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Interacting electrons on a half line coupled to impurities

Ana Lopez, Manuel Fuentes, Eduardo Fradkin

Published Jan 1, 1997• 0 citations
OpenAlex

Abstract

We generalize the bosonization methods for systems in the half line that we discussed elsewhere, to study the effects of interactions on electronic systems coupled to impurities. We introduce a model for a quantum wire coupled with a quantum dot by a purely capacitive interaction. We show that the effect on the quantum wire is a dynamical boundary condition which reflects the charge fluctuations of the quantum dot. Special attention is given to the role of the fermion boundary conditions and their interplay with the electron interactions on the bosonized effective theory.

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We present a study of the correlated transport regimes of a double quantum impurity system with mutual capacitive interactions. Such system can be implemented by a double quantum dot arrangement or by a quantum dot and nearby quantum point contact, with independently connected sets of metallic terminals. Many-body spin correlations arising within each dot-lead subsystem give rise to the Kondo effect under appropriate conditions. The otherwise independent Kondo ground states may be modified by the capacitive coupling, decisively modifying the ground state of the double quantum impurity system. We analyze this coupled system through variational methods and the numerical renormalization group technique. Our results reveal a strong dependence of the coupled system ground state on the electron-hole asymmetries of the individual subsystems, as well as on their hybridization strengths to the respective reservoirs. The electrostatic repulsion produced by the capacitive coupling produces an effective shift of the individual energy levels toward higher energies, with a stronger effect on the ``shallower'' subsystem (that closer to resonance with the Fermi level), potentially pushing it out of the Kondo regime and dramatically changing the transport properties of the system. The effective remote gating that this entails is found to depend nonlinearly on the capacitive coupling strength, as well as on the independent subsystem levels. The analysis we present here of this mutual interaction should be important to fully characterize transport through such coupled systems.

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Characterizing Quantum Matter: impurities, dots and lattices

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Quantum impurity models are a fundamental paradigm in the study of many-body quantum matter. They describe metals with actual magnetic impurities, quantum circuits with quantum dots, and are the local effective theories within dynamical mean field theory for correlated materials. They have gained prominence in recent years as a platform for treating non-Markovian open quantum systems exactly in strong coupling, and as a means for accurate simulations of quantum nanoelectronics devices. The work presented in this thesis focuses on the characterization of quantum matter through this lens. Two recurring themes are characterizing systems by their entropy signatures, and through their quench dynamics. Firstly, we investigate the non-equilibrium physics of an interacting quantum dot circuit following a sudden, externally-driven change in the dot potential. We do this by mapping the interacting problem onto a simpler Gaussian theory, which we show is exact for single-particle dynamics in equilibrium. Perhaps surprisingly, we establish that such a mapping is still possible following a quench; although this comes at the cost of numerical optimization of the effective non-interacting auxiliary fermion problem, starting from a tensor network treatment. The method provides a new way of viewing impurity quench dynamics. Secondly, we focus on entropy measurements in quantum dot circuits using a charge sensor. The equilibrium thermodynamic entropy can be extracted from such charge measurements by local Maxwell relations, and we study this connection in detail for a variety of experimentally-relevant quantum nanoelectronic device models, uncovering their entropic phase diagram. The underlying quantum impurity models are solved exactly using numerical renormalization group methods. We extend this to lattice models describing interacting materials, showing that the Maxwell relations provide a route to calculate the lattice entropy and give new physical insights. We go on to consider entropy production in the non-equilibrium setting of a quench, again showing that this can be related to practical charge measurements. In quantum circuit experiments, the dot charge is measured in practice using a secondary, capacitively coupling charge-sensor circuit. We investigate this by explicitly modelling the device together with the charge-sensor in a fully quantum mechanical treatment. This allows us to understand and quantify the measurement backaction. It also allows us to establish when the assumptions underpinning the Maxwell entropy extraction hold. Finally, we consider the strongly correlated electron physics of bilayer Hubbard models within dynamical mean field theory. Such models capture the essence of interface physics in heterostructures. In the basic bilayer, we revisit the phase diagram and study the change in character of the metal-insulator transition from continuous band-insulator gap opening to Mott quasiparticle collapse, as a function of interlayer coupling and local Coulomb interaction strength. For more complex interfaces involving additional correlated sites between layers, many-body quantum interference can be tuned, leading to different types of transition. While a pure impurity model version of this system features a second-order non-Fermi liquid Kondo critical point, full self-consistency on the lattice turns the transition first-order.

•Jan 1, 2026•0 citations

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

Weiss, Yuval

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.

arXiv:0801.3603•Jan 23, 2008•0 citations
We generalize the bosonization methods for systems in the half line that we discussed elsewhere, to study the effects of interactions on electronic systems coupled to impurities. We introduce a model for a quantum wire coupled with a quantum dot by a purely capacitive interaction. We show that the effect on the quantum wire is a dynamical boundary condition which reflects the charge fluctuations of the quantum dot. Special attention is given to the role of the fermion boundary conditions and their interplay with the electron interactions on the bosonized effective theory.
In this thesis we derive a set of bosonization rules for the problem of a fermionic interacting system living on the half line coupled to a quantum degree of freedom at a boundary. This is an appropriate model to describe the physics of an electronic system in quantum wires with impurities. First we derive the bosonization rules for free fermions on a half-line with physically sensible boundary conditions for Luttinger fermions. We use path-integral methods to calculate the bosonized fermionic currents on the half-line and derive their commutation relations for a system with a boundary. We compute the fermion determinant of the fermionic fluctuations for a system with a boundary using Forman's approach. We find that the degrees of freedom induced at the boundary do not modify the commutation relations of the bulk. We give an explicit derivation of the bosonization rules for the fermion operators for a system with boundaries. We derive a set of bosonization rules for the Fermi operators which include the explicit effect of the boundaries and of boundary degrees of freedom. As a byproduct, we calculate the one-particle Green's function and determine the effects of the boundaries on its analytic structure.
We present a study of the correlated transport regimes of a double quantum impurity system with mutual capacitive interactions. Such system can be implemented by a double quantum dot arrangement or by a quantum dot and nearby quantum point contact, with independently connected sets of metallic terminals. Many-body spin correlations arising within each dot-lead subsystem give rise to the Kondo effect under appropriate conditions. The otherwise independent Kondo ground states may be modified by the capacitive coupling, decisively modifying the ground state of the double quantum impurity system. We analyze this coupled system through variational methods and the numerical renormalization group technique. Our results reveal a strong dependence of the coupled system ground state on the electron-hole asymmetries of the individual subsystems, as well as on their hybridization strengths to the respective reservoirs. The electrostatic repulsion produced by the capacitive coupling produces an effective shift of the individual energy levels toward higher energies, with a stronger effect on the ``shallower'' subsystem (that closer to resonance with the Fermi level), potentially pushing it out of the Kondo regime and dramatically changing the transport properties of the system. The effective remote gating that this entails is found to depend nonlinearly on the capacitive coupling strength, as well as on the independent subsystem levels. The analysis we present here of this mutual interaction should be important to fully characterize transport through such coupled systems.
Quantum impurity models are a fundamental paradigm in the study of many-body quantum matter. They describe metals with actual magnetic impurities, quantum circuits with quantum dots, and are the local effective theories within dynamical mean field theory for correlated materials. They have gained prominence in recent years as a platform for treating non-Markovian open quantum systems exactly in strong coupling, and as a means for accurate simulations of quantum nanoelectronics devices. The work presented in this thesis focuses on the characterization of quantum matter through this lens. Two recurring themes are characterizing systems by their entropy signatures, and through their quench dynamics. Firstly, we investigate the non-equilibrium physics of an interacting quantum dot circuit following a sudden, externally-driven change in the dot potential. We do this by mapping the interacting problem onto a simpler Gaussian theory, which we show is exact for single-particle dynamics in equilibrium. Perhaps surprisingly, we establish that such a mapping is still possible following a quench; although this comes at the cost of numerical optimization of the effective non-interacting auxiliary fermion problem, starting from a tensor network treatment. The method provides a new way of viewing impurity quench dynamics. Secondly, we focus on entropy measurements in quantum dot circuits using a charge sensor. The equilibrium thermodynamic entropy can be extracted from such charge measurements by local Maxwell relations, and we study this connection in detail for a variety of experimentally-relevant quantum nanoelectronic device models, uncovering their entropic phase diagram. The underlying quantum impurity models are solved exactly using numerical renormalization group methods. We extend this to lattice models describing interacting materials, showing that the Maxwell relations provide a route to calculate the lattice entropy and give new physical insights. We go on to consider entropy production in the non-equilibrium setting of a quench, again showing that this can be related to practical charge measurements. In quantum circuit experiments, the dot charge is measured in practice using a secondary, capacitively coupling charge-sensor circuit. We investigate this by explicitly modelling the device together with the charge-sensor in a fully quantum mechanical treatment. This allows us to understand and quantify the measurement backaction. It also allows us to establish when the assumptions underpinning the Maxwell entropy extraction hold. Finally, we consider the strongly correlated electron physics of bilayer Hubbard models within dynamical mean field theory. Such models capture the essence of interface physics in heterostructures. In the basic bilayer, we revisit the phase diagram and study the change in character of the metal-insulator transition from continuous band-insulator gap opening to Mott quasiparticle collapse, as a function of interlayer coupling and local Coulomb interaction strength. For more complex interfaces involving additional correlated sites between layers, many-body quantum interference can be tuned, leading to different types of transition. While a pure impurity model version of this system features a second-order non-Fermi liquid Kondo critical point, full self-consistency on the lattice turns the transition first-order.
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.