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 This version replaces a withdrawn one whose reported causality violations of GW itself came from a sign error in the polarization; with the sign corrected, the computed GW shows the expected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.

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Cite

BibTeX RIS

BibTeX

@misc{atambo2026splicing,
  title = {{Splicing an exact local self-energy into $GW$: tests on small Hubbard clusters}},
  author = {Michael O. Atambo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00324},
  eprint = {2610.00324},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00324},
}

RIS

TY  - UNPB
TI  - Splicing an exact local self-energy into GW: tests on small Hubbard clusters
AU  - Michael O. Atambo
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.00324
UR  - https://arxiv.org/abs/2610.00324
AB  - 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.
ER  -

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