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Correlation effects in nanostructures

Paata Kakashvili

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

In this Ph.D. thesis I present the background and providefurther details to the work presented in the appended papers. The background consists of brief overviews of Kondo physics, Luttinger liquid theory, and conformal field theory. My own research is divided into three parts: the first deals with the interplay between Luttinger and Kondo physics (described in Papers I and II), the secondaddresses scanning tunneling microscopy (STM) response of a Luttinger liquid with an edge or an impurity (described in Paper III), and the third explores the boundary Green's function of 1D electrons in the spin-incoherent regime (described in Paper IV). In Papers I and II a setup is proposed which allows for acontrolled study of Kondo and Luttinger liquid physics. Itconsists of a quantum box, biased by a gate voltage, and side-coupled to a quantum wire by a point contact. Close to the degeneracy points of the Coulomb blockaded box the setup can be described as a Luttinger liquid interacting with an effective Kondo impurity. Using boundary conformal field theory techniques we predict that for the case of spin-polarized electrons the differential capacitance of the box will exhibit distinctive Luttinger liquid scaling with temperature and gate voltage. In the limit of zero magnetic field the Luttinger liquid behavior gets masked by two-channel Kondo screening, leading to a logarithmic scaling of the differential capacitance with temperatureand gate voltage. These effects should be possible to studyexperimentally, using the recently developed single-electrontransistor (SET) electrometer technique.In Paper III the finite-temperature local spectral weight (LSW) of a Luttinger liquid with a hard wall boundary is calculated. Close to the boundary the LSW exhibits characteristic oscillations indicative of spin-charge separation. The line shape of the LSW is also found to have a Fano-like asymmetry, a feature originating from the combined effect of electron-electron interaction and scattering off the boundary. Our results can be used to predict how edges and impurities influencethe STM response of one-dimensional electron systemsat low temperatures and voltage bias. Applications to STM onsingle-walled carbon nanotubes are also discussed.Paper IV, finally, presents a study of the spin-incoherent regime of 1D strongly interacting electrons. For sufficiently low densities the potential energy dominates the kinetic energy, and one can easily reach thespin-incoherent regime where the spin exchange energy is much less than the temperature. We have generalized the description of the spin-incoherent regime to account for the presence of a boundary. By calculating the exact Green's function we find that the charge sector critical exponent is highly sensitive to the boundary, strongly modifying the tunneling of electrons close to it. Our approach also allows for a detailed description of the crossover between boundary and bulk regimes.

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Strongly Interacting Electrons in One and Two Dimensions

Fabrizio Anfuso

In this thesis I present a theoretical study of three different\nstrongly correlated systems: Luttinger liquids, spin-1 chains and\nthe two-dimensional spin-1/2 Heisenberg model.\n\nFor every sub-field of our research I summarize the background and\nprovide further details to the work presented in the appended\npapers. Chapter 1 corresponds to paper I, Chapter 2 to paper II,\nChapter 3 and 4 to paper III and IV.\n\nIn paper I, we calculate with bosonization techniques the local\ndensity of states of a finite segment of Luttinger liquid. Besides of\nbeing of experimental relevance for Scanning Tunneling Microscopy\nexperiments on carbon nanotubes and cleaved overgrowth wires, the\nwave functions of a "Luttinger liquid in a box" very nicely\nvisualize the fundamental bosonic nature of the low energy\nexcitations.\n\nIn paper II, we analyze with field theory methods how the presence\nof a single-ion anisotropy affects the behavior of a spin-1 chain.\nWe determine in the framework of the Non Linear Sigma Model how the gaps\nof the anisotropic chain depend on an applied staggered magnetic\nfield. Our theoretical analysis is meant to model compounds like\n$\\rm R_2BaNiO_5$ that show coexistence of Haldane phase and\nlong-range order.\n\nIn paper III, we compute with quantum Monte Carlo simulations the\nlocal response to a uniform magnetic field around vacancies in the\ntwo-dimensional Heisenberg model. From the full understanding of the\nnumerical results, we can make quantitative predictions of relevance\nfor Nuclear Magnetic Resonance and susceptibility experiments. In\nparticular, we expect, even in the thermodynamic limit, a finite\nCurie contribution to the total susceptibility arising from the\nimpurities.\n\nFinally in paper IV, we consider interaction effects among\nimpurities. We compute the potential between static vacancies in\nseveral low-dimensional spin-1/2 antiferromagnets comparing\nthe numerics with exact results from conformal field theory and\nlinear spin wave theory. The potential is directly related to the\nlocal valence bond order and its magnitude is a loyal measure of the\namount of quantum correlations present into the system.

•Jan 1, 2006•1 citation

Kondo Effect in a Luttinger Liquid: Exact Results from Conformal Field Theory

Per Fröjdh, Henrik Johannesson

We report on exact results for the low-temperature thermodynamics of a spin- $\frac{1}{2}$ magnetic impurity coupled to a one-dimensional interacting electron system. By using boundary conformal field theory, we show that there are only two types of critical behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a theory with an anomalous response identical to that recently proposed by Furusaki and Nagaosa. Suppression of backscattering off the impurity leads to the same critical properties as for the two-channel Kondo effect.

arXiv:cond-mat/9505100•Physical Review Letters•Jul 10, 1995•97 citations

Correlation Effects in One-dimensional Electron Systems

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Theoretical studies suggest that an interacting one-dimensional electron system does not behave as a Fermi liquid. Instead, it is predicted to form a Luttinger liquid characterized e.g. by the absence of fermionic quasiparticles and interaction-dependent power-laws for transport quantities. So far, however, only a limited number of experiments have provided evidence for Luttinger liquid behavior. In this thesis we consider alternative methods to detect electronic correlations in one-dimensional electron systems based on our calculations of persistent currents and quantization of charge in ballistic single channel rings. The persistent current in an isolated impurity-free Luttinger liquid ring is found to be the same as for a system of non-interacting electrons. In contrast, a Luttinger liquid behavior is revealed if the ring is weakly connected to an electron reservoir and capacitively coupled to a gate electrode. The persistent current and the average charge in the ring are shown to change in a stepwise manner with the gate voltage at low temperatures. Both the step positions and the crossover temperatures are determined by the Luttinger liquid correlation parameter .alpha.(.alpha.c) which therefore can be measured. When a potential barrier is incorporated into the ring, the presence of strong interactions between electrons leads to a suppression of the persistent current and a change in its temperature dependence. A sharp maximum of the current is predicted to occur as a function of temperature. When two barriers are introduced into the ring, the possibility of resonant tunneling for the persistent current opens up. Two kinds of resonant behavior are discovered in the closed geometry: a current independent of the barrier transparency and a current analogous to the one with a single potential barrier. The persistent current in a ring containing a gas of fractionally charged excitations obeying fractional exclusion statistics is also studied. At finite temperatures, anomalous oscillations of the current are predicted as a manifestation of the statistics of the quasiparticles.

•Jan 1, 1998•0 citations

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Strongly correlated systems show a variety of intriguing properties and have been extensively studied over the years. In this thesis two strongly correlated systems are discussed: the two-dimensional antiferromagnet and the Luttinger liquid. The Luttinger liquid has attracted much interest as a paradigm of non-Fermi liquid behavior. In the first part of the thesis, basic properties of Fermi liquids are briefly described while those of Luttinger liquids are considered in more detail. The differences in spectral properties of Luttinger liquids and Fermi liquids are discussed, as are the consequences for photoemission experiments. In this context Luttinger liquids with boundaries, treated in Papers I, II, and III, are introduced. The exact single-electron Green's function at finite temperature is calculated with bosonization methods, for both periodic and finite open-boundary systems. The corresponding local spectral density is analyzed. For a system with open boundaries, the well-known zero temperature bulk behavior always crosses over to a boundary dominated regime for small energies, while at finite temperature the low energy behavior becomes quadratic. For a periodic finite-size system we also calculate the Coulomb-blockade-like oscillations and study their dependence on interaction and temperature. The two-dimensional antiferromagnet has attracted much attention due to its connection to the undoped phase of the ceramic high-temperature superconductors. In the second part of the thesis including Papers IV and V, the two-dimensional Heisenberg antiferromagnet defined on the triangular and the honeycomb lattices is studied. We use the Schwinger-boson mean-field theory to investigate the stability of the ground state with respect to frustration. For the honeycomb lattice model we find an increased Néel stability as compared to the classical case. The dynamic structure factor is considered for various temperatures and the spin-wave velocity and the susceptibility are presented as functions of frustration for spin S=1/2. In the case of the intrinsically frustrated triangular lattice model we obtain, in the limit of infinite spin, results in complete agreement with the linear spin-wave theory. In contrast to the latter, however, two of the three modes acquire a mass for finite spin. The origin of this effect is discussed.

•Jan 1, 1997•0 citations

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This thesis concerns correlation effects in quantum many-particle systems in one and two dimensions. Such systems show many exotic non-Fermi liquid phenomena, which can be treated analytically using non-perturbative field-theory methods.\nQuantum phase transitions between topologically ordered phases of matter, which do not break any symmetries, are studied. It is shown that although there is no local order parameter, a local measure from quantum information theory called reduced fidelity can detect such transitions.\nEntanglement in quantum impurity systems is also studied. The general expression for scaling corrections in entanglement entropy from boundary perturbations is derived within conformal field theory, showing that the asymptotic decay of Kondo screening clouds follow the same power-law as the impurity specific heat.\nFurthermore, the effects from spin-orbit interactions on Kondo physics in helical Luttinger liquids are studied. Such helical liquids occur on the edges of two-dimensional topological insulators. It is shown that Rashba and Dresselhaus interactions can potentially destroy Kondo singlet formation in such a system, and that the coupling to an electric field gives a mechanism to control transport properties.\nThe most recent work focuses on correlations in interacting one-dimensional Bose gases. The asymptotic expression for correlation functions in a generalized Gibbs ensemble, where all the local conservation laws appear, is obtained from Bethe Ansatz and conformal field theory.\nIn addition to the research papers with the above results, the thesis also contains an introductory text reviewing background material.

•May 8, 2013•0 citations

Luttinger-liquid physics in wire and dot geometries

Hans Peter Wächter

In a first part, I study the electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems resembling setups realized in experiments. In particular, I compute the temperature dependence of the linear conductance of a system without bulk impurities. The appearance of new temperature scales introduced by the lengths of the overhanging parts of the leads and the wire implies a conductance function which is much more complex than the simple power-law behavior obtained in earlier approaches. My results can be used to optimize the experimental setups designed for a verification of Luttinger-liquid scaling. In a second part, I suggest a setup to study Luttinger-liquid behavior in quantum wires which allows to determine the Luttinger-liquid parameter from two independent measurements: transport through a quantum dot embedded in the wire and the charge on the dot. To this end, I identified novel Luttinger-liquid power-laws in the charging of the dot. In a more technical part, I extended an adaption of the functional renormalization group such that it can be used on correlated electrons in complex geometries like the wire-lead structures described above and arbitrary dot structures coupled to Luttinger-liquid leads.

•Jan 1, 2010•0 citations
In this Ph.D. thesis I present the background and providefurther details to the work presented in the appended papers. The background consists of brief overviews of Kondo physics, Luttinger liquid theory, and conformal field theory. My own research is divided into three parts: the first deals with the interplay between Luttinger and Kondo physics (described in Papers I and II), the secondaddresses scanning tunneling microscopy (STM) response of a Luttinger liquid with an edge or an impurity (described in Paper III), and the third explores the boundary Green's function of 1D electrons in the spin-incoherent regime (described in Paper IV). In Papers I and II a setup is proposed which allows for acontrolled study of Kondo and Luttinger liquid physics. Itconsists of a quantum box, biased by a gate voltage, and side-coupled to a quantum wire by a point contact. Close to the degeneracy points of the Coulomb blockaded box the setup can be described as a Luttinger liquid interacting with an effective Kondo impurity. Using boundary conformal field theory techniques we predict that for the case of spin-polarized electrons the differential capacitance of the box will exhibit distinctive Luttinger liquid scaling with temperature and gate voltage. In the limit of zero magnetic field the Luttinger liquid behavior gets masked by two-channel Kondo screening, leading to a logarithmic scaling of the differential capacitance with temperatureand gate voltage. These effects should be possible to studyexperimentally, using the recently developed single-electrontransistor (SET) electrometer technique.In Paper III the finite-temperature local spectral weight (LSW) of a Luttinger liquid with a hard wall boundary is calculated. Close to the boundary the LSW exhibits characteristic oscillations indicative of spin-charge separation. The line shape of the LSW is also found to have a Fano-like asymmetry, a feature originating from the combined effect of electron-electron interaction and scattering off the boundary. Our results can be used to predict how edges and impurities influencethe STM response of one-dimensional electron systemsat low temperatures and voltage bias. Applications to STM onsingle-walled carbon nanotubes are also discussed.Paper IV, finally, presents a study of the spin-incoherent regime of 1D strongly interacting electrons. For sufficiently low densities the potential energy dominates the kinetic energy, and one can easily reach thespin-incoherent regime where the spin exchange energy is much less than the temperature. We have generalized the description of the spin-incoherent regime to account for the presence of a boundary. By calculating the exact Green's function we find that the charge sector critical exponent is highly sensitive to the boundary, strongly modifying the tunneling of electrons close to it. Our approach also allows for a detailed description of the crossover between boundary and bulk regimes.
In this thesis I present a theoretical study of three different\nstrongly correlated systems: Luttinger liquids, spin-1 chains and\nthe two-dimensional spin-1/2 Heisenberg model.\n\nFor every sub-field of our research I summarize the background and\nprovide further details to the work presented in the appended\npapers. Chapter 1 corresponds to paper I, Chapter 2 to paper II,\nChapter 3 and 4 to paper III and IV.\n\nIn paper I, we calculate with bosonization techniques the local\ndensity of states of a finite segment of Luttinger liquid. Besides of\nbeing of experimental relevance for Scanning Tunneling Microscopy\nexperiments on carbon nanotubes and cleaved overgrowth wires, the\nwave functions of a "Luttinger liquid in a box" very nicely\nvisualize the fundamental bosonic nature of the low energy\nexcitations.\n\nIn paper II, we analyze with field theory methods how the presence\nof a single-ion anisotropy affects the behavior of a spin-1 chain.\nWe determine in the framework of the Non Linear Sigma Model how the gaps\nof the anisotropic chain depend on an applied staggered magnetic\nfield. Our theoretical analysis is meant to model compounds like\nrmR2BaNiO5\\rm R_2BaNiO_5rmR2​BaNiO5​ that show coexistence of Haldane phase and\nlong-range order.\n\nIn paper III, we compute with quantum Monte Carlo simulations the\nlocal response to a uniform magnetic field around vacancies in the\ntwo-dimensional Heisenberg model. From the full understanding of the\nnumerical results, we can make quantitative predictions of relevance\nfor Nuclear Magnetic Resonance and susceptibility experiments. In\nparticular, we expect, even in the thermodynamic limit, a finite\nCurie contribution to the total susceptibility arising from the\nimpurities.\n\nFinally in paper IV, we consider interaction effects among\nimpurities. We compute the potential between static vacancies in\nseveral low-dimensional spin-1/2 antiferromagnets comparing\nthe numerics with exact results from conformal field theory and\nlinear spin wave theory. The potential is directly related to the\nlocal valence bond order and its magnitude is a loyal measure of the\namount of quantum correlations present into the system.
We report on exact results for the low-temperature thermodynamics of a spin- 12\frac{1}{2}21​ magnetic impurity coupled to a one-dimensional interacting electron system. By using boundary conformal field theory, we show that there are only two types of critical behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a theory with an anomalous response identical to that recently proposed by Furusaki and Nagaosa. Suppression of backscattering off the impurity leads to the same critical properties as for the two-channel Kondo effect.
Theoretical studies suggest that an interacting one-dimensional electron system does not behave as a Fermi liquid. Instead, it is predicted to form a Luttinger liquid characterized e.g. by the absence of fermionic quasiparticles and interaction-dependent power-laws for transport quantities. So far, however, only a limited number of experiments have provided evidence for Luttinger liquid behavior. In this thesis we consider alternative methods to detect electronic correlations in one-dimensional electron systems based on our calculations of persistent currents and quantization of charge in ballistic single channel rings. The persistent current in an isolated impurity-free Luttinger liquid ring is found to be the same as for a system of non-interacting electrons. In contrast, a Luttinger liquid behavior is revealed if the ring is weakly connected to an electron reservoir and capacitively coupled to a gate electrode. The persistent current and the average charge in the ring are shown to change in a stepwise manner with the gate voltage at low temperatures. Both the step positions and the crossover temperatures are determined by the Luttinger liquid correlation parameter .alpha.(.alpha.c) which therefore can be measured. When a potential barrier is incorporated into the ring, the presence of strong interactions between electrons leads to a suppression of the persistent current and a change in its temperature dependence. A sharp maximum of the current is predicted to occur as a function of temperature. When two barriers are introduced into the ring, the possibility of resonant tunneling for the persistent current opens up. Two kinds of resonant behavior are discovered in the closed geometry: a current independent of the barrier transparency and a current analogous to the one with a single potential barrier. The persistent current in a ring containing a gas of fractionally charged excitations obeying fractional exclusion statistics is also studied. At finite temperatures, anomalous oscillations of the current are predicted as a manifestation of the statistics of the quasiparticles.
Strongly correlated systems show a variety of intriguing properties and have been extensively studied over the years. In this thesis two strongly correlated systems are discussed: the two-dimensional antiferromagnet and the Luttinger liquid. The Luttinger liquid has attracted much interest as a paradigm of non-Fermi liquid behavior. In the first part of the thesis, basic properties of Fermi liquids are briefly described while those of Luttinger liquids are considered in more detail. The differences in spectral properties of Luttinger liquids and Fermi liquids are discussed, as are the consequences for photoemission experiments. In this context Luttinger liquids with boundaries, treated in Papers I, II, and III, are introduced. The exact single-electron Green's function at finite temperature is calculated with bosonization methods, for both periodic and finite open-boundary systems. The corresponding local spectral density is analyzed. For a system with open boundaries, the well-known zero temperature bulk behavior always crosses over to a boundary dominated regime for small energies, while at finite temperature the low energy behavior becomes quadratic. For a periodic finite-size system we also calculate the Coulomb-blockade-like oscillations and study their dependence on interaction and temperature. The two-dimensional antiferromagnet has attracted much attention due to its connection to the undoped phase of the ceramic high-temperature superconductors. In the second part of the thesis including Papers IV and V, the two-dimensional Heisenberg antiferromagnet defined on the triangular and the honeycomb lattices is studied. We use the Schwinger-boson mean-field theory to investigate the stability of the ground state with respect to frustration. For the honeycomb lattice model we find an increased Néel stability as compared to the classical case. The dynamic structure factor is considered for various temperatures and the spin-wave velocity and the susceptibility are presented as functions of frustration for spin S=1/2. In the case of the intrinsically frustrated triangular lattice model we obtain, in the limit of infinite spin, results in complete agreement with the linear spin-wave theory. In contrast to the latter, however, two of the three modes acquire a mass for finite spin. The origin of this effect is discussed.
This thesis concerns correlation effects in quantum many-particle systems in one and two dimensions. Such systems show many exotic non-Fermi liquid phenomena, which can be treated analytically using non-perturbative field-theory methods.\nQuantum phase transitions between topologically ordered phases of matter, which do not break any symmetries, are studied. It is shown that although there is no local order parameter, a local measure from quantum information theory called reduced fidelity can detect such transitions.\nEntanglement in quantum impurity systems is also studied. The general expression for scaling corrections in entanglement entropy from boundary perturbations is derived within conformal field theory, showing that the asymptotic decay of Kondo screening clouds follow the same power-law as the impurity specific heat.\nFurthermore, the effects from spin-orbit interactions on Kondo physics in helical Luttinger liquids are studied. Such helical liquids occur on the edges of two-dimensional topological insulators. It is shown that Rashba and Dresselhaus interactions can potentially destroy Kondo singlet formation in such a system, and that the coupling to an electric field gives a mechanism to control transport properties.\nThe most recent work focuses on correlations in interacting one-dimensional Bose gases. The asymptotic expression for correlation functions in a generalized Gibbs ensemble, where all the local conservation laws appear, is obtained from Bethe Ansatz and conformal field theory.\nIn addition to the research papers with the above results, the thesis also contains an introductory text reviewing background material.
In a first part, I study the electronic transport through a one-dimensional, finite-length quantum wire of correlated electrons (Luttinger liquid) coupled at arbitrary position via tunnel barriers to two semi-infinite, one-dimensional as well as stripe-like (two-dimensional) leads, thereby bringing theory closer towards systems resembling setups realized in experiments. In particular, I compute the temperature dependence of the linear conductance of a system without bulk impurities. The appearance of new temperature scales introduced by the lengths of the overhanging parts of the leads and the wire implies a conductance function which is much more complex than the simple power-law behavior obtained in earlier approaches. My results can be used to optimize the experimental setups designed for a verification of Luttinger-liquid scaling. In a second part, I suggest a setup to study Luttinger-liquid behavior in quantum wires which allows to determine the Luttinger-liquid parameter from two independent measurements: transport through a quantum dot embedded in the wire and the charge on the dot. To this end, I identified novel Luttinger-liquid power-laws in the charging of the dot. In a more technical part, I extended an adaption of the functional renormalization group such that it can be used on correlated electrons in complex geometries like the wire-lead structures described above and arbitrary dot structures coupled to Luttinger-liquid leads.