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Physical review. B, Condensed matter

Aharonov-Bohm oscillations with fractional period in a multichannel Wigner crystal ring

Ilya V. Krive, Arkadii A. Krokhin

Published Apr 15, 1997• 1 citation
Publisher

Abstract

We study the persistent current in a quasi-one-dimensional ring with strongly correlated electrons forming a multichannel Wigner crystal (WC). The influence of the Coulomb interaction manifests itself only in the presence of external scatterers that pin the WC. Two regimes of weak and strong pinning are considered. For strong pinning we predict the Aharonov-Bohm oscillations with fractional period. Fractionalization is due to the interchannel coupling in the process of quantum tunneling of the WC. The fractional period depends on the filling of the channels and may serve as an indicator of non-Fermi-liquid behavior of interacting electrons in quasi-one-dimensional rings.

DOI
10.1103/physrevb.55.10754

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arXiv:cond-mat/9401064•Physical review. B, Condensed matter•Jun 15, 1994•42 citations
We study the persistent current in a quasi-one-dimensional ring with strongly correlated electrons forming a multichannel Wigner crystal (WC). The influence of the Coulomb interaction manifests itself only in the presence of external scatterers that pin the WC. Two regimes of weak and strong pinning are considered. For strong pinning we predict the Aharonov-Bohm oscillations with fractional period. Fractionalization is due to the interchannel coupling in the process of quantum tunneling of the WC. The fractional period depends on the filling of the channels and may serve as an indicator of non-Fermi-liquid behavior of interacting electrons in quasi-one-dimensional rings.
Charge persistent currents in single-channel ballistic rings are investigated in various models of strongly correlated electrons (Wigner crystal, Luttinger liquid, Hubbard model, t–J model, and the fermion representation of the Heisenberg–Ising model). The primary objective of the survey is to compare the properties of Aharonov–Bohm oscillations for weakly and strongly correlated electrons and to formulate a set of persistent current characteristics that will unambiguously support the anomalous (departing from a Fermi liquid) behavior of one-dimensional electron systems. For this purpose the onset of the parity effect (dependence of the system response on whether the number of particles in the ring is odd or even) and various properties of the persistent current are analyzed in detail for model spinless fermions and real electrons. The influence of solitary impurities on the value and temperature dependence of the amplitude of the Aharanov–Bohm oscillations is also investigated. It is shown that the formation of a Wigner crystal (which is possible for weakly shielded Coulomb interaction) leads to a very pronounced anomalous (nonmonotonic) temperature dependence of the current in a nonideal ring. The properties of the charge current near the Mott metal–insulator transition are altered to the greatest degree by local types of electron–electron interaction.
We study the effect of strong electron-electron interactions on the persistent current in a multichannel ring with the aid of Bosonization and the Bethe ansatz equation discussed by Sutherland and Tsvelik. The interaction acts differently on a single and multichannel ring. If, in the first case, the current is suppressed, then in the second case the current may be enhanced. With the interaction the current is proportional to square of the number of channels. The maximum effect corresponds to medium values of the interaction, where U∼VFU\sim V_FU∼VF​. The new phenomenon of the fractional p/q- Aharonov-Bohm effect appears for interacting electrons on the multichannel ring.
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).
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.
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 influence of the interaction between electrons on the Aharonov-Bohm effect is investigated in the framework of the Hubbard model. The repulsion between electrons associated with strong correlation is compared with the case of attraction such as U-center pairing. We apply the Bethe-ansatz method and exact numerical diagonalization to the Hubbard Hamiltonian. It is shown that the quasi-half-quantum-flux periodicity occurs for any nonzero values of U for two electrons. For a large number of sites, or strong U, the quasiperiodicity becomes an exact half-quantum flux periodicity. The character of the state created on the ring is different in both cases. In U-center pairing the electrons are bound in pairs located on the same site. For strong correlations (large positive U) the electrons tend to be as far from each other as possible. We show by numerical solution of the Bethe-ansatz equations that for three electrons the flux peiodicity of the ground-state energy is equal to 1/3. The one-third periodicity may occur even for small values of the ratio U/t, for the very dilute system. It is shown analytically using the Bethe-ansatz equations for N electrons that for dilute systems with arbitrary value of the Hubbard repulsion U, the fractional Aharonov-Bohm effect occurs with period fT{\mathit{f}}_{\mathit{T}}fT​=1/N in units of the elementary flux quantum. Such a period occurs when the value Nt/LU is small, where L is the number of sites. Parity effects disappear in this fractional regime. We discuss possible experiments to detect fractional Aharonov-Bohm effects.