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arXiv:cond-mat/9711167 · Physical Review Letters

Transport through a 1D Mott-Hubbard Insulator of Finite Length

Vadim V. Ponomarenko, Naoto Nagaosa

Published Sep 14, 1998• 16 citations
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Abstract

Transport through a 1D Mott-Hubbard insulator of a finite length $L$ 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 $T>\mathrm{max}({m,T}_{L})$ and at a low energy $T,V<{T}_{L}$ ( ${T}_{L}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}v}_{c}/L$; ${v}_{c}$: 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 $V,T<{T}_{L}\mathrm{exp}(\ensuremath{-}2m/{T}_{L})$.

DOI
10.1103/physrevlett.81.2304

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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≫TL2m\ensuremath{\gg}{T}_{L}2m\ensuremath≫TL​ inside the wire, the transport is suppressed near half filling everywhere inside the gap except for a small region where V,T<TLexp(\ensuremath−2m/TL)V,T<{T}_{L}\mathrm{exp}(\ensuremath{-}2m/{T}_{L})V,T<TL​exp(\ensuremath−2m/TL​).
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 study low-energy spin and charge transport through a 1D Mott-Hubbard insulator of finite length L attached to Fermi liquid reservoirs, which, in the presence of spin accumulation, are characterized by different electrochemical potentials for electrons of opposite spin polarizations. At temperatures less than ( v c : charge velocity in the wire) and under the assumption that the Hubbard gap 2 M is large enough, M > T L , we calculate the average currents (charge and spin) and their zero-frequency correlators. The average spin (charge) current depends only on the difference (sum) of the spin-dependent voltages and even a weak electron backscattering of low rate leads to the spin current suppression at smaller than . The spin current recovers its free mode behavior at spin voltage or temperature larger than Γ. Suppression of the spin-charge correlator suggesting the appearance of spin-charge separation needs both to be larger than Γ. In the absence of the average charge current at its shot noise is proportional to the average spin backscattered current defined by V s and can be used to measure the spin accumulation in the reservoirs. The relation of these results to Kondo dot transport in the Toulouse limit is also clarified.
The use of hydrodynamic transport theory seems to indicate that the charge diffusion constant DDD of the one-dimensional (1D) half-filled Hubbard model, whose Drude weight vanishes, diverges for temperature T>0T>0T>0, which would imply anomalous superdiffusive charge transport. Here the leading term of that constant is derived for low finite temperatures ${k}_{B}T/{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}\ensuremath{\ll}1$ where ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}$ is the Mott-Hubbard gap. It only diverges in the $T\ensuremath{\rightarrow}0$ limit, being finite and decreasing upon increasing TTT within the ${k}_{B}T/{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}\ensuremath{\ll}1$ regime. Our exact results both provide valuable physical information on a complex quantum problem and bring about the interesting unsolved issue of how charge transport evolves from normal diffusive for ${k}_{B}T/{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\eta}}\ensuremath{\ll}1$ to anomalous superdiffusive in the ${k}_{B}T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit.
Nonequilibrium transport in the one dimensional Hubbard model at half-filling is studied by the time-dependent density matrix renormalization group method. It is clearly demonstrated that dielectric breakdown of the Mott insulating phase to a nonequilibrium steady state occurs when an external voltage is larger than the charge gap. From the numerically obtained current-voltage characteristics we find that the current is suppressed by the correlation effect when compared with that of a band insulator and that the current is scaled only by the charge gap for small voltages compared to the band width.
We study low energy spin and charge transport through a 1D Mott-Hubbard insulator of finite length LLL attached to Fermi liquid reservoirs characterized by different chemical potentials for electrons of opposite spin polarizations as it happens in quantum spin Hall insulators. We calculate the average currents (charge and spin) and their correlators and demonstrate how a transition induced by the reservoirs to the low energy Fermi liquid regime results in breakdown of the spin-charge separation, which is visible in the presence of the spin dependent voltages and a weak one electron scattering in the system. These calculations are carried out under assumption that the Hubbard gap 2M is large enough: M>TL≡vc/LM > T_L \equiv v_c/LM>TL​≡vc​/L (vcv_cvc​: charge velocity in the wire) and the scattering rate Γs≪TLΓ_s \ll T_LΓs​≪TL​. Relation of these results to Kondo dot transport in the Toulouse limit is also clarified.
Nonlinear transport in the one dimensional Hubbard model at half-filling under a finite bias voltage is investigated by the adaptive time-dependent density matrix renormalization group method. For repulsive on-site interaction, dielectric breakdown of the Mott insulating ground state to a current-carrying nonequilibrium steady state is clearly observed when the voltage exceeds the charge gap. It is found that by increasing the voltage further the current-voltage characteristics are scaled only by the charge gap and the scaling curve exhibits almost linear dependence on the voltage whose slope is suppressed by the electron correlation. In the case of attractive interaction the linear conductance is the perfect one 2e2/h2e^2/h2e2/h which agrees with the prediction by the Luttinger liquid theory.