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arXiv:0810.3513 · Physical Review Letters

Enhancement of Tunneling Density of States at a Junction of Three Luttinger Liquid Wires

Amit Agarwal, Sourin Das, Sumathi Rao, Diptiman Sen

Published Jul 10, 2009• 44 citations
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Abstract

We study the tunneling density of states (TDOS) for a junction of three Tomonaga-Luttinger liquid wires. We show that there are fixed points which allow for the enhancement of the TDOS, which is unusual for Luttinger liquids. The distance from the junction over which this enhancement occurs is of the order of x=v/(2omega), where v is the plasmon velocity and omega is the bias frequency. Beyond this distance, the TDOS crosses over to the standard bulk value independent of the fixed point describing the junction. This finite range of distances opens up the possibility of experimentally probing the enhancement in each wire individually.

DOI
10.1103/physrevlett.103.026401

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We study the tunneling density of states (TDOS) for a junction of three Tomonaga-Luttinger liquid wires. We show that there are fixed points which allow for the enhancement of the TDOS, which is unusual for Luttinger liquids. The distance from the junction over which this enhancement occurs is of the order of x=v/(2omega), where v is the plasmon velocity and omega is the bias frequency. Beyond this distance, the TDOS crosses over to the standard bulk value independent of the fixed point describing the junction. This finite range of distances opens up the possibility of experimentally probing the enhancement in each wire individually.
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).
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.
Power-law suppression of local electronic tunneling density of states (TDOS) in the zero-energy limit is a hallmark of the Luttinger liquid (LL) phase of the interacting one-dimensional electron system. We present a theoretical model which hosts the LL state with the surprising feature of enhancement rather than suppression in local TDOS originating from nonlocal and repulsive density-density interactions. Importantly, we find enhancement of TDOS in the manifold of parameter space where the system is stable in the renormalization group (RG) sense. We argue that enhancement of TDOS along with RG stability is possible only when the system has broken parity symmetry about the position of local TDOS enhancement. Such a model could be realized on the edge states of a bilayer quantum Hall system where both intralayer and interlayer density-density interactions are present mimicking the role of local and nonlocal interactions, respectively.
We study the transport properties of three Luttinger liquid wires (with possibly different interaction strength), connected through a Y junction, within the scattering state formalism. We first formulate the problem in current algebra language and focus on the case of a symmetric setup, for which the scattering matrix and the matrix of conductances is parametrized by two variables. For these, we derive coupled renormalization group equations, first in a ladder summation up to infinite order in the interaction. The fixed-point structure and the implicit solution of these equations is presented. It is shown that higher-order terms beyond the ladder approximation do not change the scaling behavior near the fixed points. For sufficiently strong attractive interaction, a new fixed point with unusual properties is found.
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 time-resolved transport properties of a Y junction composed of interacting one-dimensional quantum wires using a bosonization approach. In particular, we investigate the ac conductivity of the Y junction formed from finite-length Tomonaga-Luttinger liquid wires based on a plasmon scattering approach for injected charge pulses of arbitrary shapes. In addition, we calculate the tunneling current and quantum noise of the Y junction arising from pointlike tunneling impurities at the junction, including finite-temperature effects. Our results will be useful for designing nanoelectronic quantum circuits and for interpreting time-resolved experiments [Kamata et al., Nat. Nanotechnol. 9, 177 (2014)] in interacting wires and their junctions.