Spin-Fermion and Hot-Spot Theories

A spin-fermion theory keeps both the critical antiferromagnetic order parameter and the fermions near Fermi-surface points joined by the ordering wave vector. These hot spots invalidate a purely bosonic reduction: the boson is damped by the fermions while the same fluctuations destroy hot-spot quasiparticles. Controlled limits exist, but no single extrapolation automatically describes the physical two-dimensional, few-flavor problem.

Required background. Hertz–Millis Theory and Landau Damping supplies the finite-$\mathbf Q$ damping kernel; Patch Renormalization and Competing Instabilities supplies local Fermi-surface coordinates and competing channels. Helpful background. Vector Models, Auxiliary Fields, and Large-N Saddles supplies the logic and limitations of flavor expansions.

Hot-spot action and conventions

Let hot spots $\ell$ and $\bar\ell$ satisfy $\mathbf K_{\bar\ell}=\mathbf K_\ell+\mathbf Q$ modulo a reciprocal vector. Momenta $\mathbf k$ below are measured from $\mathbf K_\ell$, while the boson momentum $\mathbf q$ is measured from $\mathbf Q$. A minimal Euclidean action is

$$\begin{aligned} S={}&\sum_{\ell,\sigma}\int_k \psi_{\ell\sigma}^\dagger(k) (-i\omega+\mathbf v_\ell\cdot\mathbf k)\psi_{\ell\sigma}(k)\\ &+\frac12\int_q (r+q^2)\,\boldsymbol\phi(q)\cdot\boldsymbol\phi(-q)\\ &+g\sum_{\ell}\int_{k,q} \boldsymbol\phi(q)\cdot \psi_{\bar\ell\alpha}^\dagger(k+q) \boldsymbol\sigma_{\alpha\beta}\psi_{\ell\beta}(k)+\cdots . \end{aligned}$$

The displayed theory assumes a spin-density-wave vector, a finite angle between paired Fermi velocities, and no perfect nesting. Curvature is subleading for some hot-spot scaling questions but regularizes other singular processes and cannot be discarded globally.

Mutual boson–fermion singularities

The one-loop particle–hole bubble gives

$$D^{-1}(\mathbf q,i\Omega_n) \simeq r+q^2+\gamma|\Omega_n|,$$

with $\gamma$ set by $g^2$ and the cross product of the paired hot-spot velocities. Using this damped boson, a conventional two-dimensional one-loop calculation gives at a hot spot

$$\Sigma(i\omega)\sim i\,\operatorname{sgn}(\omega) \sqrt{\omega_0|\omega|},$$

within its stated Eliashberg-like window. Since $|\Sigma|/|\omega|$ diverges, the hot quasiparticle is destroyed. Away from a hot spot, a Fermi-liquid regime survives below a momentum-dependent crossover scale, so “the whole Fermi surface is hot” is not a consequence of this calculation.

The one-loop damping and self-energy estimates, including the kinematic window in which cold regions survive, are developed systematically by Abanov, Chubukov, and Schmalian 2003, §§ 2–4.

The same coupling enhances pairing and bond-order vertices. Metlitski and Sachdev found singular higher-loop structure and failure of a naive $1/N$ organization for the two-dimensional spin-density-wave metal Metlitski and Sachdev 2010, §§ III–VI.

What can control the theory

Several expansions reorganize the infrared:

• an $\epsilon=3-d$ expansion changes the co-dimension and tracks the Yukawa fixed point;
• a dynamically small velocity ratio can suppress classes of diagrams;
• large flavor number controls selected graphs but is not uniformly suppressive in the naive two-dimensional patch theory;
• small hot-spot angle or related kinematic limits open intermediate regimes;
• sign-problem-free lattice models test specific ultraviolet realizations but do not by themselves prove universality for all metals.

Schlief, Lunts, and Lee identified an emergent small velocity ratio in a particular $\epsilon$-organized antiferromagnetic critical metal and obtained exact limiting exponents Schlief, Lunts, and Lee 2017, pp. 021010-1–021010-15. Its assumptions should be quoted with its exponents; it is not interchangeable with a one-loop $z=2$ formula at physical velocities.

Observables and alternatives

The theory predicts anisotropic fermion self-energies, a Landau-damped spin response near $\mathbf Q$, and enhanced pairing or composite susceptibilities. Comparisons require the hot-spot locations and velocities from the same band structure, absolute momentum widths, and a temperature window above preempting superconductivity.

Broad self-energies can also arise from disorder, phonons, multiple unresolved bands, or another critical channel. A magnetic response peaked at $\mathbf Q$ does not prove that it dominates the electron decay. Strong evidence correlates the momentum dependence of the bosonic spectrum with the anisotropic fermion linewidth and tests whether one coupling scale accounts for both.

Current evidence boundary

This research-sensitive treatment was checked against primary theory literature available through 10 August 2026. The check supports several controlled limits and model-specific numerical realizations, but not a universal controlled solution for every physical $d=2$, few-flavor antiferromagnetic metal. New calculations, benchmark models, and material-level tests belong in Quantum Matter and Emergence Research rather than being folded into the settled derivation above.

Exercises

1. Why are hot spots discrete for a generic two-dimensional $\mathbf Q$?

Solution

They solve both $\varepsilon(\mathbf k)=0$ and $\varepsilon(\mathbf k+\mathbf Q)=0$. Two one-dimensional Fermi-surface curves generically intersect at isolated points. Perfect nesting can turn the intersection into an extended set and changes the theory.

2. Show that the one-loop hot-spot self-energy violates the quasiparticle criterion.

Solution

$|\Sigma(\omega)|/|\omega|\sim\sqrt{\omega_0/|\omega|}$, which diverges as $\omega\to0$. The interaction correction is larger than the bare frequency, so no pole with finite residue follows at the hot spot in this regime.

References

• Abanov, A., A. V. Chubukov, and J. Schmalian. “Quantum-Critical Theory of the Spin-Fermion Model and Its Application to Cuprates: Normal State Analysis.” Advances in Physics 52 (2003): 119–218. DOI.
• Metlitski, M. A., and S. Sachdev. “Quantum Phase Transitions of Metals in Two Spatial Dimensions. II. Spin Density Wave Order.” Physical Review B 82 (2010): 075128. DOI.
• Schlief, A., P. Lunts, and S.-S. Lee. “Exact Critical Exponents for the Antiferromagnetic Quantum Critical Metal in Two Dimensions.” Physical Review X 7 (2017): 021010. DOI.