@article{Application:6413,
      recid = {6413},
      author = {Pandey, Bipul},
      title = {Power Series Correction to Single Particle Electron  Green's Function: Application to 1D Holstein Chain},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2023-06},
      abstract = {Electrons in materials undergo numerous complex  interactions among themselves, the external fields, as well  as the constituent atomic lattice. The strength of such  many-body interactions depends on various factors such as  the electronic configuration of the host material, the  presence of doping and defects, spins of carrier and  lattice elements, etc. In the thermodynamic limit, these  interactions are often treated as bosons that interact with  the electrons in the system and manifest as side bands  (replicas or satellites) in the electronic band structure  seeping spectral weight and renormalizing the band  structure obtained from purely electronic calculations.

In  ab-initio calculations, when the strength of such  electron-boson interaction is weak, it is not only  justified to neglect these interactions completely but also  pragmatic for reasons ranging from tractability to  associated computational cost. This is because the effect  of electron-boson interaction is minute compared to the  electronic energy scale of the problem. However, in many  systems, especially organic semiconducting materials, the  bosonic vibrations (stretching modes) of the molecule are  strongly coupled with the electron. Furthermore, the  bosonic energy scale is comparable to the electronic energy  scale in these problems. Hence, neglecting the effect of  electron-boson interactions in electronic spectra in such  systems is myopic at best and catastrophic at worst.

In  the context of a single electron two orbital Holstein  system coupled to dispersionless bosons, we develop a  general method to correct single-particle Green’s function  and electronic spectral function using an integral power  series correction (iPSC) scheme. We then outline the  derivations of various flavors of cumulant approximation  through the iPSC scheme and explain the assumptions and  approximations behind them. Finally, we compute and compare  iPSC spectral function with cumulant and exact diagonalized  spectral functions and elucidate three regimes of this  problem - two that cumulant explains and one where cumulant  fails. We find that the exact and the iPSC spectral  functions match within spectral broadening across all three  regimes.

In order to scale our method to large systems, we  then develop an ODE-based Power series correction(dPSC)  formalism that goes beyond the cumulant approximation. We  implement it to a 1D Holstein chain for a wide range of  coupling strengths in a scalable and inexpensive fashion at  both zero and finite temperatures. We show that this first  differential formalism of the power series is qualitatively  and quantitatively in excellent agreement with exact  diagonalization results on the 1D Holstein chain with  dispersive bosons for a large range of electron-boson  coupling strength. We also investigate carrier mass growth  rate and carrier energy displacement across a wide range of  coupling strengths. We also present a faster second  differential formalism which is very much similar to  self-consistent cumulant formalism. We show the regime  where this method is applicable and where it diverges.  Finally, we present a heuristic argument that predicts most  of the rich satellite structure without explicit  calculation.},
      url = {http://knowledge.uchicago.edu/record/6413},
      doi = {https://doi.org/10.6082/uchicago.6413},
}