Marginal Fermi liquids from Fermi surfaces coupled via matrix boson gas

Abstract

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 ${\epsilon }_{\text{q}}^b={\lambda }_z|\text{q}{|}^z$ 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 $\Pi (\text{q},i\nu )\propto |\nu |/|\text{q}|$ 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 $\Pi (\text{q},i\nu )$ which allows us to study the interplay between $N\to \infty$ limit for which $\Sigma (i\omega )\sim \omega \ln (1/|\omega |)$, and the $\omega \to 0$ 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 $T\to 0$. This suggests that coupling the hybridization operator $u^{†}d$ to non-relativistic bosons for $d=3$ and relativistic bosons for $d=2$ provides a simple route to marginal Fermi liquid scaling.

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