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arXiv:cond-mat/0302625 · Physical review. B, Condensed matter

Luttinger liquid in a finite one-dimensional wire with box-like boundary conditions

Fabrizio Anfuso, Sebastian Eggert

Published Dec 8, 2003• 25 citations
arXivHTMLPDF

Abstract

We study a Luttinger Liquid in a finite one-dimensional wire with box-like boundary conditions by considering the local distribution of the single-particle spectral weight. This corresponds to the experimental probability of extracting a single electron at a given place and energy, which can be interpreted as the square of an electron wave function. For the noninteracting case, this is given by a standing wave at the Fermi wave vector. In the presence of interactions, however, the wave functions obtain additional structure with a sharp depletion near the edges and modulations throughout the wire. In the spinful case, these modulations correspond to the separate spin- and charge-like excitations in the system.

DOI
10.1103/physrevb.68.241301

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We study a Luttinger Liquid in a finite one-dimensional wire with box-like boundary conditions by considering the local distribution of the single-particle spectral weight. This corresponds to the experimental probability of extracting a single electron at a given place and energy, which can be interpreted as the square of an electron wave function. For the noninteracting case, this is given by a standing wave at the Fermi wave vector. In the presence of interactions, however, the wave functions obtain additional structure with a sharp depletion near the edges and modulations throughout the wire. In the spinful case, these modulations correspond to the separate spin- and charge-like excitations in the system.
We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification g=2gcg=2{g}_{c}g=2gc​, where ggg is the interaction parameter in a spinless Luttinger liquid and gc{g}_{c}gc​ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.
The electronic systems in one dimension have features quite distinct from those in higher dimensions. The onedimensional Fermi surface consists of two discrete points, while in higher dimensions it is continuous. The low-energy particle-hole pairs’ spectrum, which arises from momentum and energy conservations for wave vector between zero and twice of Fermi wave vector does not exist in one dimension. In higher dimensions, this region of missing states is filled in. The Fourier transform of one-dimensional Coulomb potential diverges. This led to wide spread belief that one-dimensional interacting electronic systems should exhibit non-Fermi liquid (known as Luttinger liquid) behaviour. However, there is still no unambiguous evidence both for the real existence or non-existence of Luttinger liquid behaviour in a realistic quantum wire. This talk briefly reviews key aspects of Fermi liquid approach and Tomonaga-Luttinger liquid approach for a onedimensional system. In this paper, structure factor, pair distribution function and electron-electron scattering, which exhibit similarity between two approaches in low energy and momentum regime, is reviewed.
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Recent advances in single atom manipulation have made it possible to create "wires" in the form of atomic scale linear structures on a semiconductor surface. If such structures are to be used in future electronic devices, it will be necessary to know to what extent they behave as one-dimensional conductors, in which the presence of electron-electron interactions in the wire lead to properties quite different from those found in higher-dimensional systems. In particular, it would be useful to know if these structures retain any of the Luttinger liquid properties that are predicted for a pure one-dimensional metal. However, experimental studies of such structures have so far yielded unclear and contradictory results. We investigate this problem theoretically by creating a highly simplified model of a wire on a surface--a chain of atoms that includes electron-electron interactions to represent the wire, coupled to a similar chain that excludes electron-electron interactions to represent, albeit crudely, the surface. We present results for the eigenstates and spectral functions for such systems that suggest that many Luttinger liquid indicators are present. However the system as a whole retains some residual Fermi liquid characteristics, and therefore merges properties of both one and higher dimensions.
We use bosonization methods to calculate the exact finite-temperature single-electron Green's function of a spinful Luttinger liquid confined by open boundaries. The corresponding local spectral density is constructed and analyzed in detail. The interplay between boundary, finite-size, and thermal effects are shown to dramatically influence the low-energy properties of the system. In particular, the well-known zero-temperature critical behavior in the bulk always crosses over to a boundary dominated regime in the vicinity of the Fermi level. Thermal fluctuations cause an enhanced depletion of spectral weight for small energies $\ensuremath{\omega}$, with the spectral density scaling as ${\ensuremath{\omega}}^{2}$ for $\ensuremath{\omega}$ much less than the temperature. Consequences for photoemission experiments are discussed.
By exactly analyzing the spin-12\frac{1}{2}21​ Luttinger liquid (LL) and numerically solving a model of a mobile impurity electron in the LL, we obtain the one-electron spectral function $A(p,\ensuremath{\omega})$ in a one-dimensional metal in an entire range of ppp at zero temperature. For ∣p∣|p|∣p∣ near the Fermi point pF{p}_{\mathrm{F}}pF​, $A(p,\ensuremath{\omega})$ is featured by two prominent peaks of spinon and (anti)holon representing spin-charge separation, but we also find an additional cusp structure between them. For $|p|\ensuremath{\gg}{p}_{\mathrm{F}}$, this structure evolves as a main peak in $A(p,\ensuremath{\omega})$ by swallowing the antiholon mode and its dispersion relation approaches that of a free electron, implying the existence of an electron excitation in the whole region, but not quite a quasiparticle in the Fermi liquid due to ever existing power-law decay of the excitation.