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Journal of Physics Condensed Matter

Magnetic resonance on a ring of aromatic molecules

Fedor Vasilievich Kusmartsev

Published May 6, 1991• 57 citations
Publisher

Abstract

The authors have calculated exactly the spectrum of electrons on a ring in a magnetic field for the case of the Hubbard interaction between electrons. They have found that very unusual resonances occur when the flux through the ring changes. The appearance of these resonances strongly depends on the total number of electrons N. They appear when the flux changes by Delta f=1/N. In analogy with two-particle bound states one can think that on a ring with strong interactions between electrons, N-particle bound states appear. Such resonances may be found in aromatic molecules and on a ring system of singly connected quantum dots for 'Coulomb blockade' conditions.

DOI
10.1088/0953-8984/3/18/014

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The authors have calculated exactly the spectrum of electrons on a ring in a magnetic field for the case of the Hubbard interaction between electrons. They have found that very unusual resonances occur when the flux through the ring changes. The appearance of these resonances strongly depends on the total number of electrons N. They appear when the flux changes by Delta f=1/N. In analogy with two-particle bound states one can think that on a ring with strong interactions between electrons, N-particle bound states appear. Such resonances may be found in aromatic molecules and on a ring system of singly connected quantum dots for 'Coulomb blockade' conditions.
We solve the one-dimensional extended Hubbard model with a magnetic flux analytically for the case of two particles. We find that bound states are ground states only for attractive interactions. We have studied the variation of the ground-state energy as a function of the flux. For the bound state, the period of the oscillation is half of a flux quantum. The dependence of the magnitude of the energy barrier on the size of the ring is discussed in connection with the possible existence of the flux quantization in the thermodynamic limit.
We find a fine structure in the Aharonov-Bohm effect, which is characterized by the appearance of a type of periodic oscillation having a smaller fractional period. The effect is illustrated with the aid of a solution of the Bethe ansatz equations for N electrons located on a Hubbard ring in a magnetic field. Specifically, we found that at low density or strong coupling on a Hubbard ring in a magnetic field oscillations with three different periods occur. Along with the conventional Aharonov-Bohm oscillations with the period equal to 1, two additional oscillations with periods equal to 1/N and M/N coexist, where M is the number of down-spin particles and the periods are measured in units of the elementary flux quantum ${\mathrm{\ensuremath{\Phi}}}_{0}$=h/e. The fine structure is due to electron-electron and Zeeman interactions. When M=N/2 there is a coexistence of the fractional 1/N and half-flux quantum periodic oscillations only. With increasing magnetic field, with the spin flips created due to Zeeman interaction, the half-flux quantum period transforms to the integer one. This transformation is not continuous but via the appearance of the fractional M/N oscillations. We discuss the relation of the described fine structure and the M/N effect to existing experiments on single and on arrays of rings, where integer and half-flux quantum periods are, respectively, observed.
The authors consider a charged particle confined to a ring encircling a magnetic flux. Supposing that the particle is trapped by a delta-function potential located on the ring, they examine the flux dependence of the energy and the wavefunction of the bound state. This offers a simple explicit model for the Aharonov-Bohm effect on a system localized, but not completely confined, to a simply connected region off a magnetic flux. The result is used to show that Berry's phase for the bound state, when the delta-function potential is slowly transported around the ring, consists not only of the Aharonov-Bohm phase proportional to the flux, but also of an extra term which reflects the incomplete localization of the state; in the limit of strong binding the extra term depends sinusoidally on the flux.
Ground-state properties of one-dimensional quantum rings with a few electrons, which interact with each other in the form of 1∕r1∕r1∕r Coulomb repulsion, are studied by exact diagonalization. For three electrons, the fully spin-polarized ground state is uniquely realized when the diameter of the ring is sufficiently large. In contrast, for four and five electrons, the fully polarized state never becomes the unique ground state, however large the diameter is. These results can be understood in terms of the multiple-spin exchange. When a perpendicular magnetic field is applied to the ring (i.e., the Aharonov-Bohm flux is introduced), the persistent current occurs, and the spin chirality is finite for a certain value of the flux. The difference between the quantum ring and the circularly arranged quantum dots is also discussed.
We investigate interacting spinless electrons on finite molecular ring networks described by a tight-binding Hubbard Hamiltonian. The interplay between lattice geometry, Coulomb interactions and discrete symmetries is analysed for rings with L=3,4,5,6L=3,4,5,6L=3,4,5,6 nodes, filled with one, two or three electrons. Special attention is devoted to the role of the network symmetries in determining the structure of the many-body spectrum and the Mulliken classification of the eigenstates. Using group-theoretical methods, we examine the evolution of the spectra in the presence of an external magnetic flux. The Zeeman effect lifts degeneracies and results in combination with the Coulomb interaction to avoided crossings in symmetry sectors. We identify a qualitatively distinction between systems with an even and odd number of particles. At half-filling, particle-hole symmetry (duality) protects selected symmetry sectors against Zeeman splitting.