Accurate characterization of correlated electronic states, as well as their evolution under external fields or in dissipative environment, is essentially important for understanding the properties of strongly correlated transition‐metal materials involving spin‐unpaired d or f electrons. This paper reviews the development and applications of a numerical simulation program, the Hierarchical Equations of Motion for QUantum Impurity with a Correlated Kernel (HEOM‐QUICK), which allows for an accurate and universal characterization of strongly correlated quantum impurity systems. The HEOM‐QUICK program implements the formally exact HEOM formalism for fermionic open systems. Its simulation results capture the combined effects of system‐environment dissipation, many‐body interactions, and non‐Markovian memory in a nonperturbative manner. The HEOM‐QUICK program has been employed to explore a wide range of static and dynamic properties of various types of quantum impurity systems, including charge or spin qubits, quantum dots, molecular junctions, and so on. It has also been utilized in conjunction with first‐principles methods such as density‐functional theory methods to study the correlated electronic structure of adsorbed magnetic molecules. The advantages in its accuracy, efficiency, and universality have made the HEOM‐QUICK program a reliable and versatile tool for theoretical investigations on strong electron correlation effects in complex materials. WIREs Comput Mol Sci 2016, 6:608–638. doi: 10.1002/wcms.1269 This article is categorized under: Structure and Mechanism > Computational Materials Science Software > Simulation Methods
Accurate characterization of correlated electronic states, as well as their evolution under external fields or in dissipative environment, is essentially important for understanding the properties of strongly correlated transition‐metal materials involving spin‐unpaired d or f electrons. This paper reviews the development and applications of a numerical simulation program, the Hierarchical Equations of Motion for QUantum Impurity with a Correlated Kernel (HEOM‐QUICK), which allows for an accurate and universal characterization of strongly correlated quantum impurity systems. The HEOM‐QUICK program implements the formally exact HEOM formalism for fermionic open systems. Its simulation results capture the combined effects of system‐environment dissipation, many‐body interactions, and non‐Markovian memory in a nonperturbative manner. The HEOM‐QUICK program has been employed to explore a wide range of static and dynamic properties of various types of quantum impurity systems, including charge or spin qubits, quantum dots, molecular junctions, and so on. It has also been utilized in conjunction with first‐principles methods such as density‐functional theory methods to study the correlated electronic structure of adsorbed magnetic molecules. The advantages in its accuracy, efficiency, and universality have made the HEOM‐QUICK program a reliable and versatile tool for theoretical investigations on strong electron correlation effects in complex materials. WIREs Comput Mol Sci 2016, 6:608–638. doi: 10.1002/wcms.1269 This article is categorized under: Structure and Mechanism > Computational Materials Science Software > Simulation Methods
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.
Abstract Many‐body open quantum systems (OQSs) have a profound impact on various subdisciplines of physics, chemistry, and biology. Thus, the development of a computer program capable of accurately, efficiently, and versatilely simulating many‐body OQSs is highly desirable. In recent years, we have focused on the advancement of numerical algorithms based on the fermionic hierarchical equations of motion (HEOM) theory. Being in‐principle exact, this approach allows for the precise characterization of many‐body correlations, non‐Markovian memory, and non‐equilibrium thermodynamic conditions. These efforts now lead to the establishment of a new computer program, HEOM for QUantum Impurity with a Correlated Kernel, version 2 (HEOM‐QUICK2), which, to the best of our knowledge, is currently the only general‐purpose simulator for fermionic many‐body OQSs. Compared with version 1, the HEOM‐QUICK2 program features more efficient solvers for stationary states, more accurate treatment of non‐Markovian memory, and improved numerical stability for long‐time dissipative dynamics. Integrated with quantum chemistry software, HEOM‐QUICK2 has become a valuable theoretical tool for the precise simulation of realistic many‐body OQSs, particularly the single atomic or molecular junctions. Furthermore, the unprecedented precision achieved by HEOM‐QUICK2 enables accurate simulation of low‐energy spin excitations and coherent spin relaxation. The unique usefulness of HEOM‐QUICK2 is demonstrated through several examples of strongly correlated quantum impurity systems under non‐equilibrium conditions. Thus, the new HEOM‐QUICK2 program offers a powerful and comprehensive tool for studying many‐body OQSs with exotic quantum phenomena and exploring applications in various disciplines. This article is categorized under: Data Science > Computer Algorithms and Programming Software > Simulation Methods Theoretical and Physical Chemistry > Statistical Mechanics
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.
A nonperturbative quantum impurity solver is proposed based on a formally exact hierarchical equations of motion (HEOM) formalism for open quantum systems. It leads to quantitatively accurate evaluation of physical properties of strongly correlated electronic systems, in the framework of dynamical mean-field theory (DMFT). The HEOM method is also numerically convenient to achieve the same level of accuracy as that using the state-of-the-art numerical renormalization group impurity solver at finite temperatures. The practicality of the HEOM+DMFT method is demonstrated by its applications to the Hubbard models with Bethe and hypercubic lattice structures. We investigate the metal-insulator transition phenomena, and address the effects of temperature on the properties of strongly correlated lattice systems.
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.
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.