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arXiv:cond-mat/9903126 · Physical review. B, Condensed matter

Transport through a finite Hubbard chain connected to reservoirs

Akira Oguri

Published May 15, 1999• 40 citations
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Abstract

The dc conductance ${g}_{N}$ through a finite Hubbard chain of size N $(=1,2,3,\dots{})$ connected to reservoirs is studied at $T=0$ in an electron-hole symmetric case. We calculate a spatial dependence of the self-energy analytically at $\ensuremath{\omega}=0$ within the second order in U, and obtain an inter-site Green's function ${G}_{N1}$, from which ${g}_{N}$ can be determined, via the Dyson equation. The results depend strongly on whether N is even or odd. For odd N, a perfect transmission occurs, and ${g}_{N}\ensuremath{\equiv}{2e}^{2}/h$ independent of the values of U. This may be attributed to a Kondo resonance appearing at $\ensuremath{\omega}=0$. For even N, ${g}_{N}$ decreases with increasing N, and converges to a finite constant which is a smooth decreasing function of U. These behaviors are essentially owing to the presence of the reservoirs, which makes a quasiparticle description valid for low-energy states at $\ensuremath{\omega}\ensuremath{\ll}\ensuremath{\Elzxh}{v}_{F}/(\mathrm{Na})$; where ${v}_{F}$ is the Fermi velocity and a is the lattice constant.

DOI
10.1103/physrevb.59.12240

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The dc conductance gN{g}_{N}gN​ through a finite Hubbard chain of size N (=1,2,3,…)(=1,2,3,\dots{})(=1,2,3,…) connected to reservoirs is studied at T=0T=0T=0 in an electron-hole symmetric case. We calculate a spatial dependence of the self-energy analytically at \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0 within the second order in U, and obtain an inter-site Green's function GN1{G}_{N1}GN1​, from which gN{g}_{N}gN​ can be determined, via the Dyson equation. The results depend strongly on whether N is even or odd. For odd N, a perfect transmission occurs, and gN\ensuremath≡2e2/h{g}_{N}\ensuremath{\equiv}{2e}^{2}/hgN​\ensuremath≡2e2/h independent of the values of U. This may be attributed to a Kondo resonance appearing at \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0. For even N, gN{g}_{N}gN​ decreases with increasing N, and converges to a finite constant which is a smooth decreasing function of U. These behaviors are essentially owing to the presence of the reservoirs, which makes a quasiparticle description valid for low-energy states at \ensuremathω\ensuremath≪\ensuremath\ElzxhvF/(Na)\ensuremath{\omega}\ensuremath{\ll}\ensuremath{\Elzxh}{v}_{F}/(\mathrm{Na})\ensuremathω\ensuremath≪\ensuremath\ElzxhvF​/(Na); where vF{v}_{F}vF​ is the Fermi velocity and a is the lattice constant.
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We study the effects of electron correlation on transport through an interacting region connected to multi-mode leads based on the perturbation expansion with respect to the inter-electron interaction. At zero temperature the conductance defined in the Kubo formalism can be written in terms of a single-particle Green's function at the Fermi energy, and it can be mapped onto a transmission coefficient of the free quasiparticles described by an effective Hamiltonian. We apply this formulation to a two-dimensional Hubbard model of finite size connected to two noninteracting leads. We calculate the conductance in the electron-hole symmetric case using the order U2U^2U2 self-energy. The conductance shows several maximums in the UUU dependence in some parameter regions of ty/txt_y/t_xty​/tx​, where txt_xtx​ (tyt_yty​) is the hopping matrix element in the xxx- (yyy-) directions. This is caused by the resonance occurring in some of the subbands, and is related with the UUU dependence of the eigenvalues of the effective Hamiltonian.
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.
The electron self-energy Sigma(k,\ensuremath{\omega}) corresponding to the one-band Hubbard Hamiltonian on clusters of the square lattice has been obtained as a function of the band filling up to second order in the interaction parameter U. The k dependence of the self-energy is completely taken into account. As a consequence, the imaginary part of the self-energy shows a linear behavior for fillings close to one electron per site whereas for large doping rates the quadratic \ensuremath{\omega} dependence characterizing Fermi liquids is recovered. The origin of this behavior has been investigated analytically: linear terms associated with nesting are shown to exist in any finite-dimensional lattice although its numerical relevance decreases very rapidly for space dimensionality larger than two. Implications of these results on the renormalization factor Z have been analyzed. From a practical point of view, we conclude that a standard second-order perturbative treatment is quantitatively precise for U values of the on-site Coulomb interaction smaller than the bandwidth of the noninteracting spectrum.
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