On the exact truncation tier of fermionic hierarchical equations of motion

Abstract

The hierarchical equations of motion (HEOM) theory is in principle exact for describing the dissipative dynamics of quantum systems linearly coupled to Gaussian environments. In practice, the hierarchy needs to be truncated at a finite tier. We demonstrate that, for general systems described by the fermionic HEOM, the (n+L̃)th-tier truncation with L̃=2NN yields the exact density operators up to the nth tier. Here, N = 2 for fermionic systems and N is the system degrees of freedom. For noninteracting systems, L̃ is further reduced by half. Such an exact termination pattern originates from the Pauli exclusion principle for fermions, and it holds true regardless of the system-environment coupling strength, the number of coupling reservoirs, or the specific scheme employed to unravel the environment memory contents. The relatively small L̃ emphasizes the nonperturbative nature of the HEOM theory. We also propose a simplified HEOM approach to further reduce the memory cost for practical calculations.

Effect of quantum resonances on local temperature in nonequilibrium open systems

“…The HEOM theory is, in principle, formally exact, and its numerical outcomes are guaranteed to be quantitatively accurate if the results converge with respect to the truncation tier of the hierarchy. 71,72 The HEOM approach has been widely used to study a variety of static and dynamic properties of strongly correlated quantum impurity systems in and out of equilibrium. 62,63,[73][74][75][76][77][78][79] In the framework of the HEOM, the influence of the noninteracting leads on the impurity is fully captured by the hybridization functions, Γ α (ω) ≡ π k |t αk | 2 δ(ω − ǫ αk ).…”

Speeding up quantum dissipative dynamics of open systems with kernel methods

“…An alternative numerical exact bench-marking approach is the hierarchical equations of motion (HEOM) [44][45][46][47][48][49][50][51][52][53] approach pioneered by Tanimura and Kubo. 54 HEOM approach captures the combined effects of system-environment dissipation, non-Markovian memory effect, and many-body correlation in a non-perturbative manner.…”

Speeding up quantum dissipative dynamics of open systems with kernel methods

“…An alternative numerical exact bench-marking approach is the hierarchical equations of motion (HEOM) [44][45][46][47][48][49][50][51][52][53] approach pioneered by Tanimura and Kubo. 54 HEOM approach captures the combined effects of system-environment dissipation, non-Markovian memory effect, and many-body correlation in a non-perturbative manner.…”