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Physical review. B, Condensed matter

Possible bistability of the persistent current of two interacting electrons in a quantum ring

L. Wendler, Vladimir M. Fomin

Published Jun 15, 1995• 28 citations
Publisher

Abstract

We calculate the energy band structure and the persistent current of two interacting electrons in a quantum ring in the presence of a magnetic flux for a strongly correlated situation: a rotating Wigner molecule with internal vibrations. We discuss a possible bistability and hysteresis in the persistent current in dependence on the magnetic flux.

DOI
10.1103/physrevb.51.17814

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Physical review. B, Condensed matter•Feb 15, 1993•30 citations
We calculate the energy band structure and the persistent current of two interacting electrons in a quantum ring in the presence of a magnetic flux for a strongly correlated situation: a rotating Wigner molecule with internal vibrations. We discuss a possible bistability and hysteresis in the persistent current in dependence on the magnetic flux.
We calculate the magnetic moment (`persistent current') in a strongly correlated electron system --- a Wigner crystal --- in a one-dimensional ballistic ring. The flux- and temperature dependence of the persistent current is shown to be essentially the same as for a system of non-interacting electrons. In contrast, by incorporating into the ring geometry a tunnel barrier, that pins the Wigner crystal, the current is suppressed and its temperature dependence is drastically changed. The competition between two temperature effects --- a reduced barrier height for macroscopic tunneling and a loss of quantum coherence --- results in a sharp peak in the temperature dependence, which for a rigid Wigner crystal appears at T∼0.5 ℏs/LT\sim 0.5\ \hbar s/LT∼0.5 ℏs/L, (sss is the sound velocity of the Wigner crystal, LLL is the length of the ring).
Using numerical exact diagonalization and analytical variational calculations, we examine the effect of an electronic two-band structure on the persistent current of an interacting disordered one-dimensional ring. Several different features in comparison to a single band model are encountered. Among them are a transition in the magnitude of current for a critical ratio of the hopping elements in the two bands, and a large increase in the rms value of the persistent current for certain values of interaction and disorder. Possible experimental implications are discussed.
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.
We use Density Functional Theory to study interacting spinless electrons on a one-dimensional quantum ring in the density range where the system undergoes Wigner crystallization. The Wigner transition leads to a drastic ``collective'' electron localization due to the Wigner crystal pinning, provided a weak impurity potential is applied. To reveal this localization we examine a persistent current in a ring penetrated by a magnetic flux. Using the DFT-OEP method we calculated the current as a function of the interaction parameter r_S. We find that in the limit of vanishing impurity potential the persistent current stays constant up to a critical value of r_S^c=2.05 but shows a drastic exponential decay for larger r_S which reflects a formation of a pinned Wigner crystal. Above r_S^c the amplitude of the electron density oscillations exactly follows the (r_S-r_S^c)^{1/2} behaviour, confirming a second-order phase transition as expected in the mean-field-type OEP approximation.
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.
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.