Ordinary metals are Fermi liquids: quasiparticles scatter off each other at a rate set by phase space, giving ħ/τ ∝ (kBT)², so resistivity rises as ρ = ρ₀ + AT². Near a quantum critical point (QCP) — a zero-temperature phase transition tuned by doping, pressure or field — that picture breaks down. Quantum fluctuations of the order parameter scatter electrons at the fastest rate quantum mechanics allows, the Planckian rate ħ/τ ≈ αkBT, giving resistivity that rises linearly in T all the way from the lowest measured temperatures. This "strange metal" behavior sits directly above the superconducting dome in cuprates and heavy-fermion compounds — the same fluctuations that scatter electrons are suspected of pairing them into the unconventional superconductor.
ħ/τ = a·(1−δ)·k_BT + b·δ²·(k_BT)²/E_F
ρ(T) = ρ₀ + (ħ/τ)·const
n = d ln ρ / d ln T (n→1 at QCP, n→2 far away)
- δ (distance from QCP) — δ=0 sits exactly on the critical doping; sliding right tunes the material into an ordinary Fermi liquid.
- Temperature — replays the same tuning parameter at different T, tracing one vertical cut through the phase diagram dome shown lower-left.
- Local exponent n — the log-log slope of the live curve; n≈1 is the T-linear "Planckian" strange metal, n≈2 is textbook Fermi-liquid behavior.
- Electron paths — the 3D lattice shows conduction electrons as moving points; denser scattering kinks (near δ=0, high T) mean a shorter mean free path and a faster momentum-relaxation rate.
Real-world relevance: this T-linear, sample-independent scattering rate — bounded near ħ/τ ≈ kBT, the "Planckian limit" — shows up in cuprate superconductors, heavy-fermion compounds, twisted bilayer graphene and pnictides, and is one of the biggest open puzzles in condensed-matter physics.