Electronic properties of the boson mode in a three-point fermion loop and the emergent SYK physics

We investigate the electronic properties of the boson mode in a three-point fermion loop. In this framwork, the single-particle excitation and the many-body local (in imaginary time and momentum space) field effects are investigated in IR or UV limits with the density fluctuation induced by external potential (or bosonic frequency). While in the limit of vanishing effect of external potential, which is equivalent to UV limit of the fermionic frequency, the conserving approximation can be applied together with the Luttinger-Ward analysis, in which case the anomalous contribution to the fermion self-energy or the expectation value of many-body interaction term, which is $g^{2}\langle \Delta^{\dag}\Delta\rangle$ ($g$ is the irreducible particle-particle vertex and $\Delta$ is the boson field operator), vanishes, and results in a Hartree-Fock type momentum- and frequency-independent fermion self-energy. The correlator $\langle\Delta^{\dag}\Delta\rangle$ is positive which can be obtained through the local moment sum rule of dynamical susceptibility.