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Aharonov-Bohm effect in Luttinger liquid

Fedor Vasilievich Kusmartsev

Published Oct 1, 1992• 0 citations
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Abstract

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.

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Soviet Journal of Low Temperature Physics•Sep 1, 1992•2 citations
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.
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 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 study the dependence of the ground state energy on an applied Aharonov-Bohm flux $\ensuremath{\Phi}$ for the Luttinger model with large momentum scattering. Employing the method of finite size bosonization, we show that for systems with a spin gap but with gapless charge degrees of freedom, the ground state energy has an exact period of hc∕2ehc∕2ehc∕2e, i.e., half a flux quantum, in the limit of large system size LLL. Finite size corrections are found to vanish exponentially in LLL. This behavior is contrasted to that of the spin gapless case, for both even and odd particle number. Generalizations to finite temperature are also discussed.
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.