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 $\int b_{ij}^{†}{u}_i^{†}{d}_j$. The bosons have a bare dispersion ${\varepsilon }_{\mathbf{q}}^b={\lambda }_z{\left|\mathbf{q}\right|}^z$ 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 $\mathrm{\Sigma }\left(i\omega \right)$ using the standard approximate boson self-energy $\mathrm{\Pi }(\mathbf{q},i\nu )\propto \left|\nu \right|/\left|\mathbf{q}\right|$ and find that $\mathrm{\Sigma }\left(i\omega \right)\sim \omega ln(N/|\omega \left|\right)$ which shows a much weaker dependence on $N$ when compared with similar results from non-SYK large-$N$ Ising-nematic models. Then we evaluate $\mathrm{\Sigma }\left(i\omega \right)$ again using a more precise form of $\mathrm{\Pi }(\mathbf{q},i\nu )$ which allows us to study the interplay between $N\to \infty$ limit for which $\mathrm{\Sigma }\left(i\omega \right)\sim \omega ln(1/|\omega \left|\right)$, and the $\omega \to 0$ limit where we recover $\mathrm{\Sigma }\left(i\omega \right)\sim \omega ln(N/|\omega \left|\right)$. 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 $NVTln(N/T)$ and $NVTln(1/T)$, respectively, for $T\to 0$. This suggests that coupling the hybridization operator $u^{†}d$ to nonrelativistic bosons for $d=3$ and relativistic bosons for $d=2$ provides a simple route to marginal Fermi liquid scaling.

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• Electron-correlation calculations
• Green's function methods
• Large-N expansion in field theory
• Sachdev-Ye-Kitaev model
• Slave bosons