Condensed Matter Ep 9: The Quantum Hall Effects

Two in the morning, the fifth of February nineteen eighty, in the high-field laboratory at Grenoble. Klaus von Klitzing is watching a chart-recorder pen that should be drawing a ramp. It is drawing a staircase. On each flat tread the Hall resistance is a fixed number — and that number turns out to be Planck's constant divided by the electron charge squared, divided by a whole number, to a precision that would eventually be measured in parts per billion and that does not care what the sample is made of. The previous conversation built the geometry: a phase that depends on the shape of a path rather than its speed, a curvature whose local density can be integrated, and a Brillouin zone that is a torus with no edges. This one spends it. The whole number on the plateau is that curvature added up over the zone and divided by two pi — the result Thouless, Kohmoto, Nightingale and den Nijs got in nineteen eighty-two, sixteen months before anyone could say what kind of object they had found. The most counter-intuitive idea in the hour is that **the dirt is essential**. A perfectly clean sample would show no plateau at all — just the ramp. Disorder localises most of the states, leaving a thin percolating thread of extended ones, and it is the localised states that give the plateau its width. Prange showed in nineteen eighty-one that when an electron gets trapped, the ones still moving carry *exactly* the current it stopped carrying. Not approximately. Exactly. Nothing in that calculation explains why — which is precisely why the topology is needed. Then the argument that produces a metal without going anywhere near the surface: the invariant is a whole number inside the sample and zero outside, integers cannot change by degrees, so the gap must close somewhere on the way out. That is the edge state, derived from the bulk. And a caution — what topology guarantees is that gapless boundary modes *exist*, not that they carry your current; imaging finds the current in the bulk in some samples and at the edges in others. Also: Landau levels and why the degeneracy is one state per flux quantum; why a topological integer cannot creep; Laughlin's gauge argument; and the honest admission that the clean band-structure derivation applies to exactly the case with zero plateau width, with the repair (twisted boundary conditions) and the cost of that repair both stated. The last third is the fractional effect, which is a different kind of thing entirely. Tsui and Störmer found a plateau at one third of a filled level in nineteen eighty-two; Laughlin wrote down a wavefunction for it in nineteen eighty-three that remains a guess with very good support rather than a derivation. Its excitations carry one third of an electron charge — measured directly by shot noise in nineteen ninety-seven, in Saclay and at the Weizmann Institute independently — and they are neither bosons nor fermions. Switch the interaction off and the state does not weaken; it ceases to exist. ——— Series: Physics - Tutorial · Grad level. Every equation spoken in full, no chalkboard required. Built by Fermi AI.