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Physical review. B, Condensed matter

Influence of the electron-electron interaction on electronic spectra and persistent currents in one-dimensional loops

Guido Burmeister, Klaus Maschke

Published Apr 12, 2002• 10 citations
Publisher

Abstract

We consider N electrons in a one-dimensional loop enclosing a static magnetic flux. The electronic positions are described in terms of collective and relative coordinates. The corresponding partition of the Hilbert space is particularly convenient to investigate the electronic spectrum in the range of large electron-electron interaction. The electronic current is determined by the collective motion of the electrons. Thus, in absence of backscattering by a one-particle potential, the electron-electron interaction does not affect the persistent current associated with the electronic ground state. This is different when backscattering is present, since then collective and relative motions are coupled. We present a detailed study of the interplay between potential backscattering and electron-electron interaction, which is revealed by the resulting ground-state currents as well as by the electronic excitation spectra.

DOI
10.1103/physrevb.65.155333

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Persistent currents in mesoscopic rings: A numerical and renormalization group study

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The persistent current in a lattice model of a one-dimensional interacting electron system is systematically studied using a complex version of the density-matrix renormalization group algorithm and the functional renormalization group method. We mainly focus on the situation where a single impurity is included in the ring penetrated by a magnetic flux. Due to the interplay of the electron-electron interaction and the impurity the persistent current in a system of N lattice sites vanishes faster than $1/N.$ Only for very large systems and large impurities our results are consistent with the Bosonization prediction obtained for an effective field theory. The results from the density-matrix renormalization group and the functional renormalization group agree well for interactions as large as half the bandwidth, even though as an approximation in the latter method the flow of the two-particle vertex is neglected. This confirms that the functional renormalization group method is a very powerful tool to investigate correlated electron systems. The method will become very useful for the theoretical description of the electronic properties of small conducting ring molecules.

arXiv:cond-mat/0209588•Physical review. B, Condensed matter•Jan 15, 2003•49 citations
We consider N electrons in a one-dimensional loop enclosing a static magnetic flux. The electronic positions are described in terms of collective and relative coordinates. The corresponding partition of the Hilbert space is particularly convenient to investigate the electronic spectrum in the range of large electron-electron interaction. The electronic current is determined by the collective motion of the electrons. Thus, in absence of backscattering by a one-particle potential, the electron-electron interaction does not affect the persistent current associated with the electronic ground state. This is different when backscattering is present, since then collective and relative motions are coupled. We present a detailed study of the interplay between potential backscattering and electron-electron interaction, which is revealed by the resulting ground-state currents as well as by the electronic excitation spectra.
We consider N strongly interacting electrons in a one-dimensional circular loop that is pierced by a time-periodic magnetic flux a(t)=a0+a1(t){a(t)=a}_{0}{+a}_{1}(t)a(t)=a0​+a1​(t) with the angular frequency $\ensuremath{\omega}.$ Similar to our previous work, where we have considered a static magnetic flux a0,{a}_{0},a0​, the electron positions are expressed in terms of collective and relative coordinates. Strong $e\ensuremath{-}e$ interaction can then be treated in a harmonic approximation for the relative motion. The presently searched solutions of the time-dependent Schr\"odinger equation for a time-periodic flux are given by the Floquet states. The Floquet states for a spatially constant one-particle potential form a complete set of Floquet-basis states, which is used to study the influence of the one-particle potential on the electronic states. While for a spatially constant one-particle potential the time-averaged observables, such as the electronic energy and the electronic current or angular momentum, depend solely on the time-averaged magnetic flux a0,{a}_{0},a0​, a spatially varying one-particle potential leads to pronounced resonances. In the case of moderate electronic relaxation, the stationary properties are determined by the Floquet state with lowest time-averaged energy. For the associated persistent angular momentum we predict jumps with heights proportional to the number of electrons N. We further show that, already by measuring the locations of these jumps in the ${(a}_{0},\ensuremath{\omega})$ plane, one could determine the number of electrons N as well as the effective $e\ensuremath{-}e$ interaction.
Theoretical studies suggest that an interacting one-dimensional electron system does not behave as a Fermi liquid. Instead, it is predicted to form a Luttinger liquid characterized e.g. by the absence of fermionic quasiparticles and interaction-dependent power-laws for transport quantities. So far, however, only a limited number of experiments have provided evidence for Luttinger liquid behavior. In this thesis we consider alternative methods to detect electronic correlations in one-dimensional electron systems based on our calculations of persistent currents and quantization of charge in ballistic single channel rings. The persistent current in an isolated impurity-free Luttinger liquid ring is found to be the same as for a system of non-interacting electrons. In contrast, a Luttinger liquid behavior is revealed if the ring is weakly connected to an electron reservoir and capacitively coupled to a gate electrode. The persistent current and the average charge in the ring are shown to change in a stepwise manner with the gate voltage at low temperatures. Both the step positions and the crossover temperatures are determined by the Luttinger liquid correlation parameter .alpha.(.alpha.c) which therefore can be measured. When a potential barrier is incorporated into the ring, the presence of strong interactions between electrons leads to a suppression of the persistent current and a change in its temperature dependence. A sharp maximum of the current is predicted to occur as a function of temperature. When two barriers are introduced into the ring, the possibility of resonant tunneling for the persistent current opens up. Two kinds of resonant behavior are discovered in the closed geometry: a current independent of the barrier transparency and a current analogous to the one with a single potential barrier. The persistent current in a ring containing a gas of fractionally charged excitations obeying fractional exclusion statistics is also studied. At finite temperatures, anomalous oscillations of the current are predicted as a manifestation of the statistics of the quasiparticles.
The persistent current and charge stiffness of a one-dimensional Luttinger liquid on a ring threaded by a magnetic flux are calculated by Monte Carlo simulation. By changing the random impurity potential strength and the electron-electron interaction, we see a crossover behavior between weak- and strong-impurity limits. For weak-impurity potentials, interactions enhance impurity effects, that is, interactions decrease the current and the stiffness. On the other hand, interactions tend to screen impurities when the impurity potential is strong. Temperature dependence of the persistent current and the charge stiffness shows a peak at a characteristic temperature, consistent with a recent single-impurity study.
We present an exact calculation of the persistent current of electrons with Hubbard interactions in disordered one-dimensional rings. Numerical simulations show that the currents exhibit an interesting dependence on the configuration and strength W of disorder, and the value of the Hubbard interaction U. Our numerical simulations show that disorder and interaction both can monotonically suppress the amplitude of the current. The interaction can sometimes enhance the current of a disordered ring depending on its configuration and degree of disorder, even though smaller than the corresponding ideal-ring value VF{\mathit{V}}_{\mathit{F}}VF​e/L. We have also studied the effect of negative U, and shown that it can influence the current as the positive U does. Based on the eigenvalue theory of a matrix, we give a simple explanation of the effects of disorder and interaction on currents. The main results are consistent with the numerical results for the case of long-range Coulomb interactions.
We address the issue of whether a system of interacting electrons confined to a one-dimensional ring can sustain a spontaneous persistent current in the absence of an externally applied flux. The current-current coupling between electrons, describing radiative back-action effects, is exactly treated in the formalism of quantum electrodynamics, where the electrons interact via the exchange of virtual photons. In addition, the instantaneous screened Coulomb potential is taken into account using the Luttinger liquid model including finite-size parity effects. The partition function is calculated exactly, with the result that the system does not possess a spontaneous persistent current. We show that, in the presence of an external flux, the amplitude of the (conventional) persistent current is reduced by quantum fluctuations of the internal transverse electromagnetic field. These corrections can be expressed in terms of the self-induction of the ring and are shown to be of first and higher order in the small dimensionless parameter \ensuremath{\alpha}vF∗{\mathit{v}}_{\mathit{F}}^{\mathrm{*}}vF∗​/c, where \ensuremath{\alpha} is the fine-structure constant and vF∗{\mathit{v}}_{\mathit{F}}^{\mathrm{*}}vF∗​ the Fermi velocity renormalized through Coulomb interactions.
The persistent current in a lattice model of a one-dimensional interacting electron system is systematically studied using a complex version of the density-matrix renormalization group algorithm and the functional renormalization group method. We mainly focus on the situation where a single impurity is included in the ring penetrated by a magnetic flux. Due to the interplay of the electron-electron interaction and the impurity the persistent current in a system of N lattice sites vanishes faster than 1/N.1/N.1/N. Only for very large systems and large impurities our results are consistent with the Bosonization prediction obtained for an effective field theory. The results from the density-matrix renormalization group and the functional renormalization group agree well for interactions as large as half the bandwidth, even though as an approximation in the latter method the flow of the two-particle vertex is neglected. This confirms that the functional renormalization group method is a very powerful tool to investigate correlated electron systems. The method will become very useful for the theoretical description of the electronic properties of small conducting ring molecules.