Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.
Investigations of strongly correlated quantum impurity systems (QIS), which exhibit diversified novel and intriguing quantum phenomena, have become a highly concerning subject in recent years. The hierarchical equations of motion (HEOM) method is one of the most popular numerical methods to characterize QIS linearly coupled to the environment. This review provides a comprehensive account of a formally rigorous and numerical convergent HEOM method, including a modeling description of the QIS and an overview of the fermionic HEOM formalism. Moreover, a variety of spectrum decomposition schemes and hierarchal terminators have been proposed and developed, which significantly improve the accuracy and efficiency of the HEOM method, especially in cryogenic temperature regimes. The practicality and usefulness of the HEOM method to tackle strongly correlated issues are exemplified by numerical simulations for the characterization of nonequilibrium quantum transport and strongly correlated Kondo states as well as the investigation of nonequilibrium quantum thermodynamics.
Recent technological advancement in scanning tunneling microscopes has enabled the measurement of spin-field and spin-spin interactions in single atomic or molecular junctions with an unprecedentedly high resolution. Theoretically, although the fermionic hierarchical equations of motion (HEOM) method has been widely applied to investigate the strongly correlated Kondo states in these junctions, the existence of low-energy spin excitations presents new challenges to numerical simulations. These include the quest for a more accurate and efficient decomposition for the non-Markovian memory of low-temperature environments and a more careful handling of errors caused by the truncation of the hierarchy. In this work, we propose several new algorithms, which significantly enhance the performance of the HEOM method, as exemplified by the calculations on systems involving various types of low-energy spin excitations. Being able to characterize both the Kondo effect and spin excitation accurately, the HEOM method offers a sophisticated and versatile theoretical tool, which is valuable for the understanding and even prediction of the fascinating quantum phenomena explored in cutting-edge experiments.
The fermionic hierarchical equations of motion (HEOM) approach has found wide application in the exploration of open quantum systems, and extensive efforts have been committed to improving its efficiency and accuracy in practical calculations. In this work, by scrutinizing the stationary-state and dynamic properties of Kondo-correlated quantum impurity systems, we show that the strength of Kondo correlation induced by the system-environment entanglement primarily determines the converged hierarchical truncation tier of the HEOM method. This complements the rule of thumb regarding the positive correlation between the height of hierarchy and system-environment coupling strength. These insights will provide useful guidelines for developing a more sophisticated fermionic HEOM method for the investigation of many-body open quantum systems.
The hierarchical equations of motion (HEOM) formalism developed recently is becoming an important method for investigating the properties and dynamical processes of quantum open systems. A comprehensive account on its establishment and recent development was given, covering theoretical formulations, numerical algorithm and practical applications. By construction HEOM resolves in a nonperturbative manner the combined effects of many-body interaction, system-environment coupling and non-Markovian memory. While it is mathematically equivalent to an exact path integral formalism, HEOM is also very convenient in computation and versatile in various systems of study. Moreover, it is shown recently that HEOM is actually a correlated system-environment dynamics theory. The practicality of HEOM to address physical and chemical problems was exemplified with examples relevant to quantum dissipation and quantum transport.
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 hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.
The hierarchical equations of motion (HEOM) approach is an accurate method to simulate open system quantum dynamics, which allows for systematic convergence to numerically exact results. To represent effects of the bath, the reservoir correlation functions are usually decomposed into summation of multiple exponential terms in the HEOM method. Since the reservoir correlation functions become highly non-Markovian at low temperatures or when the bath has complex band structures, a present challenge is to obtain accurate exponential decompositions that allow efficient simulation with the HEOM. In this work, we employ the barycentric representation to approximate the Fermi function and hybridization functions in the frequency domain. This method, by approximating these functions with optimized rational decomposition, greatly reduces the number of basis functions in decomposing the reservoir correlation functions, which further allows the HEOM method to be applied to ultralow temperature and general band structures. We demonstrate the efficiency, accuracy, and long-time stability of this decomposition scheme by applying it to the Anderson impurity model in the low-temperature regime with the Lorentzian and tight-binding hybridization functions.