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:cond-mat/0306529 · Physical Review B

Solvable regime of disorder and interactions in ballistic nanostructures: Consequences for Coulomb blockade

Ganpathy N. Murthy, R. Shankar, Damir Herman, Harsh Mathur

Published Feb 27, 2004• 19 citations
arXivHTMLPDF

Abstract

We provide a framework for analyzing the problem of interacting electrons in a ballistic quantum dot with chaotic boundary conditions within an energy ${E}_{T}$ (the Thouless energy) of the Fermi energy. Within this window we show that the interactions can be characterized by Landau Fermi-liquid parameters. When g, the Thouless number (also the dimensionless conductance when it has strong coupling to leads) of the dot, is large, we find that the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as $\stackrel{\ensuremath{\rightarrow}}{g}\ensuremath{\infty}$ (as in a $\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\ensuremath{-}N$ theory). The infinite g theory has two phases as a function of the Landau parameter ${u}_{m}$ in a channel with angular momentum m: A weak-coupling phase where constant charging and exchange interactions dominate the low-energy physics, as in previous ``universal Hamiltonian'' treatments, and a strong-coupling phase characterized by the same order parameter as in the Pomeranchuk transition in clean systems (a spontaneous interaction-induced Fermi-surface distortion), but smeared and pinned by disorder. Thus, both interactions and disorder are crucial to the existence of these phases. At finite g, the two phases and critical point evolve into three regimes in the ${u}_{m}\ensuremath{-}1/g$ plane---weak- and strong-coupling regimes separated by crossover lines from a quantum-critical regime controlled by the quantum-critical point. In this, the first of a two part series, we focus on the consequences of this picture for Coulomb Blockade experiments. We employ analytical and numerical methods to predict the statistics of single-particle levels, Coulomb Blockade peak spacings, conductance peak heights, and quasiparticle widths. We show that in the strong-coupling and quantum-critical regions, the quasiparticle acquires a width of the same order as the level spacing $\ensuremath{\Delta}$ within a few $\ensuremath{\Delta}\mathrm{'}\mathrm{s}$ of the Fermi energy due to coupling to collective excitations. In the strong-coupling regime if m is odd, the dot will (if isolated) crossover from the orthogonal to unitary ensemble for an exponentially small external flux or will (if strongly coupled to leads) break time-reversal symmetry spontaneously. For any m, the peak-spacing distribution becomes broader than expected in previous works and even has support at negative values, which in turn is correlated with small peak heights. Ballistic/chaotic quantum dots afford us unrivalled theoretical and experimental control over the problem of simultaneous disorder and interactions due to the $1/g$ expansion and our ability to vary disorder and interaction much more readily than in the bulk.

DOI
10.1103/physrevb.69.075321

Related papers

Random-matrix and quantum-critical crossovers for interacting electrons in quantum dots

Ganpathy N. Murthy

Quantum dots with a large Thouless number $g$ embody a regime where both disorder and interactions can be treated nonperturbatively using large-$N$ techniques (with $N=g$) and quantum phase transitions can be studied. Here we focus on dots where the noninteracting Hamiltonian is drawn from a crossover ensemble between two symmetry classes, where the crossover parameter introduces a tunable energy scale independent of and much smaller than the Thouless energy. We show that the quantum critical regime, dominated by collective critical fluctuations, can be accessed at this energy scale. The nonperturbative physics of this regime can only be described by the large-$N$ approach, as we illustrate with two experimentally relevant examples.

arXiv:cond-mat/0406029•Physical Review B•Oct 15, 2004•10 citations

Correlated electron physics in two-level quantum dots: Phase transitions, transport, and experiment

David E. Logan, Christopher J. Wright, Martin R. Galpin

We study correlated two-level quantum dots, coupled in effective one-channel fashion to metallic leads, with electron interactions including on-level and interlevel Coulomb repulsions, as well as the interorbital Hund's rule exchange favoring the spin-1 state in the relevant sector of the free dot. For arbitrary dot occupancy, the underlying phases, quantum phase transitions (QPTs), thermodynamics, single-particle dynamics, and electronic transport properties are considered, and direct comparison is made to conductance experiments on lateral quantum dots. Two distinct phases arise generically, one characterized by a normal Fermi liquid fixed point (FP) and the other by an underscreened (USC) spin-1 FP. Associated QPTs, which occur in general in a mixed valent regime of nonintegral dot charge, are found to consist of continuous lines of Kosterlitz-Thouless transitions, separated by first-order level-crossing transitions at high symmetry points. A ``Friedel-Luttinger sum rule'' is derived and, together with a deduced generalization of Luttinger's theorem to the USC phase (a singular Fermi liquid), is used to obtain a general result for the $T=0$ zero-bias conductance, expressed solely in terms of the dot occupancy and applicable to both phases. Relatedly, dynamical signatures of the QPT show two broad classes of behavior, corresponding to the collapse of either a Kondo resonance, or antiresonance, as the transition is approached from the Fermi liquid phase, the latter behavior being apparent in experimental differential conductance maps. The problem is studied using the numerical renormalization group method, combined with analytical arguments.

arXiv:0906.3169•Physical Review B•Sep 18, 2009•56 citations

Quantum Dots: Coulomb Blockade, Mesoscopic Fluctuations, and Qubit Decoherence

Serguei Vorojtsov

In this thesis I find an analytic expression for the conductance of a single electron transistor in the regime when temperature, level spacing, and charging energy of an island are all of the same order. I also study the correction to the spacing between Coulomb blockade peaks due to finite dot-lead tunnel couplings. I find analytic expressions for both correction to the spacing averaged over mesoscopic fluctuations and the rms of the correction fluctuations. In the second part of the thesis I discuss the feasibility of quantum dot based spin- and charge-qubits. Firstly, I study the effect of mesoscopic fluctuations on the magnitude of errors that can occur in exchange operations on quantum dot spin-qubits. Mid-size double quantum dots, with an odd number of electrons in the range of a few tens in each dot, are investigated through the constant interaction model using realistic parameters. It is found that the number of independent parameters per dot that one should tune depends on the configuration and ranges from one to four. Then, I study decoherence of a quantum dot charge qubit due to coupling to piezoelectric acoustic phonons in the Born-Markov approximation. After including appropriate form factors, I find that phonon decoherence rates are one to two orders of magnitude weaker than was previously predicted. My results suggest that mechanisms other than phonon decoherence play a more significant role in current experimental setups.

arXiv:cond-mat/0604065•Apr 4, 2006•0 citations

Numerical Renormalization Group studies of Correlation effects in Phase Coherent Transport through Quantum Dots

T. Hecht

This thesis contributes to the field of transport through quantum dots. These devices allow for a controlled study of quantum transport and fundamental physical effects, like the Kondo effect. In this thesis we will focus on dots that are well described by generalized Anderson impurity models, where the discrete levels of the quantum dot are tunnel-coupled to fermionic reservoirs. The model parameters, like level energy and width, can be tuned in experiments. Therefore these systems constitute a valuable arena for testing experiment against theory and vice versa. In order to describe these strongly correlated systems, we employ the numerical renormalization group method. This allows us to address both longstanding questions concerning experimental results and new physical phenomena in these fundamental models. This thesis consists of three major projects. The first and most extensive one is concerned with the phase of the transmission amplitude through a quantum dot. Measurements of many-electron quantum dots with small level spacing reveal universal phase behaviour, a result not fully understood for almost 10 years. Recent experiments have seen that, contrarily, for dots with only a few electrons, i.e. large level spacing, the phase depends on the mesoscopic dot parameters. Analyzing a multi-level Anderson model, we show that the generic feature of the two regimes can be reproduced in the regime of overlapping levels or well separated levels, respectively. Thereby the universal character follows from Fano-type antiresonances of the renormalized single-particle levels. Moderate temperature supports the universal character. In the mesoscopic regime, we also investigate the effect of Kondo correlations on the transmission phase. In a second project we analyze a quantum dot coupled to a superconducting reservoir. In contrast to previous belief, the energy resolution of our method is not restricted by the energy scale of the superconducting gap, leading to new insights into the method. The high resolution allows us to resolve sharp peaks in the spectral function that emerge for a certain regime of parameters. A third project deals with a quantum dot coupled to two independent channels, a system known to exhibit non-Fermi liquid behaviour. We investigate the existence of the non-Fermi liquid regime when driving the system out of the Kondo regime by emptying the dot. We find that the extent of the non-Fermi liquid regime strongly depends on the mechanisms that couple impurity and reservoirs but prevent mixing of the latter.

•Jan 1, 2008•2 citations

Disordered mesoscopic systems with interactions: Induced two-body ensembles and the Hartree-Fock approach

Y. Alhassid, Hans A. Weidenmüller, André Wobst

We introduce a generic approach to study interaction effects in diffusive or chaotic quantum dots in the Coulomb blockade regime. The randomness of the single-particle wave functions induces randomness in the two-body interaction matrix elements. We classify the possible induced two-body ensembles, both in the presence and absence of spin degrees of freedom. The ensembles depend on the underlying space-time symmetries as well as on features of the two-body interaction. Confining ourselves to spinless electrons, we then use the Hartree-Fock (HF) approximation to calculate HF single-particle energies and HF wave functions for many realizations of the ensemble. We study the statistical properties of the resulting one-body HF ensemble for a fixed number of electrons. In particular, we determine the statistics of the interaction matrix elements in the HF basis, of the HF single-particle energies (including the HF gap between the last occupied and the first empty HF level), and of the HF single-particle wave functions. We also study the addition of electrons, and in particular the distribution of the distance between successive conductance peaks and of the conductance peak heights.

arXiv:cond-mat/0406495•Physical Review B•Jul 8, 2005•21 citations

Novel quantum phases, excitations, and transport in low dimensional systems

D. Makogon

The Chapter 2 is devoted to the study of the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several point-like or extended time-dependent impurities (barriers) and find regimes of conductance enhancement. In Chapte 3 we study a set of crossed 1D systems, which are coupled with each other via tunnelling at the crossings. Concentrating on the case of a single crossing, we were able to explain recent experiments on crossed metallic and semiconducting nanotubes, which showed the presence of localized states in the region of crossing. The Chapter 4 treats the effects of disorder and interactions in a quantum Hall ferromagnet. We study the problem by projecting the original fermionic Hamiltonian into magnon states, which behave as bosons in the vicinity of the ferromagnetic ground state. The approach permits the reformulation of a strongly interacting model into a non-interacting one. The latter is a non-perturbative scheme that consists in treating the two-particle neutral excitations of the electron system as a bosonic single-particle. Indeed, the employment of bosonization facilitates the inclusion of disorder in the study of the system. It has been shown previously that disorder may drive a quantum phase transition in the Hall ferromagnet. However, such studies have been either carried out in the framework of the nonlinear sigma model, as an effective low-energy theory, or included the long-range Coulomb interaction in a quantum description only up to the Hartree-Fock level. Here, we establish the occurrence of a disorder-driven quantum phase transition from a ferromagnetic 2DEG to a spin glass phase by taking into account interactions between electrons up to the random phase approximation level in a fully quantum description. In the Chapter 5 we derive and discuss an experimentally realistic model describing ultracold atoms in an optical lattice including a commensurate, but staggered, Zeeman field. The resulting band structure is quite exotic; fermions in the third band have an unusual rounded picture-frame Fermi surface (essentially two concentric squircles), leading to imperfect nesting. We develop a generalized theory describing the spin and charge degrees of freedom simultaneously, and show that the system can develop a coupled spin-charge-density wave order. This ordering is absent in studies of the Hubbard model that treat spin and charge density separately.

•Aug 1, 2006•0 citations
We provide a framework for analyzing the problem of interacting electrons in a ballistic quantum dot with chaotic boundary conditions within an energy ET{E}_{T}ET​ (the Thouless energy) of the Fermi energy. Within this window we show that the interactions can be characterized by Landau Fermi-liquid parameters. When g, the Thouless number (also the dimensionless conductance when it has strong coupling to leads) of the dot, is large, we find that the disordered interacting problem can be solved in a saddle-point approximation which becomes exact as g\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{g}\ensuremath{\infty}g\ensuremath→​\ensuremath∞ (as in a large\ensuremath−N\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\ensuremath{-}Nlarge\ensuremath−N theory). The infinite g theory has two phases as a function of the Landau parameter um{u}_{m}um​ in a channel with angular momentum m: A weak-coupling phase where constant charging and exchange interactions dominate the low-energy physics, as in previous ``universal Hamiltonian'' treatments, and a strong-coupling phase characterized by the same order parameter as in the Pomeranchuk transition in clean systems (a spontaneous interaction-induced Fermi-surface distortion), but smeared and pinned by disorder. Thus, both interactions and disorder are crucial to the existence of these phases. At finite g, the two phases and critical point evolve into three regimes in the um\ensuremath−1/g{u}_{m}\ensuremath{-}1/gum​\ensuremath−1/g plane---weak- and strong-coupling regimes separated by crossover lines from a quantum-critical regime controlled by the quantum-critical point. In this, the first of a two part series, we focus on the consequences of this picture for Coulomb Blockade experiments. We employ analytical and numerical methods to predict the statistics of single-particle levels, Coulomb Blockade peak spacings, conductance peak heights, and quasiparticle widths. We show that in the strong-coupling and quantum-critical regions, the quasiparticle acquires a width of the same order as the level spacing \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ within a few \ensuremathΔ′s\ensuremath{\Delta}\mathrm{'}\mathrm{s}\ensuremathΔ′s of the Fermi energy due to coupling to collective excitations. In the strong-coupling regime if m is odd, the dot will (if isolated) crossover from the orthogonal to unitary ensemble for an exponentially small external flux or will (if strongly coupled to leads) break time-reversal symmetry spontaneously. For any m, the peak-spacing distribution becomes broader than expected in previous works and even has support at negative values, which in turn is correlated with small peak heights. Ballistic/chaotic quantum dots afford us unrivalled theoretical and experimental control over the problem of simultaneous disorder and interactions due to the 1/g1/g1/g expansion and our ability to vary disorder and interaction much more readily than in the bulk.
Quantum dots with a large Thouless number ggg embody a regime where both disorder and interactions can be treated nonperturbatively using large-NNN techniques (with N=gN=gN=g) and quantum phase transitions can be studied. Here we focus on dots where the noninteracting Hamiltonian is drawn from a crossover ensemble between two symmetry classes, where the crossover parameter introduces a tunable energy scale independent of and much smaller than the Thouless energy. We show that the quantum critical regime, dominated by collective critical fluctuations, can be accessed at this energy scale. The nonperturbative physics of this regime can only be described by the large-NNN approach, as we illustrate with two experimentally relevant examples.
We study correlated two-level quantum dots, coupled in effective one-channel fashion to metallic leads, with electron interactions including on-level and interlevel Coulomb repulsions, as well as the interorbital Hund's rule exchange favoring the spin-1 state in the relevant sector of the free dot. For arbitrary dot occupancy, the underlying phases, quantum phase transitions (QPTs), thermodynamics, single-particle dynamics, and electronic transport properties are considered, and direct comparison is made to conductance experiments on lateral quantum dots. Two distinct phases arise generically, one characterized by a normal Fermi liquid fixed point (FP) and the other by an underscreened (USC) spin-1 FP. Associated QPTs, which occur in general in a mixed valent regime of nonintegral dot charge, are found to consist of continuous lines of Kosterlitz-Thouless transitions, separated by first-order level-crossing transitions at high symmetry points. A ``Friedel-Luttinger sum rule'' is derived and, together with a deduced generalization of Luttinger's theorem to the USC phase (a singular Fermi liquid), is used to obtain a general result for the T=0T=0T=0 zero-bias conductance, expressed solely in terms of the dot occupancy and applicable to both phases. Relatedly, dynamical signatures of the QPT show two broad classes of behavior, corresponding to the collapse of either a Kondo resonance, or antiresonance, as the transition is approached from the Fermi liquid phase, the latter behavior being apparent in experimental differential conductance maps. The problem is studied using the numerical renormalization group method, combined with analytical arguments.
In this thesis I find an analytic expression for the conductance of a single electron transistor in the regime when temperature, level spacing, and charging energy of an island are all of the same order. I also study the correction to the spacing between Coulomb blockade peaks due to finite dot-lead tunnel couplings. I find analytic expressions for both correction to the spacing averaged over mesoscopic fluctuations and the rms of the correction fluctuations. In the second part of the thesis I discuss the feasibility of quantum dot based spin- and charge-qubits. Firstly, I study the effect of mesoscopic fluctuations on the magnitude of errors that can occur in exchange operations on quantum dot spin-qubits. Mid-size double quantum dots, with an odd number of electrons in the range of a few tens in each dot, are investigated through the constant interaction model using realistic parameters. It is found that the number of independent parameters per dot that one should tune depends on the configuration and ranges from one to four. Then, I study decoherence of a quantum dot charge qubit due to coupling to piezoelectric acoustic phonons in the Born-Markov approximation. After including appropriate form factors, I find that phonon decoherence rates are one to two orders of magnitude weaker than was previously predicted. My results suggest that mechanisms other than phonon decoherence play a more significant role in current experimental setups.
This thesis contributes to the field of transport through quantum dots. These devices allow for a controlled study of quantum transport and fundamental physical effects, like the Kondo effect. In this thesis we will focus on dots that are well described by generalized Anderson impurity models, where the discrete levels of the quantum dot are tunnel-coupled to fermionic reservoirs. The model parameters, like level energy and width, can be tuned in experiments. Therefore these systems constitute a valuable arena for testing experiment against theory and vice versa. In order to describe these strongly correlated systems, we employ the numerical renormalization group method. This allows us to address both longstanding questions concerning experimental results and new physical phenomena in these fundamental models. This thesis consists of three major projects. The first and most extensive one is concerned with the phase of the transmission amplitude through a quantum dot. Measurements of many-electron quantum dots with small level spacing reveal universal phase behaviour, a result not fully understood for almost 10 years. Recent experiments have seen that, contrarily, for dots with only a few electrons, i.e. large level spacing, the phase depends on the mesoscopic dot parameters. Analyzing a multi-level Anderson model, we show that the generic feature of the two regimes can be reproduced in the regime of overlapping levels or well separated levels, respectively. Thereby the universal character follows from Fano-type antiresonances of the renormalized single-particle levels. Moderate temperature supports the universal character. In the mesoscopic regime, we also investigate the effect of Kondo correlations on the transmission phase. In a second project we analyze a quantum dot coupled to a superconducting reservoir. In contrast to previous belief, the energy resolution of our method is not restricted by the energy scale of the superconducting gap, leading to new insights into the method. The high resolution allows us to resolve sharp peaks in the spectral function that emerge for a certain regime of parameters. A third project deals with a quantum dot coupled to two independent channels, a system known to exhibit non-Fermi liquid behaviour. We investigate the existence of the non-Fermi liquid regime when driving the system out of the Kondo regime by emptying the dot. We find that the extent of the non-Fermi liquid regime strongly depends on the mechanisms that couple impurity and reservoirs but prevent mixing of the latter.
We introduce a generic approach to study interaction effects in diffusive or chaotic quantum dots in the Coulomb blockade regime. The randomness of the single-particle wave functions induces randomness in the two-body interaction matrix elements. We classify the possible induced two-body ensembles, both in the presence and absence of spin degrees of freedom. The ensembles depend on the underlying space-time symmetries as well as on features of the two-body interaction. Confining ourselves to spinless electrons, we then use the Hartree-Fock (HF) approximation to calculate HF single-particle energies and HF wave functions for many realizations of the ensemble. We study the statistical properties of the resulting one-body HF ensemble for a fixed number of electrons. In particular, we determine the statistics of the interaction matrix elements in the HF basis, of the HF single-particle energies (including the HF gap between the last occupied and the first empty HF level), and of the HF single-particle wave functions. We also study the addition of electrons, and in particular the distribution of the distance between successive conductance peaks and of the conductance peak heights.
The Chapter 2 is devoted to the study of the transport properties of a quantum wire, described by the Tomonaga-Luttinger model, in the presence of a backscattering potential provided by several point-like or extended time-dependent impurities (barriers) and find regimes of conductance enhancement. In Chapte 3 we study a set of crossed 1D systems, which are coupled with each other via tunnelling at the crossings. Concentrating on the case of a single crossing, we were able to explain recent experiments on crossed metallic and semiconducting nanotubes, which showed the presence of localized states in the region of crossing. The Chapter 4 treats the effects of disorder and interactions in a quantum Hall ferromagnet. We study the problem by projecting the original fermionic Hamiltonian into magnon states, which behave as bosons in the vicinity of the ferromagnetic ground state. The approach permits the reformulation of a strongly interacting model into a non-interacting one. The latter is a non-perturbative scheme that consists in treating the two-particle neutral excitations of the electron system as a bosonic single-particle. Indeed, the employment of bosonization facilitates the inclusion of disorder in the study of the system. It has been shown previously that disorder may drive a quantum phase transition in the Hall ferromagnet. However, such studies have been either carried out in the framework of the nonlinear sigma model, as an effective low-energy theory, or included the long-range Coulomb interaction in a quantum description only up to the Hartree-Fock level. Here, we establish the occurrence of a disorder-driven quantum phase transition from a ferromagnetic 2DEG to a spin glass phase by taking into account interactions between electrons up to the random phase approximation level in a fully quantum description. In the Chapter 5 we derive and discuss an experimentally realistic model describing ultracold atoms in an optical lattice including a commensurate, but staggered, Zeeman field. The resulting band structure is quite exotic; fermions in the third band have an unusual rounded picture-frame Fermi surface (essentially two concentric squircles), leading to imperfect nesting. We develop a generalized theory describing the spin and charge degrees of freedom simultaneously, and show that the system can develop a coupled spin-charge-density wave order. This ordering is absent in studies of the Hubbard model that treat spin and charge density separately.