Highly efficient and accurate sum-over-poles expansion of Fermi and Bose functions at near zero temperatures: Fano spectrum decomposition scheme.

The Fano spectrum decomposition (FSD) scheme is proposed as an efficient and accurate sum-over-poles expansion of Fermi and Bose functions at cryogenic temperatures. The new method practically overcomes the discontinuity of Fermi and Bose functions near zero temperature, which causes slow convergence in conventional schemes such as the state-of-the-art Padé spectrum decomposition (PSD). The FSD scheme fragments Fermi or Bose function into a high-temperature reference and a low-temperature correction. While the former is efficiently decomposed via the standard PSD, the latter can be accurately described by several modified Fano functions. The resulting FSD scheme is found to converge overwhelmingly faster than the standard PSD method. Remarkably, the low-temperature correction supports further a recursive and scalable extension to access the near-zero temperature regime. Thus, the proposed FSD scheme, which obeys rather simple recursive relations, has a great value in efficient numerical evaluations of Fermi or Bose function-involved integrals for various low-temperature condensed physics formulations and problems. For numerical demonstrations, we exemplify FSD for the efficient unraveling of fermionic reservoir correlation functions and the exact hierarchical equations of motion simulations of spin-boson dynamics, both at extremely low temperatures.