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WIREs Comput Mol Sci
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HEOM‐QUICK : a program for accurate, efficient, and universal characterization of strongly correlated quantum impurity systems

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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

(a) The spectral function and (b) the differential conductance of parallel‐coupled two‐impurity Anderson model (TIAM), at different inter‐impurity coupling strength t 12. The parallel‐coupled TIAM model is sketched in (a). Parameters here are all in unit of Δ:W = 10, U 1 = U 2 = 10, ɛ 1 = ɛ 2 = − 5, and T = 0.5. These lines from top to bottom, correspond to t 12/Δ = 1, 0, 1.5, 2, 2.5 in (a) and t 12/Δ = 1, 1.5, 0, 2, 2.5 in (b). (Reprinted with permission from Ref . Copyright 2012 American Physical Society).
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Current versus voltage for the single‐orbital Anderson impurity model (AIM) driven by an ac voltage of amplitude eV 0 = Δ and frequency Ω. The arrows indicate the circulating directions of the loops. The dashed green curve in (a) depicts the I‐V characteristic of the steady state. The insets plot the corresponding current (in eΔ/h) versus time (in h/Δ). The other parameters are U = − 2ɛ d  = 4Δ, W = 20Δ, and T = 0.2 Δ. (Reprinted with permission from Ref . Copyright 2013 American Physical Society).
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Electric current in response to time‐dependent voltages. (a) Transient response currents driven by a ramp‐up voltage, with different values of turn‐on durationτ, which are separated at t = 0 by 3 nA. (b) Transient response current of each spin direction for τ = 50 ps. The inset shows the form of ramp‐up voltage. Other parameters are (in units of meV): ɛ = 0.5,ɛ  = 2.5, U = 4, Γ = 0.05, W = 15, T = 0.2, and Δ = 10. Note that in this figure the reservoir‐impurity coupling is denoted as Γ, and Δ is the amplitude of the ramp‐up voltage. (Reprinted with permission from Ref . Copyright 2008 Institute of Physics).
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Flowchart of the Hierarchical Equations of Motion for QUantum Impurity with a Correlated Kernel (HEOM‐QUICK) program. The program consists of several modules, including the input/output, preparation, computation, and interface modules.
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The left panel shows the static local magnetic susceptibility multiplied by temperature, χ m (ω)T (in units of g 2 μ B 2 / k B ), versus T/T K for a series of equilibrium symmetric single‐orbital Anderson impurity model (AIM) (ɛ d  = − U/2) of different U. Here, T K is the Kondo temperature, and U, T, and W are in unit of Δ. The HEOM results (scattered symbols) are compared with the latest full density‐matrix numerical renormalization group (NRG) calculations (lines) of Ref , the Figure there. The NRG calculations are for very large reservoir bandwidth W, while the Hierarchical Equations of Motion (HEOM) results are obtained with relatively small bandwidths (W = 10 and 20) for saving computational cost. The four lines from top to bottom correspond to U = 12, 8, 4, and 2, respectively. The right panel shows the steady‐state current through various symmetric single‐orbital AIMs (ɛ d  = − U/2) of different values of U. The results of functional renormalization group (fRG), time‐dependent density‐matrix renormalization group (tDMRG), iterative summation of real‐time path integral (ISPI), and real‐time quantum Monte Carlo (QMC) are extracted from Refs . Other parameters are T = 0.1Δ and W = 8U. (Reprinted with permission from Ref . Copyright 2013 American Physical Society; Ref . Copyright 2013 American Physical Society).
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Comparison between A(ω) of symmetric single‐orbital Anderson impurity model (AIM) (ɛ d  = − U/2) calculated by Hierarchical Equations of Motion (HEOM) and numerical renormalization group (NRG) methods. The parameters adopted are T = 0.2 and W = 50 (in units of Δ). The inset shows the imaginary part of the interaction self‐energy calculated from HEOM at energy close to ω = 0. The solid lines from the top down at ω = 0 correspond to U/πΔ = 0.5, 1, 3, 6 in the main panel, and U/πΔ = 3, 1, 0.5 in the inset. (Reprinted with permission from Ref . Copyright 2012 American Physical Society).
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The left panel shows the time‐dependent current calculated with proposed method in Ref (full line) and with the time‐dependent Green function method from Ref (dashed line) in response to a steplike modulation of the bias with step height μ L . The coupling is Γ L  = Γ R  = Γ/2, the temperature T = 0.05Γ, the orbital energy ɛ d  = 0, and the half width of the band is W = 30Γ. The right panel shows the transient currents through L‐reservoir, I L (t), in response to a step‐function voltage pulse applied on R‐reservoir. The lines correspond to different voltage amplitudes. Other parameters are ɛ d  = 0, Γ L  = Γ R  = 0.5Γ, T = 0.05Γ, and W = 20Γ. Note that the parameters Γ L , Γ R , and Γ in this figure are just the Δ L , Δ R , and Δ defined in section Benchmark of Hierarchical Equations of Motion for QUantum Impurity with a Correlated Kernel ( HEOM‐QUICK ), and that the μ L and Δ L in the two panels, respectively, are just the amplitude V of the step‐function voltage defined in section Benchmark of HEOM‐QUICK . (Reprinted with permission from Ref . Copyright 2005 American Physical Society; Ref . Copyright 2008 American Institute of Physics).
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Calculated (a) time‐dependent current I(t), and (b) the corresponding dynamic I‐V characteristics. Results obtained at different truncation levels L are shown for a direct comparison. The parameters adopted are W = 20, T = 0.2, ɛ d  = − U/2 = 6, V 0 = 1.5, and Ω = 0.3 (in units of Δ). (Reprinted with permission from Ref . Copyright 2013 American Physical Society).
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The spin‐up or down spectral function of an asymmetric single‐orbital Anderson impurity model (AIM) at different truncation tiers. The inset magnifies the Kondo resonance peak at ω = 0. The parameters adopted are ɛ d  = − 5, U = 15, W = 10, and T = 0.075 (in units of Δ). The four solid lines from the top down at ω = 0 correspond to L = 2, 3, 4 and 1, respectively. (Reprinted with permission from Ref . Copyright 2012 American Physical Society).
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Schematic diagram of a single‐orbital Anderson impurity model. The impurity consists of a single orbital in contact with two reservoirs (α = L and R). μ α is the chemical potential of α‐reservoir, ɛ d is the orbital energy, and U is the Coulomb interaction strength, respectively.
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The schematic diagram for the structure of the Hierarchical Equations of Motion (HEOM). The structure of the hierarchy is characterized by the parameters M in the horizontal direction and L in the vertical direction, respectively. Here, M is the number of the exponential functions to expand the reservoir correlation functions, and L denotes the truncation tier, which is determined by the e‐e interaction strength.
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(a) Spectral functions A(ω) of the two‐orbital Anderson impurity model (AIM) at various magnetic anisotropy D. (b) Differential conductance dI/dV as a function of voltage with different D. The parameters adopted here are (in unit of Δ): ɛ 1 = − 13.5, ɛ 2 = − 11.5,U 1 = 25, U 2 = 24.1, T = 0.2, and W = 49. (Reprinted with permission from Ref . Copyright 2016 American Institute of Physics).
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(a) The structure of the Au–Co(tpy‐S)2–Au junction. The blue arrows depict the stretching direction. The spin‐polarization projected density of states (PDOS) of (b)S = 1 (ground) state and (c) S = 1/2 (a local minimum) state of the junction. The spin‐polarization energy diagrams of (d) S = 1 and (e)S = 1/2 state for the cobalt 3d orbitals. A hollow arrow indicates that the orbital is fractionally occupied. (Reprinted with permission from Ref . Copyright 2016 American Institute of Physics).
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Differential conductance (dI /dV) versus bias voltage (V) at different temperatures. The scattered data are obtained by the Hierarchical Equations of Motion (HEOM) calculations, and the lines are least‐square fits to the Fano function. The parameters are (in unit of eV): ɛ d  = − 3.67, U = 5.18, Δ t  = 0.01, Δ s  = 0.4, W t  = 5.5, W s  = 5.5, and μ t  − μ s  = V. Here, s and t denote the substrate and the STM tip, respectively. (Reprinted with permission from Ref . Copyright 2014 American Institute of Physics).
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(a) Spin density distribution of the dCoPc/Au(111) adsorption system, almost located in transitional metal. (b) Spin‐resolved projected density of states (PDOS) of the Co 3d orbitals, zero energy is set as Fermi energy E F . (c) The spin‐resolved energy diagrams of Co 3d orbitals. Orbital d z 2 is the only single‐occupied orbital. (Reprinted with permission from Ref . Copyright 2014 American Institute of Physics).
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The spectral function of hypercubic and the Bethe lattices at finite temperature, confront with different Coulomb interaction. The temperature is T = 0.0125W and W is the effective bandwidth. Both of them experience a metal‐insulator transition. (Reprinted with permission from Ref . Copyright 2014 American Physical Society).
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