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arXiv:cond-mat/0504170 · The European Physical Journal B

Magnetic field effects on low dimensional electron systems: Luttinger liquid behaviour in a quantum wire

Stefano Bellucci A, Pasquale Onorato

Published May 1, 2005• 19 citations
arXivHTMLPDF
DOI
10.1140/epjb/e2005-00167-6

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It is now possible to create wires consisting of a single chain of atoms on an insulating substrate, with possible applications in nanoscale circuitry. However, experimental data on such systems are currently rather limited and as such a thorough theoretical investigation of such systems is required. In particular, it is useful to know to what extent the behaviour of such wires is one-dimensional - purely one-dimensional systems display Luttinger liquid behaviour, as opposed to the Fermi liquid behaviour characteristic of higher dimensional systems. However, conventional techniques in theoretical surface science are unable to provide a description of electronic correlations at the level required for one dimensional systems. The existing literature on such atomic wire systems is first reviewed. A discussion of physics in one spatial dimension follows, with a thorough introduction to both the Luttinger liquid and the analytical solutions of the Luttinger model. These concepts are then applied to a simplified model of an atomic wire system - two coupled chains of atoms, on just one of which electron-electron interactions are included. The chain which includes interactions thus represents the wire, while the non-interacting chain represents, albeit incompletely, the influence of the insulating substrate upon which the wire is embedded. Evidence for Luttinger liquid behaviour would be found in the excitation spectra and correlation functions of the system. A Fortran 90 code was written to implement the above, and both a description of the code and numerical results are presented. The results from excitations and correlation functions of the system indicate that, within the limitations of the model, Luttinger liquid behaviour may well survive. It is possible to calculate for larger and more complicated systems by use of the Density Matrix Renormalization Group (DMRG) algorithm, which relies on keeping only those states of the many particle system needed to describe the ground state and the low-energy excitations. The underlying theory of DMRG is discussed and followed with presentation of preliminary results based on a Hubbard model Hamiltonian.

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We form a highly simplified model of an atomic wire on a surface by the coupling of two one-dimensional chains, one with electron-electron interactions to represent the wire and one with no electron-electron interactions to represent the surface. We use exact-diagonalization techniques to calculate the eigenstates and response functions of our model, in order to determine both the nature of the coupling and to what extent the coupling affects the Luttinger liquid properties that we would expect in a purely one-dimensional system. We find that while there are indeed Luttinger liquid indicators present, some residual Fermi liquid characteristics remain.

arXiv:cond-mat/0102275•Journal of Physics Condensed Matter•May 18, 2001•12 citations
It is now possible to create wires consisting of a single chain of atoms on an insulating substrate, with possible applications in nanoscale circuitry. However, experimental data on such systems are currently rather limited and as such a thorough theoretical investigation of such systems is required. In particular, it is useful to know to what extent the behaviour of such wires is one-dimensional - purely one-dimensional systems display Luttinger liquid behaviour, as opposed to the Fermi liquid behaviour characteristic of higher dimensional systems. However, conventional techniques in theoretical surface science are unable to provide a description of electronic correlations at the level required for one dimensional systems. The existing literature on such atomic wire systems is first reviewed. A discussion of physics in one spatial dimension follows, with a thorough introduction to both the Luttinger liquid and the analytical solutions of the Luttinger model. These concepts are then applied to a simplified model of an atomic wire system - two coupled chains of atoms, on just one of which electron-electron interactions are included. The chain which includes interactions thus represents the wire, while the non-interacting chain represents, albeit incompletely, the influence of the insulating substrate upon which the wire is embedded. Evidence for Luttinger liquid behaviour would be found in the excitation spectra and correlation functions of the system. A Fortran 90 code was written to implement the above, and both a description of the code and numerical results are presented. The results from excitations and correlation functions of the system indicate that, within the limitations of the model, Luttinger liquid behaviour may well survive. It is possible to calculate for larger and more complicated systems by use of the Density Matrix Renormalization Group (DMRG) algorithm, which relies on keeping only those states of the many particle system needed to describe the ground state and the low-energy excitations. The underlying theory of DMRG is discussed and followed with presentation of preliminary results based on a Hubbard model Hamiltonian.
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.
We form a highly simplified model of an atomic wire on a surface by the coupling of two one-dimensional chains, one with electron-electron interactions to represent the wire and one with no electron-electron interactions to represent the surface. We use exact-diagonalization techniques to calculate the eigenstates and response functions of our model, in order to determine both the nature of the coupling and to what extent the coupling affects the Luttinger liquid properties that we would expect in a purely one-dimensional system. We find that while there are indeed Luttinger liquid indicators present, some residual Fermi liquid characteristics remain.