Deriving the one-electron Spectral Function for the 1D Hubbard Model

Abstract

This pre-print deals with the one dimensional Hubbard model, as described by the Pseudofermion Dynamical Theory (PDT), with the purpose of (1) deriving a novel expression for the one electron spectral function for all values of the on-site repulsion $U/t$ and filling $n \in (0,1)$, at vanishing magnetisation $m \rightarrow 0$, and (2) discover how to correctly compare the results originating from two different theoretical frameworks in the $U/t \rightarrow \infty$ limit, as a first-test of the novel expressions obtained in this paper. Thus, an exact expression of the spectral function is obtained, which is furthermore successfully compared with previously known results in the $U \rightarrow \infty$ limit. Following the PDT, the expression for the one electron spectral function factorises into a spin part and a charge part for all values of the on-site repulsion $U/t$, where the dynamical quantum objects are spin zero and $η$-spin (charge) zero singlet pairs of so-called rotated electrons, which in turn are obtained from the original electrons by a unitary transformation. The spectral function is exemplified for $U/t = 400$, with the purpose of comparing it with the same function obtained by other authors (and other means) in the $U \rightarrow \infty$ limit. The main pillars of the PDT is presented in a summarised form. For example, we will only be interested in excited energy eigenstates which originate the most significant singular features of the spectral map in the $(k,ω)$ plane, safely ignoring higher order contributions. Even though emphasis is given on step-by-step derivations where necessary, derivations that have been done elsewhere and/or do not notably contribute to the physical understanding, are sometimes avoided. Therefore, references for further study are given throughout the paper.