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arXiv:cond-mat/0503146

Spinless Hartree-Fock model of persistent current in rings with single scatterer: Comparison with correlated models

Radoslav Németh, Martin Moško

Published Mar 7, 2005• 0 citations
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Abstract

Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we study the persistent current of the interacting electron gas in a one-dimensional continuous ring containing a single $δ$ barrier. We calculate the persistent current as a function of the ring circumference, magnetic flux threading the ring, barrier strength, etc. We compare our results with the results of correlated models like the Luttinger liquid model and the Hubbard model solved by means of the renormalization group. A good agreement is found. First, the persistent current decays with the increasing ring circumference ($L$) faster than $L^{-1}$ and eventually like $L^{-α-1}$, where $α>0$ depends only on the electron-electron interaction. Second, the persistent current is a sine-shaped function of magnetic flux. This sine-like dependence and in particular the universal power law $L^{-α-1}$ have sofar been believed to arise only in the correlated many-body models. Observation of these features within the Hartree-Fock model is a surprising result.

DOI
10.48550/arxiv.cond-mat/0503146

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Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we study the persistent current of the interacting electron gas in a one-dimensional continuous ring containing a single δδδ barrier. We calculate the persistent current as a function of the ring circumference, magnetic flux threading the ring, barrier strength, etc. We compare our results with the results of correlated models like the Luttinger liquid model and the Hubbard model solved by means of the renormalization group. A good agreement is found. First, the persistent current decays with the increasing ring circumference (LLL) faster than L−1L^{-1}L−1 and eventually like L−α−1L^{-α-1}L−α−1, where α>0α>0α>0 depends only on the electron-electron interaction. Second, the persistent current is a sine-shaped function of magnetic flux. This sine-like dependence and in particular the universal power law L−α−1L^{-α-1}L−α−1 have sofar been believed to arise only in the correlated many-body models. Observation of these features within the Hartree-Fock model is a surprising result.
Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we calculate the persistent current of the interacting electron gas in a one-dimensional ring containing a single δ barrier.Our results agree with correlated models like the Luttinger liquid model and lattice model with nearest-neighbor interaction.The persistent current is a sine-like function of magnetic flux.It decays with the ring length (L) faster than L -1 and eventually like L -α-1 , where α > 0 is universal.
We study the persistent current of correlated spinless electrons in a continuous one-dimensional ring with a single weak link. We include correlations by solving the many-body Schrodinger equation for several tens of electrons interacting via the short-ranged pair interaction V(x - x'). We solve this many-body problem by advanced configuration-interaction (CI) and diffusion Monte Carlo (DMC) methods. Our CI and DMC results show, that the persistent current (I) as a function of the ring length (L) exhibits for large L the power law typical of the Luttinger liquid, I∝L−1−αI \propto L^{-1-α}I∝L−1−α, where the power ααα depends only on the electron-electron (e-e) interaction. For strong e-e interaction the previous theories predicted for ααα the formula α=(1+2αRG)1/2−1α= {(1 + 2 α_{RG})}^{1/2} - 1α=(1+2αRG​)1/2−1, where αRG=[V(0)−V(2kF)]/2πℏvFα_{RG} = [V(0)-V(2k_F)]/2π\hbar v_FαRG​=[V(0)−V(2kF​)]/2πℏvF​ is the renormalisation-group result for weakly interacting electrons, with V(q) being the Fourier transform of V(x-x'). Our numerical data show that this theoretical result holds in the continuous model only if the range of V(x - x') is small (roughly d≲1/2kFd \lesssim 1/2k_Fd≲1/2kF​, more precisely 4d2kF2<<14d^2k_F^2 << 14d2kF2​<<1). For strong e-e interaction (αRG>0.25α_{RG} > 0.25αRG​>0.25) our CI data show the power law I∝L−1−αI \propto L^{-1-α}I∝L−1−α already for rings with only ten electrons, i.e., ten electrons are already enough to behave like the Luttinger liquid. The DMC data for αRG>0.25α_{RG} > 0.25αRG​>0.25 are damaged by the so-called fixed-phase approximation. Finally, we also treat the e-e interaction in the Hartree-Fock approximation. We find the exponentially decaying I(L) instead of the power law, however, the slope of log(I(L)) still depends solely on the parameter αRGα_{RG}αRG​ as long as the range of V(x - x') approaches zero.
We present numerical results for the zero temperature persistent currents carried by interacting spinless electrons in disordered one-dimensional continuous rings. The disorder potential is described by a collection of δ-functions at random locations and strengths. The calculations are performed by a self-consistent Hartree–Fock (HF) approximation. Because the HF approximation retains the concept of single-electron levels, we compare the statistics of energy levels of noninteracting electrons with those of interacting electrons as well as of the level persistent currents. We find that the e–e interactions alter the levels and samples persistent currents and introduces a preffered diamagnetic current direction. In contrast to the analogous calculations that recently appeared in the literature for interacting spinless electrons in the presence of moderate disorder in tight-binding models we find no suppression of the persistent currents due to the e–e interactions.
Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we study tunneling of the interacting electron gas through a single δ barrier in a finite one-dimensional wire connected to contacts.Our results exhibit features known from correlated many-body models.In particular, the conductance decays with the wire length as ∝ L -2α , where the power α is universal.We also show that a similar result for a wire conductance can be extracted from the persistent current (I) through the δ barrier in a one-dimensional ring, where it is known that I ∝ L -1-α .
Persistent currents in disordered mesoscopic rings threaded by a magnetic flux are calculated using exact diagonalization methods in the one-dimensional (1D) case and self-consistent Hartree-Fock treatments for two dimensional (2D) systems. For multichannel systems, a comparative study between models of spinless or spinfull (Hubbard) fermions has been done. First, it is shown that a purely one-dimensional model can not reproduce the correct order of magnitude of the observed currents. For 2D systems, going beyond first order pertubative calculations, we show that the second harmonic of the current is {\it strongly suppressed} in the case of spinless fermion models but {\it significantly enhanced} for the Hubbard model. This reduction (resp. increase) of the second harmonic is related to a strong increase (resp. reduction) of the spacial charge density fluctuations. Our work underlines the important role of the spin degrees of freedom in the persistent currents.
Persistent currents flowing through disordered mesoscopic rings threaded by a magnetic flux are investigated. Models of fermions with on-site interactions (Hubbard model) or models of spinless fermions with nearest neighbor interactions are considered on 2D cylinders with twisted boundary conditions in one direction to account for a magnetic flux. Self-consistent Hartree-Fock methods are used to treat the electron-electron interaction beyond first order. We show that the second harmonic of the current (which is relevant in the diffusive regime) is strongly suppressed by the interaction in the case of spinless fermions while it is significantly enhanced in the Hubbard model. Our data also strongly suggest that the reduction (increase) of this harmonic is related to a strong increase (reduction) of the spatial fluctuations of the charge density.