Understanding the time evolution of strongly correlated open quantum systems (OQSs) in response to perturbations (quenches) is of fundamental importance to the precise control of quantum devices. It is, however, rather challenging in multi-impurity quantum systems because such evolution often involves multiple intricate dynamical processes. In this work, we apply the numerically exact hierarchical equations of motion approach to explore the influence of two different types of perturbations, i.e., sudden swapping of the energy levels of impurity systems and activating the inter-impurity spin-exchange interaction, on the dissipation dynamics of the Kondo-correlated two-impurity Anderson model over a wide range of energetic parameters. By evaluating the time-dependent impurity spectral function and other system properties, we analyze the time evolution of the Kondo state in detail and conclude a phenomenologically scaling relation for Kondo dynamics driven by these perturbations. The evolutionary scaling relationship is not only related to the Kondo characteristic energy TK but also significantly affected by the simultaneous non-Kondo dynamic characteristic energy. We expect these results will inspire subsequent theoretical studies on the dynamics of strongly correlated OQSs.
Understanding the time evolution of strongly correlated open quantum systems (OQSs) in response to perturbations (quenches) is of fundamental importance to the precise control of quantum devices. It is, however, rather challenging in multi-impurity quantum systems because such evolution often involves multiple intricate dynamical processes. In this work, we apply the numerically exact hierarchical equations of motion approach to explore the influence of two different types of perturbations, i.e., sudden swapping of the energy levels of impurity systems and activating the inter-impurity spin-exchange interaction, on the dissipation dynamics of the Kondo-correlated two-impurity Anderson model over a wide range of energetic parameters. By evaluating the time-dependent impurity spectral function and other system properties, we analyze the time evolution of the Kondo state in detail and conclude a phenomenologically scaling relation for Kondo dynamics driven by these perturbations. The evolutionary scaling relationship is not only related to the Kondo characteristic energy TK but also significantly affected by the simultaneous non-Kondo dynamic characteristic energy. We expect these results will inspire subsequent theoretical studies on the dynamics of strongly correlated OQSs.
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.
We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function $G(t+{t}^{\ensuremath{'}},t)$, and its associated time-dependent spectral function, $A(\ensuremath{\omega},t)$, for times t both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times $t\ensuremath{\rightarrow}{0}^{+}$, only fully develops at very long times $t\ensuremath{\gtrsim}1/{T}_{K}$, where TK is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale $1/\mathrm{\ensuremath{\Gamma}}$ during the time interval $\ensuremath{-}1/\mathrm{\ensuremath{\Gamma}}\ensuremath{\lesssim}t\ensuremath{\lesssim}+1/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is the hybridization strength. Initial and final state spectral functions are recovered in the limits $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $t\ensuremath{\rightarrow}+\ensuremath{\infty}$, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.
A hierarchical equations of motion based numerical approach is developed for accurate and efficient evaluation of dynamical observables of strongly correlated quantum impurity systems. This approach is capable of describing quantitatively Kondo resonance and Fermi-liquid characteristics, achieving the accuracy of the latest high-level numerical renormalization group approach, as demonstrated on single-impurity Anderson model systems. Its application to a two-impurity Anderson model results in differential conductance versus external bias, which correctly reproduces the continuous transition from Kondo states of individual impurity to singlet spin states formed between two impurities. The outstanding performance on characterizing both equilibrium and nonequilibrium properties of quantum impurity systems makes the hierarchical equations of motion approach potentially useful for addressing strongly correlated lattice systems in the framework of dynamical mean-field theory.
This thesis focuses on the study of dissipative quantum impurity models. Quantum impurity models describe a localized quantum system, known as the impurity, that interacts with a large quantum bath, enabling the exploration of essential phenomena such as the Kondo effect in magnetic impurities and the transport properties in quantum dots. An intriguing extension of these models is the exploration of the interplay between strong correlations and dissipation. Dissipation, often described by the Lindblad master equation in its Markovian limit, introduces challenges and opportunities as it interacts with coherent many-body effects. In this thesis, we want to investigate the non-equilibrium dynamics of quantum impurity systems influenced by both strong correlations and dissipation across four main parts. We begin by developing numerical methods, specifically a diagrammatic Monte Carlo technique, to analyze the real-time dynamics of dissipative quantum impurity models. Our findings demonstrate that local Markovian dissipation enhances the convergence of Monte Carlo sampling by constraining the process to a reduced subspace of diagrams. This effectively mitigates the sign problem and facilitates the exploration of longer time scales compared to conventional unitary approaches. In the second part, we derive a self-consistent dynamical map using the Non-Crossing Approximation to study the long-term behavior of dissipative quantum impurity models. This method effectively resums diagrams into a self-consistent Dyson equation, capturing essential non-Markovian environmental effects, including the Kondo effect. In a third part, we examine the relationship between Kondo physics and Markovian dissipation by investigating an interacting quantum dot coupled to a metallic bath under continuous charge monitoring. We identify a crossover from Kondo screening to the Quantum Zeno effect, effectively described by a non-Hermitian Kondo model in the weak dissipation regime. Finally, the last part explores the effects of particle losses on the Kondo effect and the Kondo-Zeno crossover. Preliminary results indicate that while single-particle losses reduce many-body effects, two-body losses introduce intriguing phenomena, such as spin decay and hints of the Kondo-Zeno crossover, along with an unusual spectral function exhibiting a low-frequency lineshape.
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.
The cyclic sudden switch of a magnetic impurity from a Kondo to a non-Kondo state and back was recently proposed to involve an important dissipation of the order of several kBTK per cycle. The possibility to reveal this and other electronic processes through nanomechanical dissipation by, e.g., ultrasensitive atomic force microscope (AFM) tools would represent an unusual and interesting form of spectroscopy. Here, we explore the dependence on the switching time of the expected dissipation, a quantity whose magnitude is physically expected to drop from maximum to zero between sudden and slow switching, respectively. As an application of a recently established matrix-product-state-based time-dependent variational algorithm, we study the magnetic-field-induced Kondo switching in an Anderson model of the magnetic impurity. We find, quite reasonably, that dissipation requires switching within the Kondo timescale $\ensuremath{\hbar}{({k}_{B}{T}_{K})}^{\ensuremath{-}1}$ or faster. While such a fast switching seems problematic for current AFM setups, the challenge remains open for future means to detect this dissipation by time-dependent magnetic fields, an electrostatic impurity level shift, or hybridization switching. The technical aspects revealed by this approach will be of interest for future nonequilibrium calculations.