Spin and interaction effects on charge distribution and currents in one-dimensional conductors and rings within the Hartree-Fock approximation Avraham Cohen1 , Klaus Richter2 , and Richard Berkovits1 arXiv:cond-mat/9804018v1 [cond-mat.mes-hall] 2 Apr 1998 1 The Minerva Center for the Physics of Mesoscopics, Fractals and Neural Networks, Department of Physics, Bar-Illan University, 52900 Ramat-Gan, Israel 2 Max-Planck-Institut f¨r Physik komplexer Systeme, N¨thnitzer Strasse 38, 01187 Dresden, Germany u o (January 5, 2018) Using the self–consistent Hartree-Fock approximation for electrons with spin at zero temperature, we study the effect of the electronic interactions on the charge distribution in a one-dimensional continuous ring containing a single δ scatterer. We reestablish that the interaction suppresses the decay of the Friedel oscillations. Based on this result, we show that in an infinite one dimensional conductor containing a weak scatterer, the current is totally suppressed because of a gap opened at the Fermi energy. In a canonical ensemble of continuous rings containing many scatterers, the interactions enhance the average and the typical persistent current. PACS numbers: 72.10.Fk, 73.20.Dx orders of the series. Although the dissipative conductance of the infinite conductor is suppressed by the interactions, the PC in a ring is not. This is because the conductance depends on the properties of the levels close to the Fermi energy but the PC is a thermodynamic property that depends on the response of all occupied levels. [11] Moreover, we show that once many scatterers are considered, the interactions not only do not suppress the PC, but even enhance it. We write the HF equation for electrons in a ring of radius R with angular coordinate θ and energy units h ¯ 2 /me R2 = 1 (we drop the background term) as The effects of electronic interactions on characteristic properties, such as charge fluctuations, persistent currents (PC’s) and the conductance of electronic systems are very rich and interesting. [1] They strongly depend on the strength and range of the interactions, [2,3,4] on the dimensionality of the system, and on whether the space is discrete or continuous. [5,6] Approximate calculations, like Hartree-Fock, introduce a great deal of simplifications, but at the same time many effects may be washed out. However, approximate calculations may be used to shed more light on specific problems, while keeping in mind their limitations. In this work we consider e-e interactions within the self-consistent Hartree-Fock approximation (SCHFA) for electrons with spin at zero temperature. For simplicity we assume an equal number of electrons of opposite spin states. Our aim is to study numerically the interaction effects on the charge distribution and the currents in continuous one-dimensional (1D) isolated rings and open conductors containing a single δ scatterer, [4,7,8,9,10] as well as on the PC’s in rings containing many scatterers. [11,12] Even within the Hartree-Fock approximation we recover the bosonization [4] and the density-matrix renormalizationgroup result: [10] We show that for a single scatterer in a ring the repulsive electronic interaction suppresses the decay of the charge oscillations. Based on this we show, as a central result, that for an open conductor with a weak scatterer the electronic conduction at the Fermi energy vanishes because of Bragg reflection coexisting with a gap at the Fermi energy. The zero conduction of the interacting system was obtained in Refs. [7,8,9] by exact and by renormalization group calculations. Within the first iteration of the SCHFA, it was shown [8,9] that an attempt to explain this result by a scattering perturbation series is inadequate because of logarithmic divergences of the transmission amplitude at the Fermi energy in all − 1 ∂2 R + Vdis (θ) + 2 ∂θ2 r0 −δsl′ ,sl R r0 2π 0 Ne l′ =1 2π 0 Ne l′ =1 |ψl′ (θ′ )|2 (θ − θ′ )2 + ǫ2 Ψ∗′ (θ′ )Ψl′ (θ) l ψl (θ′ )dθ′ ′ )2 + ǫ2 (θ − θ dθ′ ψl (θ) = Eψl (θ) . (1) The twisted boundary condition ψ(θ + 2π) = ψ(θ) exp(i2πφ/φ0 ) accounts for a flux φ threading the ring. φ0 ≡ hc/e is the flux quantum. Vdis (θ) is the disorder potential which may include a single or many scatterers. The first (second) integral term is the Hartree (Fock) term. The electronic wave functions Ψl (θ) ≡ ψl (θ) exp (−iθφ/φ0 ) in the Fock term are 2π periodic for any value of flux. l enumerates the energy levels together with the spin state sl . Ne is the total number of electrons in the ring. The cutoff ǫ2 allows (as in quasi 1D) using the 3D Coulomb law [5] and makes the integrations finite. The square of the distance between the particles is defined [13] by (θ − θ′ )2 ≡ min[|θ − θ′ |2 , (2π − |θ − θ′ |)2 ]. In Eq. (1), r0 ≡ ε¯ 2 /me e2 denotes the Bohr radius with h dielectric constant ε (to be distinguished from the cutoff ǫ). We define the coefficient g ≡ R/r0 to be the interaction strength. g ∼ 1 corresponds to semiconductors. 1 Ne [14] Because the sum l′ =1 Ψ∗′ (θ′ )Ψl′ (θ) represents all most a closure relation we replace, as discussed in Refs. [14] and [15], the integrodifferential equation (Eq. (1)) by an ordinary Schr¨dinger equation that we solve self– o consistently: 6.6 1 ∂2 + Vdis (θ) + gVeff (θ) ψl (θ) = Eψl (θ). 2 ∂θ2 (2) ρ(θ) − 6.8 Here Veff (θ) is given by 2π Ne l′ =1 |ψl′ (θ′ )|2 − δsl′ ,sl Re{Ψ∗′ (θ′ )Ψl′ (θ)} l (θ − θ′ )2 + ǫ2 0 6.4 dθ′ Noninteracting electrons Interacting electrons 20.03/π+0.025sin(40x+1.65π)/θ 0.35 20.03/π+0.075sin(40x+1.65π)/θ 6.2 where Re stands for real part. The spin degree of freedom is very important. For spinless electrons the interaction effect is weak because the Fock and Hartree terms tend to cancel each other due to opposite signs and similar absolute values. Taking into account the spin degree of freedom, the Hartree term is twice as large as the exchange term. Then the former dominates Veff and enhances screening; therefore we expect the interaction effects to be stronger for electrons with spin. This explains the importance of considering spin [16] in order to understand disordered interacting systems. We begin by studying the interaction effect on the charge oscillations in a ring with a single scatterer, Vdis (θ) = λδ(θ). 6.0 0.0 0.5 1.0 1.5 θ 2.0 2.5 3.0 FIG. 1. Charge oscillations per spin, along half circumference of a 1D continuous ring, induced by a weak single scatterer λδ(θ) (λ = 3.8). Thin (bold) symbols stand for noninteracting (interacting) electrons. The interactions, at self-consistency of the Hartree-Fock calculations, tend to make ρ(θ) periodic and to minimize ρ(0) further. The interaction strength is g = 3, and the flux threading the ring is φ/φ0 = 0.05 (see text). The total number of electrons per spin is 40. The curves are the estimations by the indicated formulas. (3) ˆ h with k > 0 and λ ≡ λ/(¯ 2 /me ) having units of inverse length. r(k, λ) = −iλ/(k + iλ) and t(k, λ) = k/(k + iλ). Because of time-reversal symmetry and the symmetry of the potential under coordinate inversion, t′ = t and r′ ≡ −(r/t)∗ t = r. The fluctuating density per spin is For a strong scatterer, λ ≥ Ef (Ef is the Fermi energy), the interaction effect on the decay of the charge oscillations is weak and may even be neglected because the scatterer is dominating. For a weak scatterer, λ << Ef , at the level of the SCHFA we recover the numerical result of Ref. [10] based on the density-matrix renormalization group: With increasing repulsive interaction g the decay of the Friedel oscillations is suppressed (indicating also the reliability of our SCHFA). Figure 1 depicts the decay rate for the strongest interaction for which the SCHFA still converges. As Fig. 2 shows, the effective potential tends to be periodic with half a Fermi wavelength periodicity. Both (direct and exchange) terms tend to have this periodicity which is independent of the interaction strength. This behavior holds for a larger number of electrons on a ring for a given constant charge density. The above results may be used to study the effect on the charge oscillations and on the conduction in the case ˆ of a single weak scatterer λδ(x) embedded in an infinite 1D conductor (x is the spatial coordinate). For noninteracting electrons the orthogonal wave functions, with a given spin state, are [8,9] Im takes the imaginary part of the exponential integral E1 , z ≡ 2|x|(Kf + iλ), and Kf is the Fermi wave vector. ∆ρ(0) = −2λ tan−1 (Kf /λ) is a minimum. For Kf x > 1, the asymptotic expansion of E1 implies eikx + r(k, λ)e−ikx , t(k, λ)eikx , x<0 x > 0, (4) For the SCHFA the initial Veff is calculated using the wave functions of noninteracting electrons. The charge fluctuations define the Hartree potential t′ (k, λ)e−ikx , e−ikx + r′ (k, λ)eikx , x<0 x > 0, (5) (1) φk (x) = (2) φk (x) = Kf ∆ρ(x) = 2 0 −λ2 cos 2kx + λk sin 2k|x| dk k 2 + λ2 = −2λe2λ|x| Im{E1 (−iz)}. ∆ρ(x) = − λ(Kf cos 2Kf x + λ sin 2Kf |x|) . 2 |x|(λ2 + Kf ) Using r ≡ |r|eiη and |rf | = |λ|/ (6) (7) 2 Kf + λ2 , sin ηf = 2 −k/ Kf + λ2 , one finds [8,9] ∆ρ(x) = ∞ VH (x) = gs 0 2 |rf | sin(2Kf |x| + ηf ) . |x| 1 1 ∆ρ(x′ )dx′ , + ′| |x + x |x − x′ | (8) (9) 0.25 Noninteracting Interacting 8.0 0.20 0.15 P(I) Veff(θ) 6.0 4.0 0.10 2.0 0.05 Noninteracting electrons Interacting electrons 0.0 0.0 0.5 1.0 1.5 θ 2.0 2.5 0.00 −0.6 3.0 FIG. 2. The effective potential along the half circumference of the ring of Fig. 1. The thin (bold) symbol is the noninteracting (interacting) result. Note the clear tendency of Veff (θ) at self-consistency to become periodic, and to screen the scattering potential λδ(θ) (not shown). +∞ VF (x) = − −∞ ∞ =− 0 2 i=1 2 (i)∗ (i) Re{φk (x′ )φk (x)}dk ′ dx |x − x′ | Clearly, Veff = VH +VF is a function of |x|, and will change during the iterations until self-consistency is reached. Veff is small due to a weak coupling constant (g ∼ 1). At this point we invoke an approximate self-consistency by adopting a suppression [4,10] of the decay of the Friedel oscillations, as was demonstrated above to be valid in the SCHFA. We substitute by hand the limit [4] δ = 0 in |rf | sin(2Kf |x| + ηf ) |x|δ I 0.0 0.2 0.4 λf π sin = nλf . 2 2 (13) All the states with |k| < Kf remain practically unaffected by the weak and periodic Veff . Note that consistently with Eq. (13) there is a gap [17] of order U at the Fermi energy. Thus the current vanishes at the Fermi energy. For a ring with a weak scatterer the interaction will not destroy the PC even if the current at the Fermi energy (assuming a large ring) is totally suppressed by the periodic effective potential. This follows from the fact that all occupied levels contribute to the PC, except at Ef , where Eq. (13) is assumed to be relevant. In the following we will consider the general case of a large number of scatterers in a ring. Figure 2 already shows the importance of screening for a single scatterer. This indicates that screening is of particular relevance for the case of many random scatterers: 1 1 ∆ρ(x + x′ )dx′ . (10) + |x + x′ | |x − x′ | ∆ρ(x) = −0.2 FIG. 3. The interaction effect on the statistics of the sample persistent current (in units of the PC of the clean ring of noninteracting electrons). Thin (bold) symbols stand for noninteracting (interacting) electrons for the same canonical ensemble of 201 realizations. The interaction (g = 1, φ/φ0 = 0.325) enhances the persistent current. where gs = 1 (2) for electrons without (with) spin. Our approximated Fock potential is Kf 0 −0.4 (11) for Eqs. (9) and (10), assuming that this yields a Veff close to that from the SCHFA. To carry out the integration [in Eqs. (9) and (10)], we use a cutoff that allows contributions only from |x − x′ | ≥ ǫ. This cutoff is equivalent to that used in Eq. (1). For Kf x ≫ 1, we then obtain, up to an additive constant, Ns λj δ(θ − θj ). Vdis (θ) = (14) j=1 Here the location and strength of the jth scatterer are uniformly distributed in (0, 2π) and (−Λ, Λ), respectively. Ns is the total number of scatterers in a ring. For the numerics we use Λ = 14 (in scaled units). The characteristic features of disordered noninteracting samples were, e.g., discussed by Imry and Shiren. [18] For noninteracting electrons, the localization length [14] at Ef = 200 is ξ ∼ π/2. This should reduce the average current in open conductors by a factor ∼ 1/50. The average sample PC of noninteracting electrons was reduced by factor ∼ 1/40 which is slightly greater than predicted Vef f (x) = U [gs sin(2Kf x + ηf ) − sin(4Kf x + ηf )], (12) where U ≡ −2|rf |ci (2Kf ǫ), and ci is the cosine integral. Equation (12) shows that Veff has two periodicities: λf /2 from the direct potential (λf ≡ 2π/Kf ), and λf /4 from the exchange potential. The overall periodicity is given by the larger period. The electrons at the Fermi energy exactly obey the Bragg condition [17] for total reflection, i.e., 3 tronic conduction was shown to vanish, because of Bragg reflection that coexists with a gap at the Fermi energy. This shows that, even in the HF limit the influence of the interactions on the Friedel oscillations and on conduction in one-dimension, calculated by exact and renormalization methods, may be reproduced. In rings the PC is not suppressed by the interaction. It is even enhanced in the case of many moderate scatterers due to screening. To demonstrate these effects, we considered the spin degree of freedom and used continuous conductors and rings. A. C. would like to thank A. Auerbach, D. Bar-Moshe, and B. Shapiro for valuable discussions, and A. Heinrich for his interest in this work. A. C. and K. R. would like to thank U. Eckern, P. Schwab and P. Schmitteckert for valuable comments and criticism. 0.03 0.02 0.01 0.00 −0.01 −0.02 −0.03 −0.04 −0.05 0.00 Noninteracting Interacting 0.10 0.20 φ/φ0 0.30 0.40 0.50 FIG. 4. The interaction (g = 1.75) enhances the average sample persistent current (in units of the PC of the clean ring) and introduces a preffered diamagnetic current direction. Thin (bold) symbols represent noninteracting (interacting) electrons for the same 150 realizations. I 2 , was for open conductors. The typical sample PC, reduced by factor ∼ 1/10 which indicates the importance of a statistical study. The fixed total number of electrons in a ring was 32 ± 4. Figure 3 shows the interaction effect on the sample PC statistics for an interaction coupling constant g = 1. The interaction reduces the peak, centered at zero, while broadening the distribution. Furthermore, the distribution gains more weight at negative values of the PC, which indicates a diamagnetic tendency. We found that the interaction enhances the typical PC (by factor ∼2); the average PC is neither enhanced nor suppressed. Figure 4 shows that for increasing interaction, g = 1.75, the average PC is also enhanced by factor ∼2. Figures 3 and 4 both show a clear tendency of the interaction to enhance the PC for electrons with spin. For spinless electrons the PC was found [14] to be rather unaffected by interaction. This shows an essential difference between models of electrons with or without spin. 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