Marginal Fermi liquids from Fermi surfaces coupled via matrix boson gas

TL;DR

This work proposes a controlled large-$N$ model in which two fermion species couple to an $N^2$-component matrix boson via a slave-boson–type interaction, at zero temperature and with dispersion $ ext{ε}_q^b= ext{λ}_z|q|^z$. In the special case $d=z+1$, the electronic self energy exhibits marginal Fermi liquid behavior, with Σ(iω) scaling as $-i ext{ω}$ times a logarithm that depends on frequency and $N$, and a refined treatment reveals distinct limits: at the saddle point $N o\infty$ one obtains $ ext{Σ}(i ext{ω}) o -i ext{ω} ext{ln}(1/| ext{ω}|)$, while at small frequencies with finite $N$ one recovers $ ext{Σ}(i ext{ω}) o -i ext{ω} ext{ln}(N/| ext{ω}|)$. The bosonic self-energy and specific heats acquire characteristic logarithmic corrections, leading to a bulk thermodynamic form $C_V^{ ext{fermions}}\, ext{and}\, C_V^{ ext{bosons}} o ext{const} imes T ext{ln}( ext{Λ}/T)$ with distinct $N$-dependences. Overall, the paper demonstrates a tractable, flavor-conserving large-$N$ framework for quantum-critical metals with matrix bosons that yields marginal Fermi liquid physics and clarifies the interplay of $N$, frequency, and temperature in thermodynamics, while acknowledging limitations and potential disorder effects.

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 $c_i, f_i$, each with $N$ flavors and matrix bosons $b_{ij}$ with $N^2$ components. They interact with each other via slave-boson like interaction $\int b_{ij}^{\dagger} \, c_i^{\dagger}f_j$. The bosons have a bare dispersion of $\varepsilon_{\textbf{q}}^b = λ_z |\textbf{q}|^z$ 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 $Σ(iω)$ using the standard approximate boson self energy $Π(\textbf{q}, iν) \propto |ν|/|\textbf{q}|$ and find that $Σ(iω) \sim ω\ln(N/|ω|)$ which shows a much weaker dependence on $N$ when compared with similar results from non-SYK large-$N$ Ising-nematic models. Then we evaluate $Σ(iω)$ again using a more precise form of $Π$ which allows us to study the interplay between $N \rightarrow \infty$ limit for which $Σ(iω) \sim ω\ln(1/|ω|)$, and the $ω\rightarrow 0$ limit where we recover $Σ(iω) \sim ω\ln(N/|ω|)$. 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 show nearly similar functional form $NVT \ln(N/T)$ and $NVT \ln(1/T)$ respectively for $T \rightarrow 0$.