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, manybody 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.

The precise tuning of local spin states in adsorbed organometallic molecules by a mechanically controlled scanning tunneling microscope (STM) tip has become a focus of recent experiments. However, the underlying mechanisms remain somewhat unclear. We investigate theoretically the STM tip control of local spin states in a single iron(II) porphyrin molecule adsorbed on the Pb(111) substrate. A combined density functional theory and hierarchical equations of motion approach is employed to simulate the tip tuning process in conjunction with the complete active space self-consistent field method for accurate computation of magnetic anisotropy. Our first-principles-based simulation accurately reproduces the tuning of magnetic anisotropy realized in experiment. Moreover, we elucidate the evolution of geometric and electronic structures of the composite junction and disclose the delicate competition between the Kondo resonance and local spin excitation. The understanding and insight provided by the first-principles-based simulation may help to realize more fascinating quantum state manipulations.

Several recent advancements for the hierarchical equations of motion (HEOM) approach are reported. First, we propose an a priori estimate for the optimal number of basis functions for the reservoir memory decomposition. Second, we make use of the sparsity of auxiliary density operators (ADOs) and propose two ansatzs to screen out all the intrinsic zero ADO elements. Third, we propose a new truncation scheme by utilizing the time derivatives of higher-tier ADOs. These novel techniques greatly reduce the memory cost of the HEOM approach, and thus enhance its efficiency and applicability. The improved HEOM approach is applied to simulate the coherent dynamics of Aharonov-Bohm double quantum dot interferometers. Quantitatively accurate dynamics is obtained for both noninteracting and interacting quantum dots. The crucial role of the quantum phase for the magnitude of quantum coherence and quantum entanglement is revealed.

Probes that measure the local thermal properties of systems out of equilibrium are emerging as new tools in the study of nanoscale systems. One can then measure the temperature of a probe that is weakly coupled to a bias-driven system. By tuning the probe temperature so that the expectation value of some observable of the system is minimally perturbed, one obtains a parameter that measures its degree of local statistical excitation, and hence its local heating. However, one anticipates that different observables may lead to different temperatures and thus different local heating expectations. We propose an experimentally realizable protocol to measure such local temperatures and apply it to bias-driven quantum dots. By means of a highly accurate open quantum system approach, we show theoretically that the measured temperature is quite insensitive both to the choice of observable and to the probe-system coupling. In particular, even with observables that are distinct both physically and in their degree of locality, such as the local magnetic susceptibility of the quantum dot and the global spin-polarized current measured at the leads, the resulting local temperatures are quantitatively similar for quantum dots ranging from noninteracting to Kondo-correlated regimes, and are close to those obtained with the traditional "local equilibrium" definition.