Evidence and Model Discrimination at Quantum Criticality

Quantum-critical data are underdetermined when several mechanisms produce the same power law over a limited window. Model discrimination therefore asks which observables differ, fits their full covariance, and tests predictions withheld from parameter estimation. Scaling collapse, linear resistivity, or $\omega/T$ response can support criticality; none alone identifies the critical fields or excludes noncritical alternatives.

Required background. Quantum-Critical Fans and Finite-Temperature Scaling supplies crossover and correction variables; Strange-Metal Transport and Planckian Claims supplies transport-rate limitations. Helpful background. Model Selection and Parameter Inference supplies likelihoods, priors, posterior predictive checks, and held-out validation.

Start with distinguishable hypotheses

Candidate mechanisms should be expressed as predictive models, not broad labels. Examples include:

• spin-density-wave criticality with discrete hot spots and response concentrated near $\mathbf Q$;
• nematic or gauge criticality with small-momentum scattering around an extended Fermi surface;
• Kondo-breakdown or local criticality with a reconstructed Fermi volume and local dynamical scaling;
• disorder-driven Griffiths or percolative regimes with broad distributions and sample dependence;
• electron–phonon or semiclassical multiband transport with an ordinary momentum-relaxation channel;
• a first-order or avoided transition that creates a large but finite crossover window.

Each must specify tuning parameter, dimensionality, dynamical exponent or kernel, momentum dependence, conserved quantities, disorder, finite-temperature window, and preempting orders. If two labels make no distinct prediction for the available experiment, the data cannot discriminate them.

Joint likelihood and covariance

Collect measurements into a vector $\mathbf d$ and model predictions into $\mathbf m(\boldsymbol\theta)$. With approximately Gaussian errors,

$$\chi^2(\boldsymbol\theta)= [\mathbf d-\mathbf m(\boldsymbol\theta)]^{\mathsf T} C^{-1} [\mathbf d-\mathbf m(\boldsymbol\theta)].$$

$C$ must include shared calibration, background subtraction, geometry, carrier parameters, and repeated use of the same raw spectrum. Treating correlated derived quantities as independent overstates evidence. Nuisance parameters should be common where physics requires them—for example, one critical tuning point and metric factor across several responses—rather than re-fitted independently for every plot.

Information criteria or Bayes factors penalize flexibility only relative to declared likelihoods and priors. A more robust check with modest data is predictive: fit thermodynamics and part of the scattering data, then predict held-out temperatures, momenta, fields, or a different probe.

A hierarchy of evidence

The following progression limits overclaiming:

1. Anomaly: an observable departs reproducibly from a conventional baseline.
2. Scaling regime: several points follow a stable scaling form with corrections and bounded cutoffs.
3. Critical organizer: independent crossovers extrapolate toward one zero-temperature parameter.
4. Field content: momentum, symmetry, and operator dependence select particular soft modes.
5. Mechanism: one theory jointly predicts amplitudes, exponents, transport, and competing instabilities better than alternatives.
6. Material attribution: sample dependence, disorder, multiband structure, and probe matrix elements are quantitatively included.

Evidence at one level does not inherit conclusions from a later level. For instance, Legros et al. established a systematic linear-resistivity phenomenology in overdoped cuprates Legros et al. 2019, pp. 142–147; that observation alone does not select hot-spot, local, or spatially random theories.

High-value discriminators

Momentum resolution is decisive when one model predicts hot spots and another nearly uniform angular scattering. Hall number, quantum oscillations, and spectroscopic Fermi-surface volume can test reconstruction or Kondo breakdown. Polarization and symmetry channels distinguish magnetic, nematic, and phononic modes. Disorder series separate intrinsic scattering from rare-region or momentum-relaxation effects. Optical spectral-weight transfer tests whether a dc slope is a rate or a changing Drude weight.

Cross-observable amplitude ratios are often more selective than exponents. A theory that uses the same coupling to predict a neutron linewidth, electron self-energy anisotropy, and pairing scale takes greater risk than three unrelated power-law fits. The 2025 cellular-DMFT Kondo-breakdown study, for example, predicts current scaling dominated by vertex contributions rather than direct single-particle decay Gleis et al. 2025, pp. 106501-1–106501-9; comparing both optical and one-particle rates tests that distinction.

Failure tests and reporting

Before fitting, specify observations that would disfavor each model: wrong ordering wave vector, absence of predicted Fermi-surface reconstruction, exponent drift beyond corrections, failure of a sum rule, incorrect field response, or a stronger disorder correlation than allowed. Report all searched windows and model variants to avoid selection bias. A model can be a useful effective description even when its microscopic attribution is rejected.

Primary experimental and theory sources were checked through 10 August 2026. The current literature contains multiple viable routes to overlapping quantum-critical and strange-metal phenomenology; no single transport or scaling signature closes that comparison generically. Living comparisons, new data, and benchmark calculations belong in Quantum Matter and Emergence Research.

Exercises

1. Two fitted observables share a 3% calibration uncertainty. Why is a diagonal covariance matrix inappropriate?

Solution

The same calibration shift moves both observables coherently, producing nonzero off-diagonal covariance. Counting the shifts as independent makes their combination appear more precise and can spuriously favor a model.

2. A model fits $\rho(T)$ and $C(T)$ after separate critical points are chosen. Propose a stronger test.

Solution

Fit a single shared $g_c$ and crossover scale to one observable, then predict the tuning and temperature dependence of the other, including its regular background. Holding out an additional field or momentum scan tests the model more strongly still.

References

• Gleis, A., S.-S. B. Lee, G. Kotliar, and J. von Delft. “Dynamical Scaling and Planckian Dissipation Due to Heavy-Fermion Quantum Criticality.” Physical Review Letters 134 (2025): 106501. DOI.
• Legros, A., S. Benhabib, W. Tabis, F. Laliberté, M. Dion, M. Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Z. Li, P. Auban-Senzier, N. Doiron-Leyraud, P. Fournier, D. Colson, L. Taillefer, and C. Proust. “Universal $T$-Linear Resistivity and Planckian Dissipation in Overdoped Cuprates.” Nature Physics 15 (2019): 142–147. DOI.