Holographic Models of Quantum Critical Matter and Strange Metals

Holographic strange-metal models can realize robust mechanisms—critical continua, rapid relaxation, incoherent conductivity, and scaling thermodynamics—but the same measured power law can arise from inequivalent bulk actions. A meaningful comparison therefore fixes normalizations, density, momentum relaxation, calibration observables, and temperature window before asking whether two models represent the same universality class. Neither a good scaling fit nor a black-brane horizon identifies a microscopic material.

Required background. Chemical Potential and Charged Black Branes supplies the ensemble and current normalization. Holographic Fermions and Spectral Functions supplies the causal spectral observable.

Helpful background. Evidence and Model Discrimination at Quantum Criticality supplies the experimental inference criteria. Evidence Independence, Circularity, and Double Counting explains why calibration and validation data cannot be counted twice.

Evidence cutoff: 25 July 2026. The comparison below covers primary model results available by this date; it does not assert a final classification of strange-metal experiments.

Three distinct routes to anomalous response

An Einstein–Maxwell–dilaton scaling geometry may produce

$$s(T)\propto T^{(d-\theta)/z},$$

where $d$ is the number of boundary spatial dimensions. Its conductivity exponent also depends on the scaling dimension of the charge sector, so $(z,\theta)$ alone does not fix $\rho_{\mathrm{dc}}(T)$. Such geometries organize scale covariance and entropy, while their ultraviolet completion and translation-breaking sector remain additional inputs.

A homogeneous axion model instead sets scalar sources $\phi_I=kx_I$. At weak relaxation its low-frequency conductivity has the hydrodynamic form

$$\sigma(\omega)=\sigma_Q+\frac{K}{\Gamma-i\omega},$$

with an incoherent contribution $\sigma_Q$, Drude weight $K$, and relaxation rate $\Gamma$. Temperature dependence in $K$, $\Gamma$, or $\sigma_Q$ can imitate a scaling resistivity without an $\mathrm{AdS}_2$ fermion sector.

A semi-holographic construction couples a propagating Fermi-surface field to a critical large-$N$ sector. In a refined two-coupling model, tuning a coupling ratio produces an approximately universal scaled spectral function and a linear-$T$ resistivity over a bounded window Doucot et al. 2024. The tuning and effective-theory window are part of the result, not consequences of holography in general.

A matched two-model comparison

As a first application, compare an Einstein–Maxwell–dilaton model $M_A$ with a translation-relaxing model $M_B$. Choose the same boundary units and calibrate both to the same two dimensionless quantities at $T_0$,

$$y_1=\frac{s(T_0)}{T_0^d}, \qquad y_2=T_0^{d-2}\rho_{\mathrm{dc}}(T_0).$$

Now withhold three functions from calibration:

Observable	$M_A$ mechanism	$M_B$ mechanism	Discriminating information
$s(T)/T^d$	$(z,\theta)$ scaling	charged-brane equation of state	exponent and crossover
$\rho_{\mathrm{dc}}(T)/\rho_{\mathrm{dc}}(T_0)$	charge-sector scaling	$\Gamma(T)$ and $\sigma_Q(T)$	optical spectral-weight transfer
$\mathcal A(\omega,k_F)$	optional critical fermion	absent unless added	momentum-resolved line shape

Agreement with $y_1$ and $y_2$ leaves these predictions unconstrained. A useful model comparison evaluates all three over the same temperature and frequency window, using the same density definition and contact-term subtraction. Hartnoll and Karch showed how scaling assumptions can relate strange-metal exponents Hartnoll and Karch 2015; those relations organize a scaling hypothesis but do not make its bulk realization unique.

Non-identifiability as an adversarial outcome

Suppose both models fit $\rho_{\mathrm{dc}}\propto T$ after calibration, but one transfers a narrowing Drude peak while the other maintains a broad continuum. The shared dc exponent has not identified a common mechanism. Conversely, similar optical curves can be manufactured over a finite interval by adjusting irrelevant couplings; the thermodynamic exponent or momentum-resolved spectrum may then separate the models.

Perform three adversarial tests:

1. remove one calibration observable and refit;
2. widen the temperature window until a crossover is visible;
3. compare a held-out observable sensitive to a different bulk sector.

If the model ranking reverses, report non-identifiability rather than selecting a preferred microscopic dual. If it remains stable, the result supports the tested mechanism or universality class only. Material identification additionally requires correct symmetries, charge carriers, lattice structure, sum rules, and independent observables with controlled uncertainties.

Exercises

For the Drude form above, find the dc conductivity and the integrated weight of the Drude contribution over all real frequencies.

Solution

$\sigma_{\mathrm{dc}}=\sigma_Q+K/\Gamma$. The real Drude part is $K\Gamma/(\Gamma^2+\omega^2)$, whose integral from $-\infty$ to $\infty$ is $\pi K$. Thus changing $\Gamma$ redistributes but does not by itself change the Drude weight.

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

• Doucot, Benoît, Ayan Mukhopadhyay, Giuseppe Policastro, Sutapa Samanta, and Hareram Swain. “An Effective Framework for Strange Metallic Transport.” Journal of High Energy Physics 2024, 118 (2024). DOI.
• Hartnoll, Sean A., and Andreas Karch. “Scaling Theory of the Cuprate Strange Metals.” Physical Review B 91, 155126 (2015). DOI.