SM · Collective physics · Five questions to investigate
What carries current when long-lived quasiparticles fail?
Which microscopic interactions relax electric current at the observed rate, and why do several independent transport and thermodynamic observables agree or disagree with the same mechanism?
Starting knowledge: Electric current, momentum conservation, and linear response. Quasiparticles and relaxation rates are defined below.
In a conventional metal, a quasiparticle is a long-lived excitation carrying quantum numbers similar to those of an electron, but dressed by interactions. Its lifetime is long compared with the time associated with its excitation energy, permitting a particle-like transport account. Some correlated metals show broad excitations and resistance approximately linear in temperature over important ranges where a simple Fermi-liquid treatment is inadequate.
The main question is not just how to manufacture a straight line. Ordinary electron–phonon scattering can also give roughly linear temperature dependence in its classical thermal regime. A useful strange-metal mechanism must explain the relevant temperature range, dependence on carrier density and disorder, frequency response, and consistency with thermodynamics.
There are models with computable non-Fermi-liquid behavior. In one construction, many fermion flavors couple to a critical fluctuating field through spatially random interactions. Disorder averaging and a large-number-of-flavors expansion make a particular regime tractable and yield linear resistivity with a logarithmically enhanced specific heat. These are achievements within the model; whether the required disorder and limiting assumptions capture a specific material is an additional test. [36]
A critical field describes fluctuations near a continuous transition, where a characteristic correlation length becomes large. Coupling electrons to such fluctuations can degrade individual electron excitations. That alone does not prove that electric current decays at the same rate. A clean system may preserve a quantity overlapping the current, obstructing finite direct-current resistance.
“Planckian” commonly describes an inferred time of order $\hbar/(k_BT)$. The name fixes a dimensional scale; it does not establish a universal theorem that every relaxation time is bounded below by it. Different probes measure different times, and extracting one from resistivity requires a model for the current-carrying weight.
In a simple parabolic-band continuum model with number density $n$, charge $q$, and effective mass $m$, current density and momentum density are proportional: $\mathbf j=q\mathbf P/m$. If interactions conserve total momentum and the current overlaps it, those interactions alone cannot relax this entire current component. A lattice, disorder, external coupling, or other allowed mechanism must be included where needed. On a lattice, momentum transfer by a reciprocal lattice vector, called Umklapp, can relax current under suitable kinematic conditions.
A relaxation model writes
with electric field $\mathbf E$ and momentum relaxation rate $\Gamma$. For time dependence $e^{-i\omega t}$,
$\sigma$ is conductivity, the proportionality between current and field, and $\rho_{\rm dc}$ is scalar zero-frequency resistivity in this simple isotropic setting. Real materials can have several bands and additional current components, so this expression is a diagnostic model rather than a universal identity.
Imagine collisions frequently redistribute momenta among electrons while preserving their sum. They can rapidly destroy a sharply defined single-electron excitation, yet leave the parabolic-band current unchanged. Now add weak disorder that relaxes total momentum at rate $\Gamma$. The direct-current resistance follows that slow rate even when the individual collision rate is much larger.
Let $\Gamma(T)=\Gamma_0+\alpha k_BT/\hbar$. The relaxation model gives a linear resistivity with slope
An observed $A$ determines $\alpha$ only after $m/n$ and the applicability of the model are specified. A changing current-carrying weight can imitate a changing scattering coefficient. This is why frequency-integrated conductivity, thermodynamics, and other probes matter.
Insight. Explaining a short lifetime and explaining electrical resistance are different tasks. The bridge is an explicit account of current relaxation, including the conservation laws. This distinction can eliminate an attractive mechanism before a difficult many-body calculation begins.
Calibration questions have established answers. Diagnostics ask what follows inside a specified model or comparison. Extensions are bounded proposals whose novelty and feasibility need a focused literature check. Each answer should record its assumptions and limits.
Do: Derive current–momentum proportionality for a parabolic band and the conductivity above. Examine $\Gamma\to0$ at finite and then zero frequency.
Check: The clean real conductivity contains a zero-frequency delta contribution; the order and meaning of limits must be stated.
Meaning: Establishes a conservation-law obstruction to a naive collision-rate argument.
Do: Use the slope formula to propagate uncertainties in $A$, $n$, and $m$ into $\alpha$. Repeat with a temperature-dependent current weight.
Check: Keep SI units or a fully declared natural-unit convention throughout. Identify correlated inputs instead of treating every error independently.
Meaning: Separates an empirical slope from a model-dependent relaxation time.
Do: Follow the cited random-interaction model through its conductivity calculation. Identify the term that relaxes current and compare with a spatially uniform coupling.
Check: Include the corrections required by charge conservation when relating self-energy to conductivity. Recover the stated clean-limit behavior.
Meaning: Determines necessity or sufficiency only within the specified model, not for every strange metal. [36]
Do: For a single chosen parameter set, compare the predicted temperature dependence of resistivity, optical conductivity, and specific heat, the derivative of energy with temperature.
Check: Use one parameter set and one regime. Record which observable is fitted and which is then predicted.
Meaning: A mechanism becomes informative when agreement with one measurement constrains another; a separate fit for every curve weakens the test.
Do: Choose a conventional momentum-relaxation model and one critical-interaction model. Vary only a specified disorder parameter and compare residual resistance and temperature slope.
Check: Determine whether disorder also changes carrier density, pairing, or the critical point. Hold those fixed only when the model permits it.
Meaning: Produces an experimentally motivated comparison with explicit nuisance effects. Novelty depends on the selected pair and observable.
SM-1 and SM-2 provide a low-cost entry. Proceed to SM-3 only after defining the current operator and its conserved overlaps. The connection to superconductivity is then a further question about the same system's instability, not something implied by linear resistance alone.
Unconventional superconductivity · Thermalization and the arrow of time · Turbulence and anomalous dissipation