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arXiv:cond-mat/9607004 · Journal of the Physical Society of Japan

Exactly Solvable Model for Persistent Currents in the Presence of Hard-Core Interaction in One-Dimensional Mesoscopic Ring

You‐Quan Li, Zhongshui Ma

Published Jun 15, 1996• 5 citations
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Abstract

A simplest model of the persistent current in presence of many-body interaction in a mesoscopic ring is presented. The Bethe ansatz approach is used to find exact solutions. Both cases in the absence of impurity and in the presence of impurity are solved and compared. It is indicated that both impurity and hard-core interaction do not affect the result of persistent current explicitly if there is a large number of electrons in the ring.

DOI
10.1143/jpsj.65.1519

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A simplest model of the persistent current in presence of many-body interaction in a mesoscopic ring is presented. The Bethe ansatz approach is used to find exact solutions. Both cases in the absence of impurity and in the presence of impurity are solved and compared. It is indicated that both impurity and hard-core interaction do not affect the result of persistent current explicitly if there is a large number of electrons in the ring.
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We consider a one-dimensional mesoscopic quantum ring filled with spinless electrons and threaded by a magnetic flux, which carries a persistent current at zero temperature. The interplay of Coulomb interactions and a single on-site impurity yields a nontrivial dependence of the persistent current on the size of the ring. We determine numerically the asymptotic power law for systems up to 32 00032\text{ }00032 000 sites for various impurity strengths and compare with predictions from Bethe ansatz solutions combined with bosonization. The numerical results are obtained using an improved functional renormalization-group (fRG) method. We apply the density-matrix renormalization-group (DMRG) and exact diagonalization methods to benchmark the fRG calculations. We use DMRG to study the persistent current at low electron concentrations in order to extend the validity of our results to quasicontinuous systems. We briefly comment on the quality of calculated fRG ground-state energies by comparison with exact DMRG data.
Based upon a tight-binding model Hamiltonian and by applying the Jordan-Wigner transformation, the Aharonov-Bohm (AB) effect for charged hard-core bosons in one-dimensional (1D) mesoscopic rings is investigated for the first time. By introducing a specific unitary transformation, the energy spectrum and the persistent charge current are analytically derived for ID perfect lattices. Similar to electron systems, it has been found that the energy and persistent current are periodic in AB flux with the period Φ0=hc/q. More interestingly, we have exactly shown that, in the absence of the external AB flux, the charge current can be self-sustained via the AB effect, which is in fact the ground state of the system regardless of the number of hard-core bosons being even or odd.
In this paper, we study persistent charge and spin currents in the ground state of the one-dimensional mesoscopic Hubbard ring with repulsive interaction based on its Bethe-ansatz solution. We find that the persistent charge current is suppressed by the on-site Coulomb interaction more significantly at half-filling than away from half-filling. As the system size increases the charge current decays exponentially as I ∼ L −1 e − L /ξ in the half-filling case and decays in a power law away from the half-filling case. We also find that the persistent spin current is suppressed by the on-site Coulomb interaction, and it decays algebraically as the system size increases both at half-filling and away from half-filling.
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.
We propose an effective description of the interaction between the nearest-neighboring particles in a continuum theory. The contributions of the electron-electron interaction to the persistent current in 1D strongly correlating mesoscopic rings with or without impurities are analyzed. It is shown that the nearest-neighborhood int eraction gives significant contributions to the current and correlation functions. The enhance of the theoretical value of current magnitude is observed at finite temperature in the presence of the impurity scattering. The statistical property of the persistent current over random impurity distribution is also discussed. It is found that the exponential law of the persistent current for a non-interacting system will remain in an interacting one, as long as the interactions between nonnearest-neighborhoods are excluded.