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arXiv:cond-mat/9410017

Aharonov-Bohm Effect in Luttinger Liquid and Beyond

Feodor V. Kusmartsev

Published Oct 7, 1994• 0 citations
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Abstract

In systems with the spin-charge separation, the period of the Aharonov-Bohm (AB) oscillation becomes half of the flux quantum. This effect is at least related to the fact that for the creation of the holons (spinons) are needed two electrons. The effect is illustrated on the example of the Hubbard Hamiltonian with the aid of the bosonization including topological numbers and exists also in the Luttinger liquid on two chains. The relation to a fractional 1/N- AB effect, which can be associated with a modified Luttinger liquid, is discussed.

DOI
10.48550/arxiv.cond-mat/9410017

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In systems with the spin-charge separation, the period of the Aharonov-Bohm (AB) oscillation becomes half of the flux quantum. This effect is at least related to the fact that for the creation of the holons (spinons) are needed two electrons. The effect is illustrated on the example of the Hubbard Hamiltonian with the aid of the bosonization including topological numbers and exists also in the Luttinger liquid on two chains. The relation to a fractional 1/N- AB effect, which can be associated with a modified Luttinger liquid, is discussed.
We show that in the systems with the spin-charge separation the period of the AharonovBohm oscillation becomes half of the flux quantum. The effect is related to the fact that for the creation of the holons (spinons) two electrons at least are needed. The effect is illustrated on the example of the Hubbard Hamiltonian with the aid of bosonisation technique including topological numbers. The effect exists also in the Luttinger liquid on two chains. MIRAMARE TRIESTE October 1992 On leave of absence from: Department of Theoretical Physics, Oulu University, Linnanmaa, 90570, Oulu, Finland. E-mail FEO@jussi.Oulu.Fi Permanent address: L.D. Landau Institute for Theoretical Physics, GSP-1, Kosygin 2, V-334, Moscow 117940, Russia.
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 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.
We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows us to make contact with the Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wave function is also given for that spinon state. Our construction of the spectroscopy based on fractional states leads to several unique physical insights: in the low-energy limit, we find that the Sz=0{S}_{z}=0Sz​=0 continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the one-dimensional counterpart of the Laughlin fractional Quantum Hall effect quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states that we are able to describe.
The `extended Aharonov-Bohm (AB) period' recently proposed by Kusakabe and Aoki [J. Phys. Soc. Jpn (65), 2772 (1996)] is extensively studied numerically for finite size systems of strongly correlated electrons. While the extended AB period is the system length times the flux quantum for noninteracting systems, we have found the existence of the boundary across which the period is halved or another boundary into an even shorter period on the phase diagram for these models. If we compare this result with the phase diagram predicted from the Tomonaga-Luttinger theory, devised for low-energy physics, the halved period (or shorter periods) has a one-to-one correspondence to the existence of the pairing (phase separation or metal-insulator transition) in these models. We have also found for the t-J model that the extended AB period does not change across the integrable-nonintegrable boundary despite the totally different level statistics.