Strange-Metal Transport and Planckian Claims

A strange metal displays transport or thermodynamics that resist a Landau-quasiparticle description over a declared range. “Planckian” is a more specific dimensional claim: a fitted relaxation rate is of order $k_BT/\hbar$. Neither phrase identifies a microscopic mechanism, and a linear resistivity alone does not determine a rate without an independently constrained current-carrying weight.

Required background. Metallic Non-Fermi Liquids and Quasiparticle Breakdown supplies the distinction between one-particle and transport lifetimes; Sources, Linear Response, and Kubo Formulae supplies conductivity and order of limits. Helpful background. Optical Conductivity and Sum Rules supplies spectral-weight and fitting constraints.

From conductivity to a rate

A single-component Drude parameterization is

$$\sigma(\omega)=\frac{\mathcal D}{\Gamma_{\mathrm{tr}}-i\omega}, \qquad \rho_{\mathrm{dc}}=\frac{\Gamma_{\mathrm{tr}}}{\mathcal D}.$$

$\mathcal D$ is the current-carrying Drude weight in the chosen unit convention. If $\mathcal D$ is temperature independent, $\rho=\rho_0+AT$ corresponds to $\Gamma_{\mathrm{tr}}\propto T$. If spectral weight shifts with temperature, several bands contribute, or the optical response is non-Drude, the same dc slope does not define a unique rate.

A dimensionless Planckian coefficient is

$$\alpha_{\mathrm{tr}}= \frac{\hbar\Gamma_{\mathrm{tr}}}{k_BT}.$$

Calling $\alpha_{\mathrm{tr}}$ “order one” requires stating whether the fitted rate is an angular frequency, ordinary frequency, half-width, full width, or memory-function rate; factors of $2$ and $2\pi$ otherwise masquerade as physics. The transport rate must also be distinguished from the single-particle width, energy-relaxation rate, diffusion time, and Lyapunov exponent.

Momentum relaxation is indispensable

Electron–electron interactions in a translation-invariant continuum conserve total momentum. If electric current overlaps momentum, the dc conductivity contains a delta function even when the electron spectral function is broad. Finite resistivity requires a momentum sink: lattice umklapp, disorder, phonons, boundaries, or coupling to another sector.

Critical scattering can set a fast local equilibration rate while weak disorder sets the slow momentum-relaxation rate. Alternatively, a lattice critical theory can relax current intrinsically through umklapp. These regimes have different dependences on carrier density, disorder, field, and frequency. A self-energy proportional to $T$ cannot simply be inserted into Drude theory without the appropriate vertex corrections.

What experiments constrain

Across several correlated-metal families, converting a linear resistivity slope using effective carrier parameters has produced rates of order $k_BT/\hbar$ Bruin et al. 2013, pp. 804–807. In overdoped cuprates, Legros and collaborators found a linear coefficient correlated with superfluid density across compounds Legros et al. 2019, pp. 142–147. Angle-dependent magnetoresistance in a cuprate separated an isotropic linear-in-$T$ rate from an anisotropic conventional contribution Grissonnanche et al. 2021, pp. 667–672.

These are substantive constraints, not a universal bound. Inferring $\Gamma_{\mathrm{tr}}$ depends on carrier density, effective mass, multiband decomposition, and optical spectral weight. Material trends can support common phenomenology while leaving open spin fluctuations, nematicity, local criticality, spatially random interactions, or other mechanisms.

Recent model work reinforces this distinction. A two-site cellular dynamical mean-field study of a Kondo-breakdown critical point found Planckian dynamical and current scaling driven by vertex contributions rather than a direct single-particle rate Gleis et al. 2025, pp. 106501-1–106501-9. It is a mechanism in a specified model, not a theorem applying to every linear resistivity.

Bounds, bad metals, and saturation

The uncertainty principle alone does not impose $\Gamma\le k_BT/\hbar$. Proposed chaos, viscosity, diffusion, and equilibration bounds have different assumptions and observables. A fitted $\alpha>1$ need not violate quantum mechanics, while $\alpha\simeq1$ need not reveal maximal chaos.

The Mott–Ioffe–Regel comparison $k_F\ell\sim1$ tests a semiclassical mean free path. When quasiparticles are absent, $\ell=v_F/\Gamma$ may no longer be well defined; exceeding the conventional saturation resistivity then identifies a bad-metal regime, not a literal path shorter than a lattice spacing.

A disciplined claim

A strong Planckian transport assessment reports the raw $\rho(T)$ and $\sigma(\omega,T)$, Drude or memory-function model, spectral-weight integral, carrier parameters and covariance, residual term, temperature window, disorder and field dependence, and alternative multiband fits. It tests whether the same $\Gamma_{\mathrm{tr}}$ explains dc and optical data and whether a microscopic theory predicts the observed momentum dependence.

Primary transport and theory sources were checked through 10 August 2026. They support widespread order-$k_BT/\hbar$ phenomenology in declared analyses, not a universal material-independent upper bound or unique strange-metal mechanism. New benchmark and material claims belong in Quantum Matter and Emergence Research.

Exercises

1. A Drude fit has $\mathcal D(T)=\mathcal D_0(1-bT)$ and $\Gamma=aT$. Expand $\rho$ through $T^2$.

Solution

$\rho=aT/[\mathcal D_0(1-bT)]\simeq(a/\mathcal D_0)T+(ab/\mathcal D_0)T^2$. Even a perfectly linear rate produces curvature when the Drude weight varies.

2. Convert a fitted energy width $\hbar\Gamma=2.5k_BT$ into $\alpha_{\mathrm{tr}}$.

Solution

By definition $\alpha_{\mathrm{tr}}=\hbar\Gamma/(k_BT)=2.5$. Whether this is a half-width or full-width convention must still be stated.

References

• Bruin, J. A. N., H. Sakai, R. S. Perry, and A. P. Mackenzie. “Similarity of Scattering Rates in Metals Showing $T$-Linear Resistivity.” Science 339 (2013): 804–807. DOI.
• 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.
• Grissonnanche, G., Y. Fang, A. Legros, S. Verret, F. Laliberté, C. Collignon, J. Zhou, D. Graf, P. A. Goddard, L. Taillefer, and B. J. Ramshaw. “Linear-in Temperature Resistivity from an Isotropic Planckian Scattering Rate.” Nature 595 (2021): 667–672. 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.