Splicing an exact local self-energy into $GW$: tests on small Hubbard clusters

Abstract

$GW$+DMFT-type schemes replace the local part of a $GW$ self-energy by a separately computed local self-energy. A causal double counting removes a non-causal term from the hybridization (Lee and Haule, 2017), but the resulting total self-energy has been flagged as not guaranteed to be causal (Zhu and Chan, 2021). From the Herglotz representation, the spliced self-energy is causal if and only if its spectral measure is positive semi-definite. For translation-invariant single-band systems this becomes a momentum-resolved bound: the local weight may fall short of the local $GW$ weight by at most the momentum minimum of the $GW$ weight. On finite clusters, a mismatch between $GW$ and exact poles generically produces indefinite residues. On the Matsubara axis, a negative damping eigenvalue at one frequency certifies non-causality; passing does not establish causality. We examine these diagnostics in an idealized setting, not self-consistent $GW$+DMFT: one-shot $GW$ on the exact Green's function of small Hubbard clusters with the exact local self-energy inserted. At all 528 parameter sets, our computed $GW$ and exact self-energies pass both tests, as expected. The splice fails the pointwise test at 54 points, none at half filling, all at the lowest Matsubara frequency, and already at $U/t=0.5$; there the local $GW$ damping is 1.3-9 times the exact one. A block Pick test on eight frequencies, run with a second imaginary-time code, flags 481 of the 528 sets; for the other 47 it is inconclusive. Moment checks miss all of these, because the spliced high-frequency moment is exact. The Dyson Green's function is then non-causal too, yet passes the pointwise test everywhere tested. This version replaces a withdrawn one whose reported causality violations of $GW$ itself came from a sign error in the polarization; with the sign corrected, the computed $GW$ shows the expected positivity.

arXiv commentReplaces the withdrawn version 1; Version 1 reported causality violations of GW on Hubbard clusters; these came from a sign error in the polarization of the accompanying code and are withdrawn. The paper now examines the spliced GW + exact-local self-energy on small clusters. 13 pages, 3 figures, 4 tables