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arXiv:cond-mat/9709013 · Physical Review Letters

Conductance of a Mott Quantum Wire

Oleg A. Starykh, Dmitrii L. Maslov

Published Feb 23, 1998• 8 citations
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Abstract

We consider transport through a one-dimensional conductor subjected to an external periodic potential and connected to noninteracting leads (a ``Mott quantum wire''). In the case of a strong periodic potential, the conductance is shown to jump from zero, for the chemical potential lying within the Mott-Hubbard gap, to the noninteracting value of ${2e}^{2}/h$, as soon as the chemical potential crosses the gap edge. This behavior is strikingly different from that of optical conductivity, which varies continuously with the carrier concentration. In the case of a weak potential, the perturbative correction to the conductance due to umklapp scattering is absent away from half filling.

DOI
10.1103/physrevlett.80.1694

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We consider transport through a one-dimensional conductor subjected to an external periodic potential and connected to noninteracting leads (a ``Mott quantum wire''). In the case of a strong periodic potential, the conductance is shown to jump from zero, for the chemical potential lying within the Mott-Hubbard gap, to the noninteracting value of 2e2/h{2e}^{2}/h2e2/h, as soon as the chemical potential crosses the gap edge. This behavior is strikingly different from that of optical conductivity, which varies continuously with the carrier concentration. In the case of a weak potential, the perturbative correction to the conductance due to umklapp scattering is absent away from half filling.
Suppression of electron current $ ΔI$ through a 1D channel of length LLL connecting two Fermi liquid reservoirs is studied taking into account the Umklapp interaction induced by a periodic potential. This interaction opens band gaps at the integer fillings and Hubbard gaps 2m2m2m at some rational fillings in the infinite wire: L→∞L \to \inftyL→∞. In the perturbative regime where m≪vc/Lm \ll v_c/Lm≪vc​/L (vc:v_c:vc​: charge velocity), and for small deviations δnδnδn of the electron density from its commensurate values −ΔI/V- ΔI/V−ΔI/V can diverge with some exponent as voltage or temperature V,TV,TV,T decreases above Ec=max(vc/L,vcδn)E_c=max(v_c/L,v_c δn)Ec​=max(vc​/L,vc​δn), while it goes to zero below EcE_cEc​. This results in a non-monotonous behavior of the conductance. In the case when the Umklapp interaction creates a large Mott-Hubbard gap $2m \gg T_L $ inside the wire, the transport is suppressed near half-filling everywhere inside the gap except for an exponentially small region of V,T<TLexp(−2m/TL)V,T < T_L exp(-2m/T_L)V,T<TL​exp(−2m/TL​).
We investigate the transport of interacting electrons through single-mode quantum wires whose parameters are periodically modulated on the scale of the electronic Fermi wavelength. The Umklapp scattering and backscattering of electrons can be described in terms of nonuniform quantum sine-Gordon-like models that also incorporate the effects of electronic reservoirs (electrodes) adiabatically coupled to the wire. We concentrate on weak Umklapp scattering and analyze the precursors of the Mott transition. At half-filling the temperature dependence of the extra resistance $\ensuremath{\Delta}R=R\ensuremath{-}\ensuremath{\pi}\ensuremath{\Elzxh}{/e}^{2}$ of a modulated quantum wire of length LLL changes from the interaction-dependent ``bulk'' power-law $\ensuremath{\Delta}R\ensuremath{\propto}{T}^{{4K}_{\ensuremath{\rho}}\ensuremath{-}3}$ at high temperatures $(T\ensuremath{\gg}{v}_{\ensuremath{\rho}}/L)$ to the universal $\ensuremath{\Delta}R\ensuremath{\propto}{T}^{2}$ behavior at low temperatures $(T\ensuremath{\ll}{v}_{\ensuremath{\rho}}/L).$ Away from half-filling the ``bulk'' results are qualitatively incorrect even at high temperatures ${v}_{\ensuremath{\rho}}/L\ensuremath{\ll}T\ensuremath{\ll}{T}^{*}$ despite the fact that the electron coherence in the wire is absent in this regime.
We consider the optical conductivity ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ in the metallic phase of the one-dimensional Hubbard model. Our results focus on the vicinity of half filling and the frequency regime around the optical gap in the Mott insulating phase. By means of a density-matrix renormalization group implementation of the correction-vector approach, ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ is computed for a range of interaction strengths and dopings. We identify an energy scale Eopt{E}_{\mathrm{opt}}Eopt​ above which the optical conductivity shows a rapid increase. We then use a mobile impurity model in combination with exact results to determine the behavior of ${\ensuremath{\sigma}}_{1}(\ensuremath{\omega})$ for frequencies just above Eopt{E}_{\mathrm{opt}}Eopt​ which is in agreement with our numerical data. As a main result, we find that this onset behavior is not described by a power law.
Transport through a 1D Mott-Hubbard insulator of a finite length LLL is studied beyond the perturbative approach. At a special value of the low-energy constant of the interaction we have mapped the problem onto exactly solvable models and found current vs voltage at a high temperature T>max(m,TL)T>\mathrm{max}({m,T}_{L})T>max(m,TL​) and at a low energy T,V<TLT,V<{T}_{L}T,V<TL​ ( TL=vc/L{T}_{L}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}v}_{c}/LTL​=vc​/L; vc{v}_{c}vc​: charge velocity). The result shows that for the strong interaction creating a large Mott-Hubbard gap $2m\ensuremath{\gg}{T}_{L}$ inside the wire, the transport is suppressed near half filling everywhere inside the gap except for a small region where $V,T<{T}_{L}\mathrm{exp}(\ensuremath{-}2m/{T}_{L})$.
We combine well-controlled analytical and numerical methods to determine the optical conductivity of the one-dimensional Mott-Hubbard insulator at zero temperature. A dynamical density-matrix renormalization group method provides the entire absorption spectrum for all but very small coupling strengths. In this limit we calculate the conductivity analytically using exact field-theoretical methods. Above the Lieb-Wu gap the conductivity exhibits a characteristic square-root increase. For small to moderate interactions, a sharp maximum occurs just above the gap. For larger interactions, another weak feature becomes visible around the middle of the absorption band.