Metallic Non-Fermi Liquids and Quasiparticle Breakdown

A metallic non-Fermi liquid lacks Landau quasiparticles on at least a specified part of its Fermi surface while remaining compressible and gapless. The diagnosis is analytic: it concerns the pole residue and decay rate of the retarded Green function. Unusual resistivity or heat capacity can support that diagnosis but cannot replace it, because transport and one-particle relaxation weight scattering differently.

Required background. Patch Renormalization and Competing Instabilities supplies local Fermi-surface scaling; Relevant, Marginal, and Irrelevant Directions supplies RG stability; Vector Models, Auxiliary Fields, and Large-N Saddles supplies controlled flavor limits. Helpful background. Gauge Fields, Redundancy, and Observable Content supplies gauge-invariant interpretation when the fermions are emergent.

The quasiparticle criterion

Near a Fermi momentum, write

$$G^R(\mathbf k,\omega)= \frac1{\omega-v_Fk_\perp-\Sigma^R(\mathbf k,\omega)}.$$

If the self-energy is sufficiently regular, the pole residue and width are

$$Z=\left[1-\partial_\omega\operatorname{Re}\Sigma^R (\mathbf k_F,\omega)\big|_{\omega=0}\right]^{-1}, \qquad \Gamma(\omega)=-Z\,\operatorname{Im}\Sigma^R(\mathbf k_F,\omega).$$

A Landau quasiparticle requires finite $Z$ and

$$\frac{\Gamma(\omega)}{|\omega|}\longrightarrow0 \quad\text{as}\quad \omega\to0.$$

For a Fermi liquid, $\operatorname{Im}\Sigma^R\sim-(\omega^2+\pi^2T^2)$ up to logarithms in special dimensions. A causal retarded self-energy with

$$-\operatorname{Im}\Sigma^R(\omega) \sim\lambda|\omega|^\alpha, \qquad \lambda>0,\quad 0<\alpha<1,$$

dominates the bare $\omega$ and removes a finite-residue pole. Kramers–Kronig relations fix the associated real part once the ultraviolet completion and particle–hole asymmetry are specified. The sign $\operatorname{Im}\Sigma^R\le0$ is the causality check that keeps $A=-2\operatorname{Im}G^R$ nonnegative. At $\alpha=1$, logarithms and the ratio $\Gamma/|\omega|$ distinguish marginal behavior from a sharp quasiparticle.

The spectral convention is

$$A(\mathbf k,\omega)=-2\operatorname{Im}G^R(\mathbf k,\omega), \qquad \int\frac{d\omega}{2\pi}A(\mathbf k,\omega)=1.$$

Broad low-energy weight is meaningful only after instrumental resolution, thermal broadening, disorder, and multiple bands are separated.

Mechanisms and momentum selectivity

At an antiferromagnetic critical point, isolated hot spots can lose coherence while cold portions retain quasiparticles. An Ising-nematic order parameter or transverse gauge field couples to small momentum transfers and can affect an extended Fermi surface. In a common two-dimensional patch theory the leading self-energy scales as $|\omega|^{2/3}$, but the coefficient, sign structure, and control depend on the coupled boson.

Gauge-coupled fermions illustrate why a formal large-$N$ expansion can be nonuniform: nominally higher-loop graphs acquire extra infrared singularities Lee 2009, §§ II–V. Ising-nematic patch theory similarly contains singular higher-loop structure and enhanced pairing Metlitski and Sachdev 2010, §§ III–VI. A lattice Monte Carlo realization can establish non-Fermi-liquid behavior for its model without making every continuum extrapolation identical; a two-dimensional ferromagnetic critical model provides one such primary example Xu et al. 2017, pp. 031058-1–031058-14.

A Fermi surface without quasiparticles

The absence of a pole does not necessarily erase the Fermi surface. A critical Fermi surface can be defined by a singularity or sign change of $G(\mathbf k,0)$ on a codimension-one locus, with scale-invariant spectral weight instead of a delta-function pole. For gauge-charged partons, the parton Green function is not itself gauge invariant; physical thermodynamics and composite response must be used.

Compressibility, quantum oscillations, momentum-space singularities, and Luttinger constraints probe different aspects. Quantum oscillations can coexist with strong scattering over a window and do not alone establish zero-temperature quasiparticles.

One-particle decay is not dc transport

Small-angle scattering can strongly broaden a spectral function while relaxing little electrical current. In a translation-invariant continuum, interactions can conserve total momentum and leave a Drude delta function even with no quasiparticles. A finite dc resistivity requires momentum relaxation through a lattice umklapp process, disorder, phonons, boundaries, or coupling to another sector.

Thus a self-energy exponent cannot be inserted directly into a Drude rate without vertex and momentum-relaxation analysis. Optical conductivity, thermal transport, Hall response, and the one-particle spectrum should be calculated within one declared order of limits.

Current evidence boundary

Primary theory and model evidence were checked through 10 August 2026. They establish several mechanisms and controlled or sign-problem-free realizations of quasiparticle breakdown, but do not make “non-Fermi liquid” a unique microscopic model or equate every anomalous metal with one universal exponent. Ongoing model-to-material work belongs in Quantum Matter and Emergence Research.

Exercises

1. Classify $\operatorname{Im}\Sigma^R=-C|\omega|^{3/2}$ with finite $Z$.

Solution

$\Gamma/|\omega|\propto|\omega|^{1/2}\to0$, so the excitation is asymptotically sharp. The decay is nonanalytic but still compatible with a quasiparticle.

2. Why can forward scattering give a large spectral width but small resistivity?

Solution

The one-particle lifetime counts scattering events, whereas current relaxation weights their momentum change. For angle $\theta\ll1$, the transport factor $1-\cos\theta\simeq\theta^2/2$ suppresses forward-scattering contributions.

References

• Lee, S.-S. “Low-Energy Effective Theory of Fermi Surface Coupled with U(1) Gauge Field in $2+1$ Dimensions.” Physical Review B 80 (2009): 165102. DOI.
• Metlitski, M. A., and S. Sachdev. “Quantum Phase Transitions of Metals in Two Spatial Dimensions. I. Ising-Nematic Order.” Physical Review B 82 (2010): 075127. DOI.
• Xu, X. Y., K. Sun, Y. Schattner, E. Berg, and Z. Y. Meng. “Non-Fermi Liquid at $(2+1)d$ Ferromagnetic Quantum Critical Point.” Physical Review X 7 (2017): 031058. DOI.