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arXiv:cond-mat/0211303 · Physical Review Letters

Zeeman Splitting of Zero-Bias Anomaly in Luttinger Liquids

A. V. Shytov, L. I. Glazman, Oleg A. Starykh

Published Jul 22, 2003• 5 citations
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Abstract

Tunneling density of states (DOS) in Luttinger liquid has a dip at zero energy, commonly known as the zero-bias anomaly. In the presence of a magnetic field, in addition to the zero-bias anomaly, the DOS develops two peaks separated from the origin by the Zeeman energy. We show that these finite-bias anomalies are characterized by a power-law behavior of the DOS and the differential conductance, and find the corresponding exponents at arbitrary strength of the electron-electron interaction. The developed theory is applicable to various kinds of quantum wires, including carbon nanotubes.

DOI
10.1103/physrevlett.91.046801

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Tunneling density of states (DOS) in Luttinger liquid has a dip at zero energy, commonly known as the zero-bias anomaly. In the presence of a magnetic field, in addition to the zero-bias anomaly, the DOS develops two peaks separated from the origin by the Zeeman energy. We show that these finite-bias anomalies are characterized by a power-law behavior of the DOS and the differential conductance, and find the corresponding exponents at arbitrary strength of the electron-electron interaction. The developed theory is applicable to various kinds of quantum wires, including carbon nanotubes.
The conductance of a weakly interacting electron gas in the presence of a single scatterer is found at arbitrary strength of the scattering potential. At weak interaction, a simple renormalization group approach can be used instead of the standard bosonization procedure. Our technique allows us to take into account the backscattering of electrons that leads to deviations of the low-temperature conductance from the behavior typical for Luttinger liquid. In the presence of magnetic field, the backscattering may give rise to a peak in differential conductance at bias equal to the Zeeman energy.
We calculate the low-energy tunneling density of states $\ensuremath{\nu}(\ensuremath{\epsilon},T)$ of an NNN-channel disordered wire, taking into account the electron-electron interaction nonperturbatively. The finite scattering rate $1/\ensuremath{\tau}$ results in a crossover from the Luttinger liquid behavior at higher energies, $\ensuremath{\nu}\ensuremath{\propto}{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}$, to the exponential dependence $\ensuremath{\nu}(\ensuremath{\epsilon},T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0)\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}{\ensuremath{\epsilon}}^{*}/\ensuremath{\epsilon})$ at low energies, where ${\ensuremath{\epsilon}}^{*}\ensuremath{\propto}1/(N\ensuremath{\tau})$. At finite temperature TTT, the tunneling density of states depends on the energy through the dimensionless variable $\ensuremath{\epsilon}/\sqrt{{\ensuremath{\epsilon}}^{*}T}$. At the Fermi level $\ensuremath{\nu}(\ensuremath{\epsilon}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0,T)\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\sqrt{{\ensuremath{\epsilon}}^{*}/T})$.
We calculate the conductances of a three-way junction of spinless Luttinger-liquid wires as functions of bias voltages applied to three independent Fermi-liquid reservoirs. In particular, we consider the setup that is characteristic of a tunneling experiment, in which the strength of electron-electron interactions in one of the arms of the junction (``tunneling tip'') is different from that in the other two arms (which together form a ``main wire''). The scaling dependence of the two independent conductances on bias voltages is determined within a fermionic renormalization-group approach in the limit of weak interactions. The solution shows that, in general, the conductances scale with the bias voltages in an essentially different way compared to their scaling with the temperature TTT. Specifically, unlike in the two-terminal setup, the nonlinear conductances cannot be generically obtained from the linear ones by simply replacing TTT with the ``corresponding'' bias voltage or the largest one. Remarkably, a finite tunneling bias voltage prevents the interaction-induced breakup of the main wire into two disconnected pieces in the limit of zero TTT and a zero source-drain voltage.
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).
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.