Recent advances in fermionic hierarchical equations of motion method for strongly correlated quantum impurity systems

Figure  1.  Schematic diagram for the hierarchal structure of the HEOM. Each green discs represents a density operator. $ \rho^{(0)} $ is the reduced-system density operator and $ \rho^{(n>0)} $ is the $ n $th-tier ADOs. Purple double-arrows denote the couplings between two density operators. The size of the hierarchy space is set by the horizontal M and vertical L dimensional parameters. M is the number of basis functions to unravel the environmental memory and L denotes the truncation tier. Reprinted with permission from Ref. [114]. Copyright 2018, American Institute of Physics.

Figure  2.  (a) The expansion of the Fermi function based on the PSD scheme (green/blue dotted line), the FSD sheme (yellow/red solid line), and the exact results (empty triangles). (b) The low-temperature correction part in the FSD scheme (red solid line) expanded by the Fano functions and the numerical error (blue solid line). The original and reference temperatures are in unit of $ \Gamma $: $ T = 0.001 $, $ T_0 = 0.1 $. Reprinted with permission from Ref. [98]. Copyright 2019, American Institute of Physics.

Figure  3.  Impurity spectral function $ A(\omega) $ of an SIAM at different truncation tiers $ L $. The scattered data are obtained by the HEOM calculations with zero-value terminator and adiabatic terminator, respectively. Parameters here are all in unit of $ \Gamma $: $ T = 0.04 $, $\epsilon_{\uparrow} = \epsilon_{\downarrow} = -U/2 = -4$, and $ W = 20 $. The inset shows the Kondo resonance peak in the vicinity of $ \omega = 0 $. Reprinted with permission from Ref. [100]. Copyright 2021, American Institute of Physics.

Figure  4.  (a) The time evolution of ac voltage-driven electric current. The arrows represent the time of divergence start. The parameters are (in units of $ \Delta $): $ U = -2\epsilon_d = 12 $, $ \Delta_L = \Delta_R = 0.5 $, $ T = 0.05 $, $ W = 20 $, $ eV_0 = 1.5 $, and $ \omega = 0.3 $. $ \chi $ represents different parameter sets adopted for the FSD scheme, among which $ \chi = 1000 $ (marked by the star in the legend) is the most accurate. Reprinted with permission from Ref. [22]. Copyright 2020, American Institute of Physics. (b) The variation of real-time electric current $ I_L(t) $ driven by ac voltage, flowing from impurity to left reservoir. The results are calculated by HEOM-adia and HEOM-zero at different truncation tier $L$, respectively. The arrows mark the time from which the results start to diverge. The inset shows the long-time evolution of $ I_L(t) $ calculated by HEOM-adia truncated at $ L $ = 4. The parameters adopted are all in unit of $ \Gamma $: $ V_0 = 3.0 $, $ \omega_0 = 1.2 $, $ T = 0.1 $, $\epsilon_{\uparrow} = \epsilon_{\downarrow} = -U/2 = -12$, and $ W = 40 $. Reprinted with permission from Ref. [100]. Copyright 2021, American Institute of Physics.

Figure  5.  The $ dI/dV $ spectra calculated by the DFT+HEOM method for the (a) FePc/FePc/Pb(111) and (b) FePc/ CoPc/Pb(111) composites at temperature T = 60 K. Reprinted with permission from Ref. [140]. Copyright 2018, Royal Society of Chemistry.

Figure  6.  The differential conductance $ dI/dV $ spectra for the SIAM corresponding to the tip/FeOEP/Pb(111) composite with different tip displacement (a) $\Delta z = -350\;{\rm{pm}}$, (b) $ \Delta z = -450\;{\rm{pm}} $. The inset of (b) shows the schematic diagram for the energy levels of local spin states. Reprinted with permission from Ref. [81]. Copyright 2018, American Chemical Society.

Figure  7.  (a) Calculated and (b) experimentally measured $ dI/dV $ spectra at several tip displacements $ z $. (c) The splitting width $ \Delta V $ between two asymmetric peaks verus the difference of hybridization strengths $ \Delta E $. The gray line gives a linearly fitting result. Reprinted with permission from Ref. [82]. Copyright 2022, American Chemical Society.

Figure  8.  The distribution of local temperature $ T^{\ast}_i $ along a linear chain consisting of four impurities based on the ZCC and the LMPC under a thermal bias with (a) a strong impurity-lead coupling strength $ \Delta_\alpha = 0.5\Delta \,(\alpha = L, R) $ and (b) a weak impurity-lead coupling strength $ \Delta_\alpha = 0.0125\Delta $. $ t $ represents the coupling strength between two adjacent impurities. The inset shows that the impurity spectral function $ A(\omega) $ calculated by the HEOM method at different $ \Delta_\alpha $ and $ t $. Reprinted with permission from Ref. [25]. Copyright 2021, American Physical Society.