Marginal Fermi liquids from Fermi surfaces coupled via matrix boson gas

We propose a model of metallic critical point which we study at T = 0 in the large- N limit. We start with two species of fermions u i , d i , each with N flavors and a gas of matrix bosons b i j with N 2 components. The fermions interact with each other via the intermediate boson as ∫ b i j † u i † d j . The bosons have a bare dispersion ɛ q b = λ z | q | z and we study the problem in d spatial dimensions. We show that for d = z + 1 , the electronic self-energy shows marginal Fermi liquid behavior. We first evaluate the fermionic self-energy Σ ( i ω ) using the standard approximate boson self-energy Π ( q , i ν ) ∝ | ν | / | q | and find that Σ ( i ω ) ∼ ω ln ( N / | ω | ) which shows a much weaker dependence on N when compared with similar results from non-SYK large- N Ising-nematic models. Then we evaluate Σ ( i ω ) again using a more precise form of Π ( q , i ν ) which allows us to study the interplay between N → ∞ limit for which Σ ( i ω ) ∼ ω ln ( 1 / | ω | ) , and the ω → 0 limit where we recover