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arXiv:cond-mat/0101002 · Physical review. B, Condensed matter

Quasiparticle description for transport through a small interacting system

Akira Oguri

Published Feb 22, 2001• 44 citations
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Abstract

We study the effects of electron correlation on transport through a small interacting system connected to reservoirs using an effective Hamiltonian which describes the free quasiparticles of a Fermi liquid. The effective Hamiltonian is defined microscopically with the value of the self-energy at $\ensuremath{\omega}=0.$ Specifically, we apply the method to a Hubbard chain of finite size N $(=1,2,3,\dots{}),$ and calculate the self-energy within the second order in U in the electron-hole-symmetric case. When couplings between the chain and the reservoirs on the left and right are small, the conductance for even N decreases with increasing N, showing a tendency toward a Mott-Hubbard insulator. This is caused by the off-diagonal element of the self-energy, and this behavior is qualitatively different from that in the special case examined in previous work. We also study the effects of the asymmetry in the two couplings. While a perfect transmission due to the Kondo resonance occurs for any odd N in the symmetric coupling, the conductance for odd N decreases with increasing N in the case of asymmetric coupling.

DOI
10.1103/physrevb.63.115305

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We study the effects of electron correlation on transport through a small interacting system connected to reservoirs using an effective Hamiltonian which describes the free quasiparticles of a Fermi liquid. The effective Hamiltonian is defined microscopically with the value of the self-energy at \ensuremathω=0.\ensuremath{\omega}=0.\ensuremathω=0. Specifically, we apply the method to a Hubbard chain of finite size N (=1,2,3,…),(=1,2,3,\dots{}),(=1,2,3,…), and calculate the self-energy within the second order in U in the electron-hole-symmetric case. When couplings between the chain and the reservoirs on the left and right are small, the conductance for even N decreases with increasing N, showing a tendency toward a Mott-Hubbard insulator. This is caused by the off-diagonal element of the self-energy, and this behavior is qualitatively different from that in the special case examined in previous work. We also study the effects of the asymmetry in the two couplings. While a perfect transmission due to the Kondo resonance occurs for any odd N in the symmetric coupling, the conductance for odd N decreases with increasing N in the case of asymmetric coupling.
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 $\ensuremath{\omega}=0$ 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 ${g}_{N}\ensuremath{\equiv}{2e}^{2}/h$ independent of the values of U. This may be attributed to a Kondo resonance appearing at $\ensuremath{\omega}=0$. For even N, 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 $\ensuremath{\omega}\ensuremath{\ll}\ensuremath{\Elzxh}{v}_{F}/(\mathrm{Na})$; where vF{v}_{F}vF​ is the Fermi velocity and a is the lattice constant.
We study the low-energy properties of a Hubbard chain of finite size N_C connected to two noninteracting leads using the numerical renormalization group (NRG) method. The results obtained for N_C = 3 and 4 show that the low-lying eigenstates have one-to-one correspondence with the free quasi-particle excitations of a local Fermi liquid. It enables us to determine the transport coefficients from the fixed-point Hamiltonian. At half-filling, the conductance for even N_C decreases exponentially with increasing U showing a tendency towards the development of a Mott-Hubbard gap. In contrast, for odd N_C, the Fermi-liquid nature of the low-energy states assures perfect transmission through the Kondo resonance. Our formulation to deduce the conductance from the fixed-point energy levels can be applied to various types of interacting systems.
In order to extend the Landauer formulation of quantum transport to correlated fermions, we consider a spinless system in which charge carriers interact, connected to two reservoirs by non-interacting one-dimensional leads. We show that the mapping of the embedded many-body scatterer onto an effective one-body scatterer with interaction-dependent parameters requires to include parts of the attached leads where the interacting region induces power law correlations. Physically, this gives a dependence of the conductance of a mesoscopic scatterer upon the nature of the used leads which is due to electron interactions inside the scatterer. To show this, we consider two identical correlated systems connected by a non-interacting lead of length LC. We demonstrate that the effective one-body transmission of the ensemble deviates by an amount A/LC from the behavior obtained assuming an effective one-body description for each element and the combination law of scatterers in series. A is maximum for the interaction strength U around which the Luttinger liquid becomes a Mott insulator in the used model, and vanishes when U ↦0 and U ↦∞. Analogies with the Kondo problem are pointed out.
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 Fermi liquid theory is a central concept in modern condensed matter physics used to describe conventional metals. A state of this universality class consists of well-defined quasiparticles, which occupy a finite number of states within a Fermi volume defined in momentum space. According to the Luttinger’s theorem, such a state encloses a Fermi volume proportional to the electron density modulo the filled bands if no symmetry is broken. In the past decades strong deviations from Fermi liquid theory have been observed in the scaling behavior of thermodynamic observables in different materials, among which are for instance cuprates and iron pnictides. This thesis is about two prototypical systems that can not be described by Fermi liquid theory. The first part of this work investigates two-dimensional effective models, which are in a Mott insulating state at half-filling and have a finite conductivity when doped with holes. In such a Mott insulating state, the charge carriers are strongly localized due to the Coulomb repulsion. Hence, strong correlations are assumed to have a strong impact on the formation of the ground state, also in the regime of small hole concentration. In a first project we have used a so-called spinon dopon mean-field theory to represent the two-dimensional Fermi-Hubbard model with strong on-site repulsive interaction in effective degrees of freedom, in which the holes can be embedded into a quantum spin liquid. The corresponding SU(2) invariant ground state belongs to the class of fractionalized Fermi liquids. In a second project we investigate a quantum dimer model, an effective model based on a Hilbert space spanned by short range singlets and bound states of holes and spins. The focus here is on the calculation of the hole-part of the electron spectral function by using exact diagonalization and its comparison with two analytic methods, a diagrammatic computation based on the Bethe-Salpeter equation and a so-called two-mode approximation. The electron spectral function shows a similar analytic form in momentum space between nodal and antinodal point when compared to results from photoemission spectroscopy experiments on cuprates. Furthermore, in a subsequent work we calculate the exact ground state of the quantum dimer model along a certain parameter line. In order to analyze the behavior of the elctron spectral function when increasing the density of holes, we investigate the Fermi-Hubbard model in a current project using a dynamical mean-field approach. To solve the 4-site cluster impurity problem, we use a numerical renormalization group approach. However, numerical limitations force us to restrict the analysis to spin-polarized baths. According to the spectral data, the system is similar to the SU(2) invariant case at half-filling in a Mott insulating state and posseses a momentum-selective energy gap at finite doping. The topology of the Fermi surface shows a Lifshitz transition when increasing the hole concentration. Here, the curvature of the Fermi surface changes from electron- to hole-like. For comparison we apply the dynamical mean-field theory also to the quantum dimer model and observe that the electron spectral functions at finite doping are qualitatively similar to that of the two-dimensional Fermi-Hubbard model. The second half of the work is about Tomonaga-Luttinger liquid theory, which is used to describe the low-energy effective degrees of freedom of one-dimensional systems. Here, we first discuss a conceptional extension of the operator-based bosonization theory for one-dimensional systems. This extension is especially suited for inhomogeneous one-dimensional systems. First, we investigate a one-dimensional system with a local interation potential and compute an exact solution of the single particle propagator at T = 0. The critical exponent of the single particle propagator has an unconventional form as a function of the microscopic Tomonaga-Luttinger parameters, which is not covered by the original Luttinger paradigm postulated by F. Duncan M. Haldane. In a second project on one-dimensional systems, we study the impact of scattering processes among bosonic low-energy excitations on the thermalization process. Such scattering processes are irrelevant on large length scales, however strongly affect the dynamics. In our analytic analysis we focus on a experimental setup, where a one-dimensional Bose gas is instantaneously splitted in two identical, however strongly correlated, halves of one-dimensional electronic systems. In the following, the corresponding non-equilibrium state runs through multiple regimes in time, such as a metastable prethermalization regime. However, above a certain threshold in time such scattering processes cause an effective thermalization of the system. In order to demonstrate this, we compute the kinetic equation in the Keldysh field integral formalism from a diagrammatic expansion based on a self-consistent Born approximation.
We study the strong correlation effects in the vicinity of the Mott metal-insulator transition using coupled clean or disordered Hubbard chains with a infinitely large coordinate number D⊥→∞D_{\perp}\to\inftyD⊥​→∞ in the direction perpendicular to the chains and with a long-range transverse hopping. Strong electron correlation effects are treated partially non-perturbatively with the use of the exact results for the 1D Hubbard model. In the case of clean systems, the thermodynamic and transport quantities which characterize the Mott transition from the Fermi liquid state, such as the specific heat coefficient, the Drude weight, and the compressibility, are obtained as functions of hole-doping δ=1−nδ=1-nδ=1−n (nnn, electron density) by the systematic expansion in terms of the inverse of the transverse hopping range l⊥l_{\perp}l⊥​. We find that the δδδ-dependence of these quantities shows non-universal behaviors with exponents depending on the strength of electron-electron interaction. In the presence of disorder, it is shown that the frequency-dependence of the dynamical conductivity obeys the Mott's law, implying the possibility of {\it Mott glass state} for δ→0δ\to 0δ→0, provided that there exists a finite range interaction.