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Monte Carlo Second- and Third-Order Many-Body Green’s Function Methods with Frequency-Dependent, Nondiagonal Self-Energy
journal contribution
posted on 2019-10-17, 15:43 authored by Alexander
E. Doran, So HirataWe fully develop the Monte Carlo
many-body Green’s function
(MC-GF) method with the following enhancements: (1) The truncation
order of the perturbation expansion of the Dyson self-energy is raised
from the second order (MC-GF2) to the third order (MC-GF3) with the
aid of a computerized procedure to enumerate and transform all 84
third-order Goldstone diagrams into Monte Carlo integrable expressions
and then into central processing unit (CPU)/graphical processing unit
(GPU)-parallel computer codes. (2) An efficient algorithm is proposed
that computes all off-diagonal and diagonal elements of the MC-GF2
and MC-GF3 self-energy matrices by common subexpression elimination.
(3) The frequency-independent approximation is lifted by introducing
a method that computes frequency derivatives of the MC-GF2 and MC-GF3
self-energies up to any arbitrarily high order at nearly no additional
computational cost. (4) The imaginary-time integration in the Laplace-transformed
expressions of the self-energy is carried out stochastically (instead
of using a quadrature in the previous implementations), resulting
in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker
convergence acceleration scheme is analyzed numerically, and the guidelines
are established to select an optimal number of walkers for maximal
efficiency. When such an optimal number is used, the cost per sample
is constant of molecular size on either many CPUs or many GPUs. (6)
The computational cost to obtain a binding energy within a given statistical
uncertainty is observed to increase as (tentatively) O(n4) and O(n5) of molecular
size (n is the number of electrons) with and without
the diagonal approximation, respectively, as opposed to O(n5) and O(n6) of
the corresponding deterministic algorithms. With this method applied
to the electron binding energies of C60, we show that the
third-order corrections to the self-energies are much greater in electron
binding energies than in ground-state energies. They display a sign
of oscillatory convergence toward experimental results, not necessarily
improving the agreement with increasing perturbation order, justifying
MC-GF3 and motivating even higher-order methods.