dramatically altered and nonmonotonic dependence may result. Drag between wires of equal and unequal density are discussed, as well as the effects of weak disorder in the wires. We speculate that weak disorder may play an important role in extracting information about quantum wires in real drag experiments.
We study the transconductance for two coupled one-dimensional wires or edge states described by Luttinger liquid models. The wires are assumed to interact over a finite segment. We find for the interaction parameter g=1/2g\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2g=1/2 that the drag rate is finite at zero temperature, which cannot occur in a Fermi-liquid system. The zero temperature drag is, however, cut off at low temperature due to the finite length of the wires. We also consider edge states in the fractional quantum Hall regime, and suggest that low temperature enhancement of the drag effect might be seen in this regime.