  Expansion many-body methods correspond to solving complex tensor networks.
The (iterative) solving of the network and the (repeated) storage of the
unknown tensors requires a computing power growing polynomially with the size
of basis of the one-body Hilbert space one is working with. Thanks to current
computer capabilities, ab initio calculations of nuclei up to mass $A\sim100$
delivering a few percent accuracy are routinely feasible today. However, the
runtime and memory costs become quickly prohibitive as one attempts (possibly
at the same time) (i) to reach out to heavier nuclei, (ii) to employ
symmetry-breaking reference states to access open-shell nuclei and (iii) to aim
for yet a greater accuracy. The challenge is particularly exacerbated for
non-perturbative methods involving the repeated storage of (high-rank) tensors
obtained via iterative solutions of non-linear equations. The present work
addresses the formal and numerical implementations of so-called importance
truncation (IT) techniques within the frame of one particular non-perturbative
expansion method, i.e., Gorkov Self-Consistent Green's Function (GSCGF) theory,
with the goal to eventually overcome above-mentioned limitations. By a priori
truncating irrelevant tensor entries, IT techniques are shown to reduce the
storage to less than 1% of its original cost in realistic GSCGF calculations
performed at the ADC(2) level while maintaining a 1% accuracy on the
correlation energy. The future steps will be to extend the present development
to the next, e.g., ADC(3), truncation level and to SCGF calculations applicable
to doubly open-shell nuclei.
