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arXiv:2511.02039 · Physical review. B./Physical review. B

Marginal Fermi liquids from Fermi surfaces coupled via matrix boson gas

Vibhu Mishra

Published Feb 11, 2026• 0 citations
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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 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

DOI
10.1103/1pyq-7cpy

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The problem of continuous quantum phase transitions in metals involves critical bosons coupled to a Fermi surface. We solve the theory in the limit of a large number, NB{N}_{B}NB​, of bosonic flavors, where the bosons transform in the adjoint representation (a matrix representation), while the fermions are in the fundamental representation (a vector representation) of a global SU(NB)SU({N}_{B})SU(NB​) flavor symmetry group. The leading large NB{N}_{B}NB​ solution corresponds to a non-Fermi liquid coupled to Wilson-Fisher bosons. In a certain energy range, the fermion velocity vanishes---resulting in the destruction of the Fermi surface. Subleading 1/NB1/{N}_{B}1/NB​ corrections correspond to a qualitatively different form of Landau damping of the bosonic critical fluctuations. We discuss the model in $d=3\ensuremath{-}\ensuremath{\epsilon}$ but because of the additional control afforded by large NB{N}_{B}NB​, our results are valid down to d=2d=2d=2. In the limit $\ensuremath{\epsilon}\ensuremath{\ll}1$, the large NB{N}_{B}NB​ solution is consistent with the renormalization group analysis of Fitzpatrick et al. [Phys. Rev. B 88, 125116 (2013)].
We study the fermion-boson coupled system in d=3−ϵ space dimensions near the quantum phase transition; infinite many boson modes locating on a sphere become critical simultaneously, which is dubbed “critical boson surface” (CBS). The fermions on the Fermi surface can be scattered to nearby points locating on a boson ring in the low-energy limit. The large number of the boson scattering channels N renders the well-known Landau damping effect largely suppressed. We propose an effective theory for a single point on the Fermi surface and the associated critical boson ring induced by the boson scattering channels. Based on the effective model, one-loop renormalization group analysis is performed with asymptotic ϵ-expansion. The fermion self-energy and Yukawa interaction vertex are dressed with ϵ poles and are largely enhanced due to the presence of critical boson ring. The imaginary part of the self-energy exhibits a linear-in frequency dependence and the real part gives a vanishing quasiparticle weight in the low-energy limit, which signatures the celebrated “marginal Fermi liquid” behavior. We find a novel non-Fermi liquid fixed point, at which the critical properties show features associated with the CBS.
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We study the problem of disorder-free metals in the vicinity of a quantum critical point in d=3+1d=3+1d=3+1 dimensions. We begin with perturbation theory in the Yukawa coupling between the electrons and undamped bosons (order-parameter fluctuations) and show that the perturbation expansion breaks down below energy scales where the bosons get substantially Landau damped. Above this scale, however, we find a regime in which low-energy fermions obtain an imaginary self-energy that varies linearly with frequency, realizing the ``marginal Fermi liquid'' phenomenology [Varma et al., Phys. Rev. Lett. 63, 1996 (1989)]. We discuss a large-NNN theory in which the marginal Fermi liquid behavior is enhanced while the role of Landau damping is suppressed, and show that quasiparticles obtain a decay rate parametrically larger than their energy.