Development Of Fermionic Hierarchical Equations Of Motion Approach And Program For Low-Temperature Environments

The Hierarchical Equation of Motion(HEOM)approach still faces the following two challenges in the study of the dynamics of open quantum systems at low temperatures.First,within the theoretical framework of HEOM,the environment correlation function is decomposed by an exponential basis function.The rationality of the decomposition scheme determines the accuracy of the description of the environment and the computational cost.Second,the many-body effect within the system requires a complex hierarchy while the number of unknowns grows exponentially with the size of the hierarchy,i.e.,the "exponential wall" problem.To address these two challenges,we aim to improve the efficiency of HEOM method to make it more applicable to many-body open fermionic systems in lowtemperature environments.First,to accurately characterize the effect of the lowtemperature environment on the dynamics of the system,an environmental spectral decomposition scheme based on the time-domain Prony fit is used.By testing the nointeracting electron reservoir model,the Prony scheme can accurately reproduce the behavior of the environmental correlation function at low temperatures with a small number of basis functions,compared with the previous spectral decomposition schemes.To accurately describe the significant many-body effects within the system at low temperatures,we refer to the Bonn-Oppenheimer approximation and propose an adiabatic truncation scheme that adiabatically separates the dissipative modes with faster rates from the other dissipative modes in the dissipative process.This method optimizes the hierarchical structure,significantly reduces the number of unknown elements,and decreases computational consumption and time.Through the numerical testing on the single impurity Anderson model,the adiabatic truncation scheme yields sufficiently accurate numerical convergence results with less computational effort and alleviates the numerical instability during the long-time dynamic evolution.For complex systems with large degrees of freedom,the subspace HEOM method is developed by omitting the high-energy eigenstates of the system by projection.Moreover,we develop a perturbation solver in the HEOM space,and numerical tests show that the scheme provides an effective new way to solve the steady state of open quantum systems.Combining these theoretical advances and efficient numerical algorithms,we have developed the basic part of the HEOM-QUICK2 program to realize the accurate solving of steady-state properties and real-time simulation of the dynamics of strongly correlated open quantum systems in low-temperature environments.To validate the HEOMQUICK2 program,we simulate the long-time evolution of Kondo states in quantum impurities,the evolution of localized spin states within nanomagnets and local systems involving low-energy spin excitation.We compare in detail the numerical computational performance of the previous version of the program with that of the HEOMQUICK2 program for these systems.It is found that the HEOM-QUICK2 program not only surpasses in numerical accuracy but also has higher computational efficiency.The HEOM-QUICK2 program is a powerful and efficient numerical tool for the study of low-temperature strongly correlated systems.