Holographic Models of Non-Fermi Liquids and Locally Critical Metals

An $\mathrm{AdS}_2$ near-horizon region can generate a momentum-dependent infrared scaling exponent and a nonanalytic fermion self-energy. Matching that inner solution to the full geometry explains when a sharp Fermi surface survives and when it does not. It is a mechanism inside a specified large-$N$ finite-density model, not a microscopic identification of a strange metal.

Required background. Holographic Fermions and Spectral Functions supplies the spinor source–response prescription. Chemical Potential and Charged Black Branes supplies the charged background and ensemble.

Helpful background. SYK Models and Local Quantum Criticality provides an independent realization of time-dominated scaling. Strange-Metal Transport and Planckian Claims supplies the experimental and many-body claim boundaries.

Evidence cutoff: 25 July 2026. The model classifications and comparison claims below include primary literature available by this date.

Infrared scaling in an AdS₂ throat

An extremal charged black brane often approaches

$$ds^2=L_2^2\frac{d\tau^2-d\zeta^2}{\zeta^2}-h_{xx}(r_h)d\mathbf x^2, \qquad h_{xx}(r_h)>0, \qquad A_\tau=\frac{e_2}{\zeta}.$$

Spatial momentum is then a parameter in the $\mathrm{AdS}_2$ mass. For a minimally coupled probe spinor, the infrared dimension is

$$\delta_k=\frac12+\nu_k, \qquad \nu_k=\sqrt{L_2^2\left(m^2+\frac{k^2}{h_{xx}(r_h)}\right)-q^2e_2^2}.$$

The infalling $\mathrm{AdS}_2$ solution gives, at zero temperature,

$$\mathcal G_k^R(\omega)=c(k)e^{-i\pi\nu_k}\omega^{2\nu_k},$$

where $c(k)$ is fixed by the infrared normalization and gamma functions. This response scales in time but retains nontrivial momentum labels: “local criticality” does not mean that every spatial correlation is local.

If $\nu_k$ becomes imaginary, the solution is log-periodic in frequency. This oscillatory region warns that the simple probe description can be unstable or strongly mixed; it is not a conventional quasiparticle regime.

Inner–outer matching

Choose an overlap region satisfying $\omega\ll r-r_h\ll\mu$. Solving the zero-frequency outer equation and matching it to the inner response yields

$$G_R(\omega,k)= \frac{b_+^{(0)}(k)+b_-^{(0)}(k)\mathcal G_k^R(\omega)+O(\omega)} {a_+^{(0)}(k)+a_-^{(0)}(k)\mathcal G_k^R(\omega)+O(\omega)}.$$

A Fermi momentum satisfies $a_+^{(0)}(k_F)=0$. Expanding about it gives the first application,

$$G_R(\omega,k)\simeq \frac{h_1}{k-k_F-v_F^{-1}\omega-h_2e^{i\gamma}\omega^{2\nu_{k_F}}}.$$

For $\nu_{k_F}>1/2$, the analytic $\omega$ term dominates and the excitation can be parametrically sharp, although its decay need not have Fermi-liquid scaling. For $\nu_{k_F}<1/2$, the infrared self-energy dominates and the width is comparable to the excitation energy. At $\nu_{k_F}=1/2$, logarithms produce the marginal case. The complete matching calculation and this classification were developed by Faulkner et al. 2011, §§ III–V.

Where the matching claim ends

The formula assumes a parametrically long $\mathrm{AdS}_2$ throat, sufficiently small $\omega/T$ or the corresponding finite-temperature scaling variable, and a probe sector whose backreaction is negligible. Irrelevant deformations of the throat determine the crossover scale and can destabilize the extremal entropy. Lattice-scale momentum structure, a conserved Fermi volume, and electron quantum numbers are not supplied by the infrared exponent.

As an adversarial check, fit the power $2\nu_{k_F}$ inside progressively larger frequency windows. Once the upper edge approaches the throat-to-UV crossover, the inferred exponent drifts because analytic outer-region terms compete with $\mathcal G^R$. A stable-looking exponent in one decade therefore does not prove a scale-invariant microscopic metal. A second check is to add the leading irrelevant deformation: if it removes the apparent scaling before the observable window, the undeformed throat was not predictive there.

The warranted conclusion is narrow: the geometry realizes a semi-local critical sector coupled to a Fermi-surface pole and predicts its low-energy line shape under declared matching assumptions. Similar exponents can arise from inequivalent microscopic or effective descriptions.

Exercises

Determine which term controls the pole denominator as $\omega\to0$ for $\nu_{k_F}=1/3$ and $\nu_{k_F}=2/3$.

Solution

For $\nu=1/3$, $\omega^{2\nu}=\omega^{2/3}$ exceeds $\omega$ at small frequency and controls the denominator. For $\nu=2/3$, $\omega$ exceeds $\omega^{4/3}$ and the analytic term controls the dispersion.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Faulkner, Thomas, Hong Liu, John McGreevy, and David Vegh. “Emergent Quantum Criticality, Fermi Surfaces, and $\mathrm{AdS}_2$.” Physical Review D 83, 125002 (2011). DOI.