On the exact truncation tier of fermionic hierarchical equations of motion

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 + (L) over tilde) th-tier truncation with (L) over tilde = 2N(sigma)N(nu) yields the exact density operators up to the nth tier. Here, N-sigma = 2 for fermionic systems and N-nu is the system degrees of freedom. For noninteracting systems, (L) over tilde 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 (L) over tilde emphasizes the nonperturbative nature of the HEOM theory. We also propose a simplified HEOM approach to further reduce the memory cost for practical calculations. Published by AIP Publishing.