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arXiv:0705.4552 · Europhysics Letters (EPL)

Tunneling into strongly biased Tomonaga-Luttinger liquid

Maxim Trushin, Alexander L. Chudnovskiy

Published Mar 12, 2008• 11 citations
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Abstract

We calculate the tunneling density of states for a Tomonaga-Luttinger liquid placed under a strong bias voltage. For the tunneling through a side-coupled point contact, one can observe the power law singularities in the tunneling density of states separately for the right- and left-movers despite the point-like tunnel contact. Deviations of the nonequilibrium tunneling exponents from the equilibrium case are discussed.

DOI
10.1209/0295-5075/82/17008

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We calculate the tunneling density of states for a Tomonaga-Luttinger liquid placed under a strong bias voltage. For the tunneling through a side-coupled point contact, one can observe the power law singularities in the tunneling density of states separately for the right- and left-movers despite the point-like tunnel contact. Deviations of the nonequilibrium tunneling exponents from the equilibrium case are discussed.
It is well known that the pristine bulk of an interacting one-dimensional system in Tomonaga-Luttinger liquid (TLL) phase shows power law suppression of quasiparticle tunneling amplitude for all values of TLL parameter ggg, in the zero-energy limit. We perform a density matrix renormalization group (DMRG) study of a fully symmetric Y junction of TLL wires and observe an anomalous enhancement of the tunneling density of states (TDOS) in the vicinity of the junction for both (a) interacting bosons case and (b) interacting fermions case, when g>1g>1g>1. We also observe suppression of TDOS for g<1g<1g<1 for both bosonic and fermionic cases. We find that the TDOS enhancements follow different power laws for bosonic and fermionic cases which suggests that these represent distinct fixed points, owing to statistical correlations that play an important role at the Y junction. Analysis of static conductance for the junction indicates that the fixed point for 1<g<31<g<31<g<3 resembles the mysterious MMM fixed point of Y junction predicted by Oshikawa, Chamon, and Affleck [J. Stat. Mech. (2006) P02008]. We also show that the TDOS enhancement spans over a length scale of $\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1}$ from the junction, for 1<g<31<g<31<g<3.
The transport through a barrier is studied for a Tomonaga-Luttinger liquid of finite length connected to reservoirs. An effective action for the phase variable at the barrier is derived for spatially varying electric field. In the dc limit only the total voltage drop between the left and right reservoirs appears in the action. We discuss crossovers of the renormalization group flow and hence of the temperature and wire length dependence of the conductance, taking into account the location of the barrier.
We calculate the tunneling density of states (TDOS) in a dissipationless three-wire junction of interacting spin-1/2 electrons, and find an anomalous enhancement of the TDOS in the zero-bias limit, even for repulsive interactions for several bosonic fixed points. This enhancement is physically related to the reflection of holes from the junction for incident electrons, and it occurs only in the vicinity of the junction ($x<{v}_{\mathrm{min}}/2\ensuremath{\omega}$, where vmin{v}_{\mathrm{min}}vmin​ is the minimum of the velocity of charge or spin excitations and $\ensuremath{\omega}$ is the bias frequency), crossing over to the bulk value, which is always suppressed at larger distances. The TDOS exponent can be directly probed in an STM experiment by measuring the differential tunneling conductance as a function of either the bias voltage or temperature as done in C. Blumenstein et al., Nat. Phys. 7, 776 (2011).
As voltage decreases d.c. condctivity of a Tomonaga-Luttinger liquid wire collapses to a small value determined by the length of the wire and its contacts with the leads. In condition that voltage drop (V) mostly occurs across a tunnel barrier inside the wire the tunneling density of states and, hence, the differential conductivity are shown to exhibit an interference structure resulted from the transition of the Luttinger liquid quasiparticles into free electrons at the exits from the wire. The finite length correction to the scale-invariant V2/g−2V^{2/g-2}V2/g−2 dependence of the conductivity oscillates as a function of voltage with periodicities related to both rigth and left traversal times.
We study the current produced in a Tomonaga-Luttinger liquid by an applied bias and by weak, pointlike impurity potentials which are oscillating in time. We use bosonization to perturbatively calculate the current up to second order in the impurity potentials. In the regime of small bias and low pumping frequency, both the dc and ac components of the current have power-law dependences on the bias and pumping frequencies with an exponent $2K\ensuremath{-}1$ for spinless electrons, where KKK is the interaction parameter. For K<1∕2K<1∕2K<1∕2, the current grows large for special values of the bias. For noninteracting electrons with K=1K=1K=1, our results agree with those obtained using Floquet scattering theory for Dirac fermions. We also discuss the cases of extended impurities and of spin-1∕21∕21∕2 electrons.