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 $b_{ij}$ with $N^2$ components. The fermions interact with each other via the intermediate boson as $\int b_{ij}^{\dagger} \, u_i^{\dagger}d_j$. The bosons have a bare dispersion $\varepsilon_{\textbf{q}}^b = \lambda_z |\textbf{q}|^z$ 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 $\Sigma(i\omega)$ using the standard approximate boson self energy $\Pi(\textbf{q}, i\nu) \propto |\nu|/|\textbf{q}|$ and find that $\Sigma(i\omega) \sim \omega \ln(N/|\omega|)$ which shows a much weaker dependence on $N$ when compared with similar results from non-SYK large-$N$ Ising-nematic models. Then we evaluate $\Sigma(i\omega)$ again using a more precise form of $\Pi(\textbf{q}, i\nu)$ which allows us to study the interplay between $N \rightarrow \infty$ limit for which $\Sigma(i\omega) \sim \omega \ln(1/|\omega|)$, and the $\omega \rightarrow 0$ limit where we recover $\Sigma(i\omega) \sim \omega \ln(N/|\omega|)$. We also use the full bosonic self energy to obtain the correction to the bosonic specific heat as $\frac{T}{N} \ln(1/T)$. Since there are $N^2$ bosons and $N$ fermions, the bulk heat capacity for both fermions and bosons shows nearly identical functional form $NVT \ln(N/T)$ and $NVT \ln(1/T)$ respectively for $T \rightarrow 0$. This suggests that coupling the hybridization operator $u^{\dagger}d$ to non-relativistic bosons for $d=3$ and relativistic bosons for $d=2$ provides a simple route to marginal Fermi liquid scaling.