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:0801.3603

Influence of spin and interactions on quantum dots and nano-wires

Weiss, Yuval

Published Jan 23, 2008• 0 citations
arXivHTMLPDF

Abstract

In this dissertation we use sophisticated numerical methods in order to examine ground-state (GS) properties of two types of quantum systems with electron electron interactions: A quantum dot (QD) and a nano-wire. In the first half of the work we study a system of a single level coupled to a one-dimensional wire with interacting spinless electrons, when the wire is either clean or disordered. We utilize the density-matrix renormalization-group (DMRG) method to investigate the influence of the level on several thermodynamic properties of the clean interacting wire, which can be in one of two phases: Tomonaga-Luttinger liquid and charge density wave phases. When the wire is disordered, we investigate the Friedel oscillations, exploring the difference between the two phases and comparing them to the clean non-interacting case, for which we develop an exact formula for the oscillations. In the second half of the dissertation we study two cases of an isolated two-dimensional QD. We begin by an investigation of a new numerical method, the particle-hole DMRG (PH-DMRG), which is used to calculate the GS energy of a disordered QD consisting of interacting spinless electrons. We show that this method is much more accurate than the Hartree-Fock method, and we suggest an improvement of the algorithm, which reduces the error rate by almost 30 percents. Finally we study the magnetization of a QD with spin 1/2 electrons, in the presence of spin-orbit coupling and interactions. We calculate the g-factor and the expectation values of the spin operators in the GS, and find that when the QD is occupied by an even number of electrons, the GS can have a finite magnetization.

DOI
10.48550/arxiv.0801.3603

Related papers

Tomonaga-Luttinger parameters for quantum wires

Wolfgang Häusler, Lars Kecke, Allan H. MacDonald

The low-energy properties of a homogeneous one-dimensional electron system are completely specified by two Tomonaga-Luttinger parameters ${K}_{\ensuremath{\rho}}$ and ${v}_{\ensuremath{\sigma}}.$ In this paper we discuss microscopic estimates of the values of these parameters in semiconductor quantum wires that exploit their relationship to thermodynamic properties. Motivated by the recognized similarity between correlations in the ground state of a one-dimensional electron liquid and correlations in a Wigner crystal, we evaluate these thermodynamic quantities in a self-consistent Hartree-Fock approximation. According to our calculations, the Hartree-Fock approximation ground state is a Wigner crystal at all electron densities and has antiferromagnetic order that gradually evolves from spin-density wave to localized in character as the density is lowered. Our results for ${K}_{\ensuremath{\rho}}$ are in good agreement with weak-coupling perturbative estimates ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}$ at high densities, but deviate strongly at low densities, especially when the electron-electron interaction is screened at long distances. ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}\ensuremath{\sim}{n}^{1/2}$ vanishes at small carrier density n, whereas we conjecture that ${K}_{\ensuremath{\rho}}\ensuremath{\rightarrow}1/2$ when $\stackrel{\ensuremath{\rightarrow}}{n}0,$ implying that ${K}_{\ensuremath{\rho}}$ should pass through a minimum at an intermediate density. Observation of this nonmonotonic dependence could be used to measure the effective interaction range in a realistic semiconductor quantum wire geometry. In the spin sector we find that the spin velocity decreases with increasing interaction strength or decreasing n. Strong correlation effects make it difficult to obtain fully consistent estimates of ${v}_{\ensuremath{\sigma}}$ from Hartree-Fock calculations. We conjecture that ${v}_{\ensuremath{\sigma}}{/v}_{\mathrm{F}}\ensuremath{\propto}{n/V}_{0},$ where ${V}_{0}$ is the interaction strength, in the limit $\stackrel{\ensuremath{\rightarrow}}{n}0.$

arXiv:cond-mat/0108290•Physical review. B, Condensed matter•Feb 7, 2002•43 citations

Friedel oscillations in disordered quantum wires: Influence of electron-electron interactions on the localization length

Y. Weiss, Moshe Goldstein, Richard Berkovits

The Friedel oscillations caused by an impurity located at one edge of a disordered interacting quantum wire are calculated numerically. The electron density in the system's ground state is determined using the density-matrix renormalization-group method, and the Friedel oscillation data are extracted using the density difference between the case in which the wire is coupled to an impurity and the case where the impurity is uncoupled. We show that the power-law decay of the oscillations occurring for an interacting clean one-dimensional sample described by the Luttinger liquid theory is multiplied by an exponential decay term due to the disorder. Scaling of the average Friedel oscillations by this exponential term collapses the disordered sample data on the clean results. We show that the length scale governing the exponential decay may be associated with the Anderson localization length and thus be used as a convenient way to determine the dependence of the localization length on disorder and interactions. The localization length decreases as a function of the interaction strength, in accordance with previous predictions.

arXiv:cond-mat/0611217•Physical Review B•Feb 28, 2007•18 citations

Theoretical study of interacting electrons in one dimension - ground states and experimental signatures

Trinanjan Datta

This dissertation focuses on a theoretical study of interacting electrons in one dimension. The research elucidates the ground state (zero temperature) electronic phase diagram of an aluminum arsenide quantum wire which is an example of an interacting one dimensional electron liquid. Using one dimensional field theoretic methods involving abelian bosonization and the renormalization group we show the existence of a spin gapped quantum wire with electronic ground states such as charge density wave and singlet superconductivity. The superconducting state arises due to the unique umklapp interaction present in the aluminum arsenide quantum wire bandstructure discussed in this dissertation. It is characterized by Cooper pairs carrying a finite pairing momentum. This is a realization of the Fulde-Ferrell-Larkin- Ovchinnikov state which is known to lead to inhomogeneous superconductivity. The dissertation also presents a theoretical analysis of the finite temperature single hole spectral function of the one dimensional electron liquid with gapless spin and charge modes (Luttinger liquid). The hole spectral function is measured in angle resolved photoemission spectroscopy experiments. The results predict a kink in the effective electronic dispersion of the Luttinger liquid. A systematic study of the temperature and interaction dependence of the kink provides an alternative way to detect spincharge separation in one dimensional systems where the peak due to the spin part of the spectral function is suppressed.

arXiv:0711.0120•Nov 1, 2007•0 citations

DMRG method for the linear DC-conductance of one-dimensional correlated systems

Jan-Moritz Bischoff

Interesting discoveries have recently been made in the area of solid state physics and nano technology regarding electronic systems that are confined to reduced spatial dimensions. Under these circumstances the quantum nature of the electrons begins to surface. In the one-dimensional case this class of systems is therefore called quantum- or nano wires. The conductance in such quantum wires becomes quantized and is dominated by electronic correlations. The goal of this thesis was to develop a fast and efficient procedure to describe a realistic transport measurement for such systems. The method presented here is in- spired by the work of Dan Bohr et al. who used the Density Matrix Renormalization Group algorithm(DMRG) to evaluate the Kubo formula for the linear response to an applied potential. In a similar way the procedure of this thesis also computes the Kubo formula using the dynamical DMRG in order to calculate the low frequency response to an AC source-drain voltage of finite systems. Making use of a special finite size scaling allows to extrapolate the results into the thermodynamic limit and to finally obtain the linear DC-conductance. First, the theoretical framework for the new method was developed. This was fol- lowed by testing the new approach in different model systems. The first model com- prised of a tight-binding chain that was filled with spinless, noninteracting fermions. In subsequent steps the model was expanded by introducing a homogeneous in- teraction in the Luttinger liquid regime followed by inhomogeneous interaction for different chain sections. The potential impact of impurities was assessed in the above models. Lastly, the two cases of the Hubbard chain of spin- 2 1 electrons and of the Holstein model in one dimension, where Einstein phonons interact with spinless fermions, were studied. The results of the proposed method are consistent with Luttinger liquid and Lan- dauer scattering theory. In the Luttinger liquid regime the influence of phonons to the charge structure factor was confirmed. A dependency on the length of the quantum wire in the form of dissipation effects for the conductance was not observed.

•Jan 1, 2019•0 citations

Electro-electron interactions and correlations in atomic-scale wires: Luttinger liquids and their spectral signatures

L. K. Dash

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.

•Jan 1, 2002•0 citations

Numerical Renormalization, Doped Antiferromagnets, and Statistics - Topics in Low Dimensional Physics

Martin Andersson

The first part of the thesis deals with so called numerical renormalization group methods, and in particular the density-matrix renormalization group (DMRG). This method performs extremely well for large one-dimensional quantum systems like spin-chains and Hubbard models. We discuss general features of the quantum states obtained by using this method for large systems. One finds that the method works better, i.e. faster convergence and higher accuracy, when applied to systems having an energy gap between the ground state and the first excited state. An investigation of the behavior of the DMRG for a simple gapless system is done, mainly focusing on the properties of correlation functions. The second part of the thesis deals with topics related to superconductivity. It starts with a discussion of magnetic Josephson junctions and the physical processes governing the behavior of such a system. We also introduce and discuss models of high-temperature superconductors, in particular the t-J model and its relation to stripes. Starting from the t-J model on a two dimensional square lattice, we discuss an approximation where the spin-configuration is assumed static. We discuss the properties of the fictitious magnetic field experienced by the electrons due to the static spin-configuration. Configurations being close to ferromagnetic and antiferromagnetic turn out to have very different properties. Furthermore, we derive an effective model for this system and look for different phases of this model. In particular we investigate whether the system takes advantage of a spin-generated fictitious flux. Uniform phases are discussed in a Hartree-Fock approximation and we also consider stripes between antiferromagnetically ordered regions, these being modeled in a self-consistent Hartree approximation. Finally, and inspired by the fictitious fluxes, there is a chapter discussing the statistics of particles living on a circle. It is shown that there are two free parameters in the quantization of such a system and we also sort out the relation between fermions and hard-core bosons in one dimension.

•Jan 1, 2001•1 citation
In this dissertation we use sophisticated numerical methods in order to examine ground-state (GS) properties of two types of quantum systems with electron electron interactions: A quantum dot (QD) and a nano-wire. In the first half of the work we study a system of a single level coupled to a one-dimensional wire with interacting spinless electrons, when the wire is either clean or disordered. We utilize the density-matrix renormalization-group (DMRG) method to investigate the influence of the level on several thermodynamic properties of the clean interacting wire, which can be in one of two phases: Tomonaga-Luttinger liquid and charge density wave phases. When the wire is disordered, we investigate the Friedel oscillations, exploring the difference between the two phases and comparing them to the clean non-interacting case, for which we develop an exact formula for the oscillations. In the second half of the dissertation we study two cases of an isolated two-dimensional QD. We begin by an investigation of a new numerical method, the particle-hole DMRG (PH-DMRG), which is used to calculate the GS energy of a disordered QD consisting of interacting spinless electrons. We show that this method is much more accurate than the Hartree-Fock method, and we suggest an improvement of the algorithm, which reduces the error rate by almost 30 percents. Finally we study the magnetization of a QD with spin 1/2 electrons, in the presence of spin-orbit coupling and interactions. We calculate the g-factor and the expectation values of the spin operators in the GS, and find that when the QD is occupied by an even number of electrons, the GS can have a finite magnetization.
The low-energy properties of a homogeneous one-dimensional electron system are completely specified by two Tomonaga-Luttinger parameters ${K}_{\ensuremath{\rho}}$ and ${v}_{\ensuremath{\sigma}}.$ In this paper we discuss microscopic estimates of the values of these parameters in semiconductor quantum wires that exploit their relationship to thermodynamic properties. Motivated by the recognized similarity between correlations in the ground state of a one-dimensional electron liquid and correlations in a Wigner crystal, we evaluate these thermodynamic quantities in a self-consistent Hartree-Fock approximation. According to our calculations, the Hartree-Fock approximation ground state is a Wigner crystal at all electron densities and has antiferromagnetic order that gradually evolves from spin-density wave to localized in character as the density is lowered. Our results for ${K}_{\ensuremath{\rho}}$ are in good agreement with weak-coupling perturbative estimates ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}$ at high densities, but deviate strongly at low densities, especially when the electron-electron interaction is screened at long distances. ${K}_{\ensuremath{\rho}}^{\mathrm{pert}}\ensuremath{\sim}{n}^{1/2}$ vanishes at small carrier density n, whereas we conjecture that ${K}_{\ensuremath{\rho}}\ensuremath{\rightarrow}1/2$ when $\stackrel{\ensuremath{\rightarrow}}{n}0,$ implying that ${K}_{\ensuremath{\rho}}$ should pass through a minimum at an intermediate density. Observation of this nonmonotonic dependence could be used to measure the effective interaction range in a realistic semiconductor quantum wire geometry. In the spin sector we find that the spin velocity decreases with increasing interaction strength or decreasing n. Strong correlation effects make it difficult to obtain fully consistent estimates of ${v}_{\ensuremath{\sigma}}$ from Hartree-Fock calculations. We conjecture that ${v}_{\ensuremath{\sigma}}{/v}_{\mathrm{F}}\ensuremath{\propto}{n/V}_{0},$ where V0{V}_{0}V0​ is the interaction strength, in the limit $\stackrel{\ensuremath{\rightarrow}}{n}0.$
The Friedel oscillations caused by an impurity located at one edge of a disordered interacting quantum wire are calculated numerically. The electron density in the system's ground state is determined using the density-matrix renormalization-group method, and the Friedel oscillation data are extracted using the density difference between the case in which the wire is coupled to an impurity and the case where the impurity is uncoupled. We show that the power-law decay of the oscillations occurring for an interacting clean one-dimensional sample described by the Luttinger liquid theory is multiplied by an exponential decay term due to the disorder. Scaling of the average Friedel oscillations by this exponential term collapses the disordered sample data on the clean results. We show that the length scale governing the exponential decay may be associated with the Anderson localization length and thus be used as a convenient way to determine the dependence of the localization length on disorder and interactions. The localization length decreases as a function of the interaction strength, in accordance with previous predictions.
This dissertation focuses on a theoretical study of interacting electrons in one dimension. The research elucidates the ground state (zero temperature) electronic phase diagram of an aluminum arsenide quantum wire which is an example of an interacting one dimensional electron liquid. Using one dimensional field theoretic methods involving abelian bosonization and the renormalization group we show the existence of a spin gapped quantum wire with electronic ground states such as charge density wave and singlet superconductivity. The superconducting state arises due to the unique umklapp interaction present in the aluminum arsenide quantum wire bandstructure discussed in this dissertation. It is characterized by Cooper pairs carrying a finite pairing momentum. This is a realization of the Fulde-Ferrell-Larkin- Ovchinnikov state which is known to lead to inhomogeneous superconductivity. The dissertation also presents a theoretical analysis of the finite temperature single hole spectral function of the one dimensional electron liquid with gapless spin and charge modes (Luttinger liquid). The hole spectral function is measured in angle resolved photoemission spectroscopy experiments. The results predict a kink in the effective electronic dispersion of the Luttinger liquid. A systematic study of the temperature and interaction dependence of the kink provides an alternative way to detect spincharge separation in one dimensional systems where the peak due to the spin part of the spectral function is suppressed.
Interesting discoveries have recently been made in the area of solid state physics and nano technology regarding electronic systems that are confined to reduced spatial dimensions. Under these circumstances the quantum nature of the electrons begins to surface. In the one-dimensional case this class of systems is therefore called quantum- or nano wires. The conductance in such quantum wires becomes quantized and is dominated by electronic correlations. The goal of this thesis was to develop a fast and efficient procedure to describe a realistic transport measurement for such systems. The method presented here is in- spired by the work of Dan Bohr et al. who used the Density Matrix Renormalization Group algorithm(DMRG) to evaluate the Kubo formula for the linear response to an applied potential. In a similar way the procedure of this thesis also computes the Kubo formula using the dynamical DMRG in order to calculate the low frequency response to an AC source-drain voltage of finite systems. Making use of a special finite size scaling allows to extrapolate the results into the thermodynamic limit and to finally obtain the linear DC-conductance. First, the theoretical framework for the new method was developed. This was fol- lowed by testing the new approach in different model systems. The first model com- prised of a tight-binding chain that was filled with spinless, noninteracting fermions. In subsequent steps the model was expanded by introducing a homogeneous in- teraction in the Luttinger liquid regime followed by inhomogeneous interaction for different chain sections. The potential impact of impurities was assessed in the above models. Lastly, the two cases of the Hubbard chain of spin- 2 1 electrons and of the Holstein model in one dimension, where Einstein phonons interact with spinless fermions, were studied. The results of the proposed method are consistent with Luttinger liquid and Lan- dauer scattering theory. In the Luttinger liquid regime the influence of phonons to the charge structure factor was confirmed. A dependency on the length of the quantum wire in the form of dissipation effects for the conductance was not observed.
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 first part of the thesis deals with so called numerical renormalization group methods, and in particular the density-matrix renormalization group (DMRG). This method performs extremely well for large one-dimensional quantum systems like spin-chains and Hubbard models. We discuss general features of the quantum states obtained by using this method for large systems. One finds that the method works better, i.e. faster convergence and higher accuracy, when applied to systems having an energy gap between the ground state and the first excited state. An investigation of the behavior of the DMRG for a simple gapless system is done, mainly focusing on the properties of correlation functions. The second part of the thesis deals with topics related to superconductivity. It starts with a discussion of magnetic Josephson junctions and the physical processes governing the behavior of such a system. We also introduce and discuss models of high-temperature superconductors, in particular the t-J model and its relation to stripes. Starting from the t-J model on a two dimensional square lattice, we discuss an approximation where the spin-configuration is assumed static. We discuss the properties of the fictitious magnetic field experienced by the electrons due to the static spin-configuration. Configurations being close to ferromagnetic and antiferromagnetic turn out to have very different properties. Furthermore, we derive an effective model for this system and look for different phases of this model. In particular we investigate whether the system takes advantage of a spin-generated fictitious flux. Uniform phases are discussed in a Hartree-Fock approximation and we also consider stripes between antiferromagnetically ordered regions, these being modeled in a self-consistent Hartree approximation. Finally, and inspired by the fictitious fluxes, there is a chapter discussing the statistics of particles living on a circle. It is shown that there are two free parameters in the quantization of such a system and we also sort out the relation between fermions and hard-core bosons in one dimension.