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.
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 thesis, we describe methods for solving quantum impurity models out of equilibrium, and consider several applications of these methods. Quantum impurity models describe systems in which a small subsystem (the impurity) is coupled to a large external environment (the bath). These models are useful in describing a wide variety of physical systems, including quantum dots, absorbed adatoms, and molecular junctions. They are also important for the study of lattice models, such as the Hubbard model, where they appear as auxiliary problems within the framework of dynamical mean field theory (DMFT). Many interesting phenomena in these systems, such as transport through a quantum dot, or optical pumping of a Mott insulator, occur away from equilibrium. Theoretical description of these phenomena requires methods that are capable of handling both real time evolution and strong correlation effects presenting a formidable challenge. In this thesis, we describe several methods for the solution of non-equilibrium quantum impurity models which meet this challenge. These methods are based on an expansion in the coupling between the impurity and the bath known as the hybridization expansion. Chapter 1 gives an introduction to the problems addressed in this thesis. Chapter 2 provides a brief overview of Keldysh formalism for nonequilibrium problems. Chapter 3 describes the bare hybridization expansion for the Anderson impurity model as well as its bold extensions. In particular we derive the non-crossing approximation (NCA), the one-crossing approximation (OCA), the inchworm expansion, and a bold expansion for the impurity Green's function. Chapter 4 describes the implementation of a quantum Monte Carlo algorithm based on the inchworm expansion. The rest of the thesis describes applications of these algorithms to problems of physical interest. Chapters 5 and 6 apply the methods previously described to understanding the Kondo effect in quantum dot systems. The first chapter examines the behavior of the Kondo effect under an applied voltage bias, a classic non-equilibrium strongly-correlated problem. The second chapter examines different ways of measuring the cloud of electrons which screen the impurity spin in the Kondo regime. Chapters 7 and 8 apply the same methods to the Hubbard model using the Dynamical mean-field theory (DMFT) mapping. The first chapter studies the Hubbard model in equilibrium using the numerically exact inchworm quantum Monte Carlo method. The second chapter studies the possibility of dynamic switching between the metal and Mott-insulator states of the Hubbard model using the one-crossing approximation (OCA). Finally chapter 9 gives some overall conclusions and suggests future directions for the development of hybridization expansion methods.
Quantum impurity models with strong electron correlations, such as the paradigmatic Anderson Impurity Model (AIM), are central to our understanding of a range of physical phenomena including local moment formation, Coulomb blockade and Kondo screening. They describe magnetic atoms and molecules on surfaces, quantum dot circuits, and correlated materials through dynamical mean field theory. The physics of such systems in strongly non-equilibrium conditions is particularly complex and challenging to capture, whereas Gaussian models of free fermions can be easily solved. Here we show that the time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system. The method allows us to understand interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.
In recent years, nano-sized systems involving local charge or spin states have received wide interests because of their potential application in emerging fields such as quantum information and quantum computation. For instance, organometallic molecular complexes may serve as building blocks of quantum storage devices, because the spin-unpaired d or f electrons at transition metal centers may be employed to construct spin qubits. Moreover, if the system contains strong electron-electron interaction, the involving local quantum states are subject to prominent electron correlation effects (such as the Kondo effect). Theoretically, spatially confined nano-systems are often described by quantum impurity models. Thus the accurate prediction of the intrinsic properties of quantum impurity systems and the deep understanding on the response and evolution of local quantum states under external fields or in dissipative environment are fundamentally important for the design and fabrication of quantum devices. The accurate characterization of quantum coherence, correlation, and entanglement in quantum impurity systems remains a great challenge. Enormous efforts have been made to achieve this goal. A variety of theoretical methods have been developed, including the numerical renormalization group method, the quantum Monte Carlo method, and many others. However, all the existing methods are subject to certain limitations regarding accuracy or efficiency. Therefore, we choose to view this problem from a new perspective—the perspective of open quantum systems. Over the past decade, we have developed a formally exact quantum dissipation theory, the hierarchical equations of motion (HEOM) theory, for fermionic open systems. The HEOM theory captures the combined effects of system-environment dissipation, many-body interaction, and non-Markovian memory in a nonperturbative manner. It is capable of addressing static and dynamic responses of system observables in both equilibrium and nonequilibrium situations. We have implemented the HEOM method in our self-designed computer program HEOM-QUICK. We have also devised a series of advanced algorithms which substantially enhance the numerical efficiency of HEOM. The HEOM-QUICK program thus provides an accurate, efficient, and versatile theoretical tool for the investigation of strongly correlated quantum impurity systems. We have applied the HEOM method to study a variety of problems associated with quantum impurity systems. These include the intrinsic properties of quantum impurity models and lattice models, the transient current response of quantum dots to ac voltages, the thermopower and local heating effect in nonequilibrium quantum dots, and the tuning of local spin states in adsorbed molecular magnets. In particular, the HEOM method has been combined with density functional theory (DFT) method, and thus allows for first-principles-based simulation on the precise tuning of local spin states in adsorbed magnetic molecules. Numerical simulations have discovered or reproduced many important quantum phenomena that are essential for relevant experiments, including the Kondo effect, Mott metal-insulator transition, quantum memristive effect, etc. This paper gives a brief overview of our development of the HEOM method for fermionic open systems, as well as its applications to various strongly correlated quantum impurity systems.
Generalized quantum impurity models—which feature a few localized and strongly correlated degrees of freedom coupled to itinerant conduction electrons—describe diverse physical systems, from magnetic moments in metals to nanoelectronics quantum devices such as quantum dots or single-molecule transistors. Correlated materials can also be understood as self-consistent impurity models through dynamical mean-field theory. Accurate simulation of such models is challenging, especially at low temperatures, due to many-body effects from electronic interactions, resulting in strong renormalization. In particular, the interplay between local impurity complexity and Kondo physics is highly nontrivial. A common approach, which we further develop in this work, is to consider instead a simpler effective impurity model that still captures the low-energy physics of interest. The mapping from a bare to an effective model is typically done perturbatively, but even this can be difficult for complex systems, and the resulting effective model parameters can nevertheless be quite inaccurate. Here we develop a nonperturbative, unsupervised machine learning approach to systematically obtain low-energy effective impurity-type models, based on the renormalization-group framework. The method is shown to be general and flexible, as well as accurate and systematically improvable. We benchmark the method against exact results for the Anderson impurity model, and we provide an outlook for more complex models beyond the reach of existing methods. Published by the American Physical Society 2024
This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.
This thesis contributes to the field of strongly correlated electron systems with studies in two distinct fields thereof: the specific nature of correlations between electrons in one dimension and quantum quenches in quantum impurity problems. In general, strongly correlated systems are characterized in that their physical behaviour needs to be described in terms of a many-body description, i.e. interactions correlate all particles in a complex way. The challenge is that the Hilbert space in a many-body theory is exponentially large in the number of particles. Thus, when no analytic solution is available - which is typically the case - it is necessary to find a way to somehow circumvent the problem of such huge Hilbert spaces. Therefore, the connection between the two studies comes from our numerical treatment: they are tackled by the density matrix renormalization group (DMRG) [1] and the numerical renormalization group (NRG) [2], respectively, both based on matrix product states. The first project presented in this thesis addresses the problem of numerically finding the dominant correlations in quantum lattice models in an unbiased way, i.e. without using prior knowledge of the model at hand. A useful concept for this task is the correlation density matrix (CDM) [3] which contains all correlations between two clusters of lattice sites. We show how to extract from the CDM, a survey of the relative strengths of the system’s correlations in different symmetry sectors as well as detailed information on the operators carrying long-range correlations and the spatial dependence of their correlation functions. We demonstrate this by a DMRG study of a one-dimensional spinless extended Hubbard model [4], while emphasizing that the proposed analysis of the CDM is not restricted to one dimension. The second project presented in this thesis is motivated by two phenomena under ongoing experimental and theoretical investigation in the context of quantum impurity models: optical absorption involving a Kondo exciton [5, 6, 7] and population switching in quantum dots [8, 9, 10, 11, 12, 13, 14, 15]. It turns out that both phenomena rely on the various manifestations of Anderson orthogonality (AO) [16], which describes the fact that the response of the Fermi sea to a quantum quench (i.e. an abrupt change of some property of the impurity or quantum dot) is a change of the scattering phase shifts of all the single-particle wave functions, therefore drastically changing the system. In this context, we demonstrate that NRG, a highly accurate method for quantum impurity models, allows for the calculation of all static and dynamic quantities related to AO and present an extensive NRG study for population switching in quantum dots.