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International Journal of Modern Physics B

PHASE LOCKING EFFECT OF PERSISTENT CURRENTS IN MESOSCOPIC ANDERSON–HUBBARD RINGS

Fuk-yeung Tsoi, Shi‐Dong Liang

Published Feb 20, 2000• 1 citation
Publisher

Abstract

The effects of the Anderson impurities as well as weak electron–electron interactions on the persistent current in a mesoscopic ring are studied by using the Anderson–Hubbard model. In this system, the single-particle energy spectrum possesses a flat b and and the s-d hybridization plays an interesting role in the phase dependence of the persistent current. In the weak s-d hybridization, the phases can be locked for different electron numbers. This phase locking effect originates from the existence of the flat band. In addition, it is found from the mean field approach that weak electron–electron interactions lead to the spin splitting of the energy b and and the shift of the electron number for the phase locking effect.

DOI
10.1142/s0217979200000480

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A tight-binding model of electron dynamics in mesoscopic normal rings is studied using boundary conformal field theory. The partition function is calculated in the low energy limit and the persistent current generated as a function of an external magnetic flux threading the ring is found. We study the cases where there are defects and electron–electron interactions separately. The same temperature scaling for the persistent current is found in each case and the functional form can be fitted, with a high degree of accuracy, to experimental data.
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.
The persistent current and spin-charge separation are studied in a one-dimensional (1D) Hubbard ring with side sites connected to the even sites of the ring. At half filling, the persistent current is always paramagnetic and has a period of flux quantum. Away from half filling, as the onsite repulsion UUU increases, the sign of the current may oscillate depending on the total number of electrons Ne{N}_{e}Ne​ and the hopping ttt between the ring site and the side site. For small hopping ttt, the number of electrons ${N}_{e}^{\ensuremath{'}}$ in the ring sites plays an important role in determining the sign of the current. In contrast to the 1D Hubbard model, a weak or a moderate UUU can strongly enhance the persistent current while the hopping ttt always suppresses the current. The study on the transmittance of the ring with two connected leads indicates that in the large UUU and small ttt case, the transmittance varies with a period $1∕{N}_{e}^{\ensuremath{'}}$ of a flux quantum, demonstrating a spin-charge separation in this system.
We study the persistent current and the Drude weight of a system of spinless fermions, with repulsive interactions and a hopping impurity, on a mesoscopic ring pierced by a magnetic flux, using a density matrix renormalization group (DMRG) algorithm for complex fields. Both the Luttinger liquid (LL) and the charge density wave (CDW) phases of the system are considered. Under a Jordan-Wigner transformation, the system is equivalent to a spin-1∕21∕21∕2 XXZXXZXXZ chain with a weakened exchange coupling. We find that the persistent current changes from an algebraic to an exponential decay with the system size, as the system crosses from the LL to the CDW phase with increasing interaction UUU. We also find that in the interacting system, the persistent current is invariant under the impurity transformation $\ensuremath{\rho}\ensuremath{\rightarrow}1∕\ensuremath{\rho}$, for large system sizes, where $\ensuremath{\rho}$ is the defect strength. The persistent current exhibits a decay that is in agreement with the behavior obtained for the Drude weight. We find that in the LL phase the Drude weight decreases algebraically with the number of lattice sites NNN, due to the interplay of the electron interaction with the impurity, while in the CDW phase, it decreases exponentially, defining a localization length which decreases with increasing interaction and impurity strength. Our results show that the impurity and the interactions always decrease the persistent current, and imply that the Drude weight vanishes in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ in both phases.
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.