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
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
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.
An essential role for the description of correlated quantum many-particle systems is played by impurity problems. These consist of a small number of localized orbitals, where the electrons are subject to a Coulomb-interaction, coupled to noninteracting bath degrees of freedom. Impurity models are used for the description of magnetic impurities in metals, of quantum points in nanostructures, and in the context of the dynamical mean field theory (DMFT). For their solution a large number of numerical methods exists. We mention here quantum Monte Carlo methods, the numerical renormalization group, or the exact diagonalization method. In this thesis we introduce a new approach to impurity problems, which is based on the functional renormalization group (fRG). In contrast to the methods mentioned above this approach is approximative, but it can be applied with comparatively lower numerical effort. The aim of this thesis is to examine the advantages and limits of this method. An important role in this connection is played by the single impurity Anderson model (SIAM), which is already understood very well with other methods and is seen as a prototype for models with strong local correlations. The spin fluctuations of this model are governed by the so-called Kondo energy scale and a central question of this thesis is whether this scale can be reproduced with our approach. We begin with a short introduction to the Green's function formalism for correlated quantum many particle systems. Then the fRG flow equations for the one-particle irreducible vertex functions are deduced, and several approximation schemes are discussed. In chapter 4 we introduce the fRG scheme used in this thesis. At first the SIAM is mapped to a semi-infinite chain, in which the interacting orbital is given by the first site of this chain. The system is then subdivided into two parts. The first part, which is called 'core', consists of the interacting orbital and the first L bath sites. The remaining bath sites form the second part. At the beginning both parts are decoupled from each other such that the core can be solved exactly. Starting with this exact solution the coupling between the core and the remaining bath sites is switched on slowly in a renormalization group flow. This way it is possible to calculate the one-particle and two-particle correlation functions on the correlated site. We call this flow scheme 'hybridization flow'. The flow equations are formulated in an effective theory on the first bath site outside the core, which turns out to be advantageous compared to other implementations of the hybridization flow. In chapter 5 this hybridization flow method is applied to the SIAM with semi-elliptic bath density of states. We discuss the differences which arise for different core-sizes (L = 0,1,2,3) and for different truncations of the fRG flow equations. In the cases L=0 and L=2 the local Fermi liquid properties of the SIAM and the associated Kondo scale can not be reproduced in both approximation schemes. For the core sizes L=1 and L=3 the ground state of the core is already a spin singlet state in the beginning of the flow and the Fermi liquid properties are reproduced. The Kondo energy scale, which determines the width of the central quasi-particle peak and the size of the local spin-susceptibility, is however not accurately resolved. This is partly due to the finite temperature used in our implementation such that for larger interaction strength the temperature is above the Kondo energy scale. The application of the hybridization flow method to quantum impurity problems in the context of the DMFT is discussed in chapter 6. In this approach lattice models with a local interaction, like the Hubbard model, are mapped to an effective impurity problem for a single lattice site embedded in a dynamical mean field representing the influence of the other electrons. This scheme is exact in infinitely many dimensions and a nonperturbative approximation method for finite-dimensional systems. In particular the Mott metal-insulator transition in the Hubbard model is successfully described by this method. One class of quantities, which is easily accessible from the hybridization flow scheme, is given by the local two-particle correlation functions. These play an essential role in non-local extensions of the DMFT. Therefore we mainly focus on these quantities. At first we apply our flow scheme to the Hubbard model on the Bethe lattice in infinite dimensions and calculate the local one-particle irreducible vertex functions in the insulating and the metallic phase in good agreement with previous calculations, which use the exact diagonalization method as impurity solver. After this we extend our flow scheme to a cluster DMFT method, which includes short-range antiferromagnetic correlations and calculate again the local and non-local vertex functions for the two-dimensional Hubbard model. We compare these to the results of the single-site DMFT.
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.
The unusual low temperature behaviour of metals and metal alloys, such as heavy fermion systems, Mott insulators and unconventional superconductors, has been a central topic of scientific investigation for over half a century. Due to strong correlations between the many particles, the solution to most theoretical models of these systems is difficult because standard perturbation methods break down. In certain cases, non-perturbative renormalization group approaches can be used to access the low energy behaviour and take into account the effects of strong correlations. However, these methods do not work for several important and interesting classes of model. These include lattice models with finite dimension, models with impurities of high degeneracy and steady state transport through quantum dots. The aim of this thesis is therefore to develop a new and general renormalization group approach that will lend itself to wide application in models where existing techniques fail. The method explored here utilises renormalized perturbation theory (RPT) in conjunction with scaling equations and collective excitations. It is tested on the single impurity Anderson model, the paradigmatic model used to understand magnetic impurities within a non-magnetic host metal. The Anderson model is simple yet exhibits interesting features due to strong correlation, such as a single renormalized low energy scale called the Kondo temperature. This model is also well understood as it has been solved using the numerical renormalization group (NRG), which provides ample numerical data against which this new technique is compared. Finally, a preliminary study is given for this method when it is applied to a wider class of models.
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.
The construction of good effective models is an essential part of understanding and simulating complex systems in many areas of science. It is a particular challenge for correlated many-body quantum systems displaying emergent physics. We propose a machine learning approach that optimizes an effective model based on an estimation of its partition function. The success of the method is demonstrated by application to the single impurity Anderson model and double quantum dots, where nonperturbative results are obtained for the old problem of mapping to effective Kondo models. We also show that an alternative approach based on learning minimal models from observables may yield the wrong low-energy physics. On the other hand, learning minimal models from the partition function recovers the correct low-energy physics but may not reproduce all observables.