Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0704.1445 · Physical Review B

Deformed Wigner crystal in a one-dimensional quantum dot

Yasha Gindikin, Vladimir A. Sablikov

Published Jul 31, 2007• 26 citations
arXivHTMLPDF

Abstract

The spatial Fourier spectrum of the electron density distribution in a finite one-dimensional system and the distribution function of electrons over single-particle states are studied in detail to show that there are two universal features in their behavior, which characterize the electron ordering and the deformation of Wigner crystal by boundaries. The distribution function has a $\ensuremath{\delta}$-like singularity at the Fermi momentum ${k}_{F}$. The Fourier spectrum of the density has a steplike form at the wave vector $2{k}_{F}$, with the harmonics being absent or vanishing above this threshold. These features are found by calculations using exact diagonalization method. They are shown to be caused by Wigner ordering of electrons, affected by the boundaries. However, the common Luttinger liquid model with open boundaries fails to capture these features because it overestimates the deformation of the Wigner crystal. An improvement of the Luttinger liquid model is proposed, which allows one to describe the above features correctly. It is based on the corrected form of the density operator conserving the particle number.

DOI
10.1103/physrevb.76.045122

Related papers

A strongly correlated electron state at one-dimensional quantum points

Ya. V. Gindikin, Vladimir A. Sablikov

The method of exact diagnoalization is applied to find the distribution of electrons over single-particle states at a 1D quantum point and the Fourier spectrum of the electron density. It is shown that the distribution function contains an unexpected signularity at the Fermi level. It is found that (i) the singularity results from the Wigner electron ordering and (ii) the Wigner ordering involves suppression of spatial Fourier harmonics of the electron density observed when the interaction potential grows over the entire range of wave vectors except for the double Fermi wave vector. It is demonstrated that the distribution function calculated within the Luttinger model also has a singularity at the Fermi level and this singularity is of an irregular shape.

Journal of Communications Technology and Electronics•Jun 1, 2007•0 citations

Many particle aspects of a quantum wire

Ankit Sharma

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.

Indian Journal of Pure & Applied Physics•Feb 1, 2006•0 citations

Wigner crystallization in quasi-one-dimensional quantum wire

Ankush Girdhar, Vinod Ashokan

Abstract We study the ground-state properties of quasi-one-dimensional paramagnetic electron fluid using variational quantum Monte Carlo method. The electrons are transversally confined using the harmonically regularized Coulomb potential with long range order. In this work we have investigated the crossover from liquid to crystalline phase in the finite width paramagnetic quantum wire. The calculated pair correlation function shows the oscillations of period $$2r_\text {s} a_0$$ 2 r s a 0 , where $$a_0$$ a 0 is the Bohr’s radius and $$r_\text {s}$$ r s is Wigner Seitz radius or electron-density parameter which corresponds to a $$k=4k_\text {F}$$ k = 4 k F peak with $$k_\text {F}$$ k F being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as $$r_\text {s}$$ r s increases or as the wire width b decreases.

Discover Materials•May 27, 2023•1 citation

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

Fabrizio Anfuso, Sebastian Eggert

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.

arXiv:cond-mat/0302625•Physical review. B, Condensed matter•Dec 8, 2003•25 citations

Correlation functions of higher-dimensional Luttinger liquids

Lorenz Bartosch, Peter Kopietz

Using higher-dimensional bosonization we study correlation functions of fermions with singular forward scattering. Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)] we consider density-density interactions in $d$ dimensions that diverge in Fourier space for small momentum transfers |q| as $|\mathbf{q}{|}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}=2(d\ensuremath{-}1).$ In this case the single-particle Green's function shows Luttinger-liquid behavior. However, in contrast to $d=1,$ in higher dimensions the singularity of the momentum distribution ${n}_{\mathbf{k}}$ for wave vectors k close to the Fermi wave vector ${k}_{F}$ is characterized by a different exponent than the singularity of the density of states \ensuremath{\rho}(\ensuremath{\omega}) for frequencies \ensuremath{\omega} close to zero. We also calculate the irreducible polarization ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\mathbf{q},\ensuremath{\omega})$ for $|\mathbf{q}|\ensuremath{\approx}{2k}_{F}.$ Whereas in the one-dimensional Tomonaga-Luttinger model ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\ifmmode\pm\else\textpm\fi{}{2k}_{F}+q,\ensuremath{\omega})$ exhibits anomalous scaling, we show that in $d>1$ the leading singular corrections cancel. Thus, even though singular density-density interactions in $d>1$ lead to Luttinger liquid behavior of the single-particle Green's function, gauge invariant correlation functions such as the polarization or the conductivity show conventional Fermi-liquid behavior. We discuss consequences for the effect of disorder on higher-dimensional Luttinger liquids.

arXiv:cond-mat/9805005•Physical review. B, Condensed matter•Feb 15, 1999•12 citations

One-dimensional Fermi liquids

Johannes Voit

We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.

arXiv:cond-mat/9510014•Reports on Progress in Physics•Sep 1, 1995•1,282 citations
The spatial Fourier spectrum of the electron density distribution in a finite one-dimensional system and the distribution function of electrons over single-particle states are studied in detail to show that there are two universal features in their behavior, which characterize the electron ordering and the deformation of Wigner crystal by boundaries. The distribution function has a \ensuremathδ\ensuremath{\delta}\ensuremathδ-like singularity at the Fermi momentum kF{k}_{F}kF​. The Fourier spectrum of the density has a steplike form at the wave vector 2kF2{k}_{F}2kF​, with the harmonics being absent or vanishing above this threshold. These features are found by calculations using exact diagonalization method. They are shown to be caused by Wigner ordering of electrons, affected by the boundaries. However, the common Luttinger liquid model with open boundaries fails to capture these features because it overestimates the deformation of the Wigner crystal. An improvement of the Luttinger liquid model is proposed, which allows one to describe the above features correctly. It is based on the corrected form of the density operator conserving the particle number.
The method of exact diagnoalization is applied to find the distribution of electrons over single-particle states at a 1D quantum point and the Fourier spectrum of the electron density. It is shown that the distribution function contains an unexpected signularity at the Fermi level. It is found that (i) the singularity results from the Wigner electron ordering and (ii) the Wigner ordering involves suppression of spatial Fourier harmonics of the electron density observed when the interaction potential grows over the entire range of wave vectors except for the double Fermi wave vector. It is demonstrated that the distribution function calculated within the Luttinger model also has a singularity at the Fermi level and this singularity is of an irregular shape.
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.
Abstract We study the ground-state properties of quasi-one-dimensional paramagnetic electron fluid using variational quantum Monte Carlo method. The electrons are transversally confined using the harmonically regularized Coulomb potential with long range order. In this work we have investigated the crossover from liquid to crystalline phase in the finite width paramagnetic quantum wire. The calculated pair correlation function shows the oscillations of period 2rsa02r_\text {s} a_02rs​a0​ 2 r s a 0 , where a0a_0a0​ a 0 is the Bohr’s radius and rsr_\text {s}rs​ r s is Wigner Seitz radius or electron-density parameter which corresponds to a k=4kFk=4k_\text {F}k=4kF​ k = 4 k F peak with kFk_\text {F}kF​ k F being Fermi wave vector in the static structure factor at finite electron density. It is a signature of the onset of Wigner crystallization. It is found that the cross-over from liquid to a quasi-Wigner phase is due to enhancement of the electron correlation effects as rsr_\text {s}rs​ r s increases or as the wire width b decreases.
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.
Using higher-dimensional bosonization we study correlation functions of fermions with singular forward scattering. Following Bares and Wen [Phys. Rev. B 48, 8636 (1993)] we consider density-density interactions in ddd dimensions that diverge in Fourier space for small momentum transfers |q| as $|\mathbf{q}{|}^{\ensuremath{-}\ensuremath{\eta}}$ with $\ensuremath{\eta}=2(d\ensuremath{-}1).$ In this case the single-particle Green's function shows Luttinger-liquid behavior. However, in contrast to d=1,d=1,d=1, in higher dimensions the singularity of the momentum distribution nk{n}_{\mathbf{k}}nk​ for wave vectors k close to the Fermi wave vector kF{k}_{F}kF​ is characterized by a different exponent than the singularity of the density of states \ensuremath{\rho}(\ensuremath{\omega}) for frequencies \ensuremath{\omega} close to zero. We also calculate the irreducible polarization ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\mathbf{q},\ensuremath{\omega})$ for $|\mathbf{q}|\ensuremath{\approx}{2k}_{F}.$ Whereas in the one-dimensional Tomonaga-Luttinger model ${\ensuremath{\Pi}}_{\ensuremath{\star}}(\ifmmode\pm\else\textpm\fi{}{2k}_{F}+q,\ensuremath{\omega})$ exhibits anomalous scaling, we show that in d>1d>1d>1 the leading singular corrections cancel. Thus, even though singular density-density interactions in d>1d>1d>1 lead to Luttinger liquid behavior of the single-particle Green's function, gauge invariant correlation functions such as the polarization or the conductivity show conventional Fermi-liquid behavior. We discuss consequences for the effect of disorder on higher-dimensional Luttinger liquids.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.