Fermi Podcast

A plot on a laboratory wall: resistivity against temperature for a copper-oxide superconductor above the temperature where it superconducts. The line is straight. It is straight down through a hundred kelvin, through ten, and as far down as anyone can push it with the superconductivity switched off by a magnet. That is the whole problem, and it has been on the wall for four decades. The third conversation in this series proved — not fitted, proved — that a metal's resistivity must rise as the *square* of temperature. Two Pauli squeezes on the available phase space, and no interaction strength anywhere in the argument, which is why it holds for copper and for the heavy-fermion compounds where electrons are a thousand times heavier than free ones. A straight line is not a small deviation from that. It is a different law. So this finale rebuilds the theorem on air and then asks which assumption dies. The answer is precise, and it is the episode's central takeaway: **the excitation is sharper than the energy it carries**. That is what makes a quasiparticle a thing at all — a bump narrow enough to have a name. In a strange metal the width equals the energy. There is no small parameter, no long-lived excitation, and therefore no Fermi surface to count seats around. The apparatus that explains every ordinary metal does not fail by a bit; it has nothing to describe. Also: why a Mott insulator is not a band insulator — a material every band calculation calls a metal that refuses to conduct, because the electrons jam each other rather than running out of states; the Zhang–Rice singlet; why doping one is so hard, since a moving hole drags a string of broken magnetic bonds behind it and pays for every link; the pseudogap and the Fermi arcs; quantum criticality; **Planckian dissipation**, the scattering rate at the fastest anything is allowed to relax; the Hall-number jump that says the carrier count changes from the doping to one-plus-the-doping across a particular concentration; and **quantum spin liquids** — order with no order parameter, the last callback to the broken-symmetry conversation. Then how the field is actually attacking it: **moiré materials** as designer strong correlation, where two graphene sheets twisted by about one degree produce a superconductor you can tune with a knob; and cold-atom and quantum-computer simulation of the Hubbard model, with a straight account of what those machines have and have not yet shown — including that the pure two-dimensional Hubbard model appears *not* to superconduct until one further hopping term is added. The series ends where it has to. Nothing is resolved; high-temperature superconductivity is unsolved after forty years. But the closing observation is the one all twelve conversations were built toward: every time this subject has understood something, it did so by finding a new object — a macroscopic phase, a dressed excitation, an order parameter, a diverging length, an integer belonging to a band — and never by working harder on the thing written in the Hamiltonian. The strange metal is not unsolved because the equations are difficult. The Hamiltonian is three lines long. It is unsolved because nobody has yet found the object. ——— Series: Physics - Tutorial · Grad level. Every equation spoken in full, no chalkboard required. Built by Fermi AI.

A metre of copper wire obeys a rule that never lets you down: halve the length, halve the resistance. Cut it to a centimetre and the resistance falls by a hundred. Cut it to a micron and it falls again, faithfully. Now swap the copper for a sheet of electrons buried a hundred nanometres under the surface of a gallium arsenide chip, cooled to a few hundredths of a degree above absolute zero, and squeeze the channel shut with a pair of gates. The resistance stops falling smoothly. It comes down a staircase — and the treads are at values built out of nothing but the electron's charge and Planck's constant. This conversation is about what happens when a conductor gets small enough that Ohm's law stops being true, and it starts by having the answer guessed before it is derived. Conductance is current over voltage; that is charge squared divided by energy times time; and energy times time is an action. So the natural unit of conductance is the electron charge squared over Planck's constant, and you can get that far with no physics at all — which is the point, and also the limit. It gives you the currency and not the amount. The amount is Landauer's, and it inverts what resistance means. Conductance is not an obstruction to be overcome; it is a **transmission probability**. A perfect channel does not have zero resistance — it has about thirteen thousand ohms, and that resistance does not live in the channel at all. It lives in the join, where a wide reservoir has to squeeze its current into a countable number of ways through. Which is also why the number is universal: every one of those experiments got the same step height out of a different chip. Then the hero: pinch two constrictions nearly shut and the puddle between them becomes a **quantum dot** — an artificial atom, and the word is meant literally. It has discrete levels for the same particle-in-a-box reason an atom does, three orders of magnitude larger in size and getting on for five orders smaller in energy. Its shell closings come at two, six and twelve, and it obeys Hund's rule. Put one more electron on and you pay a charging energy; that price is why the current comes in sharp peaks separated by dead zones, and why sweeping gate against bias draws diamonds you can read the addition energy off with a ruler. The last act is the Kondo effect, one of very few strong-correlation problems that is genuinely solved — you can compute the resistivity, susceptibility and specific heat at every temperature, each a universal function of one measured number. In a bulk metal a magnetic impurity makes resistance *rise* as you cool. In a dot it does the opposite, because the impurity is not beside the road, it *is* the road — and the conductance climbs to the maximum a single channel is allowed. Ends where every tool in the hour fails: the zero point seven anomaly, which has been argued about since nineteen ninety-six. ——— Series: Physics - Tutorial · Grad level. Every equation spoken in full, no chalkboard required. Built by Fermi AI.

In nineteen eighty-eight Duncan Haldane drew a honeycomb lattice on a sheet of paper and threaded a magnetic field through it that adds up to exactly nothing — up through the middle of each hexagon, down again around the edges, net zero. There is no field to speak of, and yet the model carries the same quantised Hall conductance the previous conversation spent an hour earning from an eighteen-tesla magnet. It took about twenty-five years before anyone built one. That is the opening move, and the harder prize follows immediately. Keep time-reversal symmetry — do not break it internally, do not apply anything — and the total curvature over the Brillouin zone is forced to zero, so the Chern number is zero and by the previous hour's argument there should be nothing left to say. There is. The invariant that survives is not a whole number but a parity: **even or odd**, a Z-two count, and odd is a material that insulates through its bulk and conducts only on its skin, with nothing in the room switched on. What replaces the magnet is **spin–orbit coupling** — an electron moving through a crystal's electric field sees, in its own frame, a magnetic field whose direction depends on which way it is going. Opposite-moving electrons see opposite fields. So the two spins get opposite Hall responses that cancel in total while leaving something behind at the edge: a **helical** pair of channels, one spin running each way, locked to their directions. The most satisfying idea in the hour is why that edge resists dirt. To backscatter, an electron must reverse its direction, which means reversing its spin — and the two paths around a non-magnetic impurity differ by a full rotation of a spin one-half, which contributes a minus sign. The two reflection amplitudes cancel exactly. Not "are suppressed". Cancel. But the protection is only as good as the symmetry it rests on: bring in a magnetic impurity and it is gone. That is what "symmetry-protected" buys you, and exactly what breaks it. Then the experiments, in order: Kane and Mele's two two-thousand-and-five papers on graphene where the effect is far too small to see; Bernevig, Hughes and Zhang predicting the right material — a mercury telluride quantum well past a critical thickness where the bands invert; König and colleagues at Würzburg measuring it in two thousand and seven and finding the conductance plateau the theory demanded; then the three-dimensional versions, and the single Dirac cone on the surface of bismuth selenide seen directly with photoemission. The last section is the Majorana mode and the topological qubit — a state that is its own antiparticle, split across the two ends of a wire so that no local disturbance can read it. The theory is solid; the experimental record is not. That is stated plainly, retractions included, without hype and without performing scepticism about it. ——— Series: Physics - Tutorial · Grad level. Every equation spoken in full, no chalkboard required. Built by Fermi AI.

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.

Walk a spear from the North Pole down to the equator, a quarter of the way round, and back up — keeping it always pointing as straight ahead as the ground allows, never twisting it in your hands. It comes home turned through a right angle. Nothing rotated it. The rotation is a property of the journey. This is the most abstract hour in the series and it says so out loud, then refuses to leave the ground: one spin, one magnetic field, one hand moving the field slowly round a loop. The claim to be earned is that the spin comes back carrying a phase that has nothing to do with how long you took. Go round twice as slowly and it is unchanged — which is exactly what a physicist's instinct says is impossible, because the obvious phase, the one from the energy, scales with duration. Alice reasons her way to that wrong answer on air before it is corrected. The distinction, once you have it, is hard to un-see. The dynamical phase is a sum over *when*. The leftover is a sum over *where* — each term fixed by two neighbouring settings of the apparatus, with no time in it anywhere. That is why slowness cannot stretch it. Berry wrote it down in nineteen eighty-four; Pancharatnam had the same quantity for polarised light in nineteen fifty-six, aged twenty-two, in a paper communicated by his uncle C. V. Raman and then ignored for thirty years. Then the geometry proper. The phase is the area swept out on the sphere of directions — half the solid angle for a spin one-half — which immediately raises the question the rest of the series turns on: a loop divides a sphere into two regions, so which one is the area? The answer only works because the two disagree by exactly two pi, and a phase cannot tell the difference. That ambiguity is not a blemish. It is the seed of an integer. Also: why the arbitrary phase you are free to assign at every point cancels round a closed loop and nowhere else; the degeneracy that acts like a magnetic monopole in parameter space, and how far the Aharonov–Bohm analogy may honestly be taken before it stops; anomalous velocity — how a phase, which is not a trajectory, nevertheless pushes a moving electron sideways, and why monolayer molybdenum disulphide has curvature everywhere while its bilayer has none; the Zak phase and why a crystal's Brillouin zone has no edges but is a torus, its walls being gluings rather than boundaries. It ends where the geometry stops being a theorem about a doughnut: von Klitzing's plateaus, flat to parts in a billion, and Thouless and company showing two years later that the whole number on the plateau is this curvature added up over the zone. ——— Series: Physics - Tutorial · Grad level. Every equation spoken in full, no chalkboard required. Built by Fermi AI.