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

TL;DR

This paper explores the electronic properties of boson modes in a three-point fermion loop, revealing how external potentials influence single-particle excitations and many-body effects, with implications for SYK physics.

Contribution

It introduces a detailed analysis of boson modes in fermion loops, connecting local moment sum rules with fermionic self-energy behavior in different limits.

Findings

In the UV limit, the fermion self-energy becomes momentum- and frequency-independent.

The correlator $raket{ riangle^{ odag ext{ extdagger}} riangle}$ is positive, consistent with sum rule constraints.

The framework links bosonic fluctuations to emergent SYK-like physics.

Abstract

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.