Charges in Fields

In 1980 Klaus von Klitzing cooled a scrap of silicon to near absolute zero, threaded it with a strong magnetic field, and measured the Hall resistance — the sideways voltage a current develops in a field, known since 1879. Classically it should rise smoothly with B. Instead it climbed a staircase: flat plateaus at exactly

The integer $\nu$ here is the filling factor: loosely, the number of electrons the sample holds per quantum of magnetic flux threading it — equivalently, how many of the quantized energy shelves we build in §13.3 are completely filled. For now just hold it as “a thing that is counted”; §13.3 makes the count exact.

Exactly, and universally: the same plateaus in silicon or gallium arsenide, in clean samples or dirty ones, this year or next, to parts per billion. It is the most reproducible measurement in electrical science — so reproducible that when the SI was rebuilt in 2019 around fixed values of $h$ and $e$, this effect became an exact primary realisation of the ohm. (The ohm itself remains a derived unit, defined through the fixed constants; what 2019 changed is that a von Klitzing measurement now realises it exactly, with no calibration chain back to a physical artefact.) Planck's constant, sitting in a resistance standard in a drawer in every national metrology lab.

Numbers that exact cannot come from the messy details of a material; dirt varies, geometry varies, everything varies — except something that is counted. Our job in this chapter: find what a magnetic field gives an electron to count. The answer will be Chapter 7's ladder wearing a magnet for a coat, and on the way we will be forced to confront the strangest bookkeeping device in physics — the vector potential — and discover that a piece of it is shockingly, measurably real.

How does a magnetic field enter the Schrödinger equation? Not through a potential energy — magnetic forces do no work — but through the momentum, by the rule called minimal coupling:

Take the rule on two credentials. First, Ehrenfest: compute $d⟨\vec{v}⟩/dt$ from (13.2) and out comes exactly the Lorentz force $q\vec{v}\times \vec{B}$ (exercise 2 — the operator algebra is honest work but nothing new). Second, and deeper: it is the only local coupling that respects the field's built-in redundancy. The same $\vec{B}$ is described by infinitely many potentials — add any gradient, $\vec{A}\to \vec{A}+\nabla \chi$, and the curl is untouched. Physics must not care. And with (13.2) it doesn't, because the wavefunction volunteers to absorb the change:

A local, position-dependent phase on ψ — invisible to every probability — soaks up the entire redefinition (check it: the gradient in $\hat{\vec{p}}$ acting on the phase factor produces exactly $+q\nabla \chi$, cancelling the shift — exercise 1). This dance of A and the phase of ψ is called gauge invariance, and it is not a quirk: it is the organizing principle that, pushed further, generates all of particle physics. For now, see the redundancy with your own eyes:

Now solve (13.2) for a uniform field — electron in the plane, B out of the page. Chapter 7 promised its algebra would keep paying dividends; here is one nobody sees coming. (The ladder we are about to build, and the gauge we build it in, both carry the name of Lev Landau, who found these levels in 1930 at twenty-two — the same Soviet theorist whose ten-volume Course became the century's benchmark for physics.) Choose the Landau gauge $\vec{A}=(0,Bx)$:

Nothing depends on y, so ${\hat{p}}_y$ is conserved: write $\psi =e^{iky}\phi (x)$ and the second term becomes $\frac{1}{2}m{\omega }_c^2(x-x_0{)}^2$ with ${\omega }_c=qB/m$ and center $x_0=\hslash k/qB$ — a harmonic oscillator, exact, no approximation, one copy for every k. The free electron's continuum collapses onto the ladder:

That independence is the story. Each rung is not one state but a vast shelf of them — every allowed k, i.e. every center position $x_0$ across the sample. Count them in a box of area $L_x{L}_y$ (exercise 4) and the shelf holds exactly one state per flux quantum threading the sample:

This counting is not abstract — it is read off a real device. Von Klitzing's sample was a Hall bar: a flat two-dimensional electron sheet carrying a current $I$ along its length, threaded by a perpendicular field $B$ out of the plane, with the transverse Hall voltage $V_H$ measured across its width. The Hall resistance $R_{xy}=V_H/I$ is what climbs the staircase of (13.1) — and the counting we just did fixes each tread.

There is von Klitzing's counting: the filling factor ν in (13.1) is the number of completely filled shelves, an integer because shelves fill one flux quantum at a time — geometry and dirt never enter the count. (Honesty requires one more ingredient we only name: disorder pins the filling between shelves over whole ranges of B, stretching points into plateaus. The quantization unit, though, is pure Landau arithmetic.) Play the counting game:

Step back and ask why counting shelves deserved to rebuild the ohm. The plateaus are flat because between two filled shelves there is an energy gap: disorder localizes the stray states sitting in it, so over a whole range of $B$ no current-carrying state changes occupation and the resistance cannot budge. They are exact because $\nu$ is an integer being counted, not a material property being measured — it is a topological invariant, deaf to sample shape, dirt, and geometry, which is why silicon and gallium arsenide report the same number to parts per billion. And that is the deeper thing they opened. Before 1980 we sorted phases of matter by the symmetry they broke; the quantum Hall state breaks none — it is labelled by an integer with no local order parameter behind it, the first specimen of topological matter. The catalog it founded — topological insulators, the fractional quantum Hall effect and its fractionally charged excitations, the topological phases chased today for fault-tolerant quantum computing — is one of the largest frontiers in physics, and it grew from a staircase nobody expected in a dirty scrap of silicon.

One loose end, sharpened into a lab experiment below: we chose a gauge to solve the problem, and the solution wears that choice on its face — plane waves in y, oscillators in x. Choose the symmetric gauge instead and you get circular states labeled by angular momentum, from a radial equation that looks nothing like (13.4). Gauge invariance stakes its reputation on the claim that the energies come out identical anyway. That is a testable claim.

If A is mere bookkeeping, here is a puzzle that kept physicists arguing from 1959 into the 1980s. Run Chapter 1's two-slit experiment with electrons, but hide a thin solenoid between the paths — field B confined entirely inside it, zero everywhere the electron can go. Chapter 8 tells us what each path's arrow does in a vector potential: it turns by $(q/\hslash )\int \vec{A}\cdot d\vec{l}$ along the path. The two arms therefore arrive with a relative twist

by Stokes' theorem — the loop integral of A is the flux through the loop, wherever that flux hides. The fringes must shift, one full fringe per flux quantum, though no electron ever feels a force:

For decades skeptics hunted for leaks — stray fields, edge effects. In 1986 Tonomura's group sealed a microscopic magnet inside a superconducting shield (which expels every trace of field) and the fringes shifted anyway, exactly as (13.7) demands. So is A “real”? The honest resolution is surgical: the value of A at a point is bookkeeping — no experiment can see it, as the gauge widget showed. But the loop integral of A is gauge-invariant, and nature reads it. Electromagnetism, as quantum mechanics experiences it, lives not in fields at points but in phases around loops — the same lesson the AB ring below states as a spectrum, and the seed of every modern gauge theory.

The lab Rust-QP/ch13-landau tests this chapter's two big claims the hard way. First, gauge invariance as arithmetic: solve the Landau problem in two gauges whose equations share nothing — and demand one spectrum. (The tridiagonal eigenvalues come from Sturm-sequence bisection, exact for these matrices and a thousandfold faster than a dense solver; and the symmetric gauge hides a trap that bit us — the m = 0 centrifugal term is attractive at exactly the critical fall-to-center strength, and the naive grid manufactured a spurious state at E ≈ −1382):

1/// H = p_x^2/2 + (k - B x)^2 / 2 on a wide grid. Returns lowest `count`.
2fn landau_gauge_levels(b: f64, k: f64, count: usize) -> Vec<f64> {
3    let x_max = 14.0 / b.sqrt();
4    let build = |n: usize| -> (Vec<f64>, Vec<f64>, Vec<f64>) {
5        let dx = 2.0 * x_max / n as f64;
6        let diag: Vec<f64> = (0..n)
7            .map(|i| {
8                let x = -x_max + (i as f64 + 0.5) * dx;
9                1.0 / (dx * dx) + 0.5 * (k - b * x) * (k - b * x)
10            })
11            .collect();
12        (diag, vec![-0.5 / (dx * dx); n - 1], vec![1.0; n])
13    };
14    richardson_levels(3000, count, build)
15}
16
17/// With psi = e^{i m phi} R(r), the radial equation is self-adjoint against
18/// the plane's geometric weight r dr. In conservative (Sturm-Liouville) form
19///   -(1/2)(r R')' + [ m^2/(2r) + B^2 r^3/8 - (mB/2) r ] R = E ( r R ),
20/// i.e. a generalized problem A R = E W R with p = r/2, W = diag(r), and the
21/// eigenFUNCTION R ~ r^{|m|} smooth at the origin (an integer power). We solve
22/// it with the generalized Sturm bisection and Richardson-extrapolate to
23/// O(dr^4). This replaces the older substitution u = sqrt(r) R: that mapped
24/// the problem to an ordinary (W = I) one but introduced a half-integer cusp
25/// u ~ r^{|m|+1/2}, whose non-smoothness pinned the error at O(dr^2) — the grid
26/// could be refined but Richardson bought nothing (halving dr only quartered
27/// the error, the tell-tale of a stalled 2nd order).
28///
29/// m = 0 stays special. Its centrifugal coefficient m^2 - 1/4 = -1/4 sits at
30/// the critical fall-to-center strength, so the r-radial form is delicate; the
31/// clean cure is the change of variable t = B r^2/2, in which
32///   -(t R')' + (t/4) R = (E/B) R  (an ordinary W = I problem)
33/// has the perfectly smooth eigenfunctions e^{-t/2} L_n(t) — Richardson then
34/// reaches O(dt^4) as well.
35fn symmetric_gauge_levels(b: f64, m: i32, count: usize) -> Vec<f64> {
36// … (the radial builders, then the same Sturm bisection + Richardson)
37
38// … the referees — two operators, one ladder, and degeneracy in k:
39row("k-degeneracy: spread across k (exact 0; gate only)", referee.landau_k_spread, 1e-12),
40// … (m-degeneracy) …
41row("gauge invariance: Landau vs symmetric", referee.cross_gauge_err, 2e-9),

Second, the AB effect as a pair of certificates: the fringe curve is built from numerical line integrals of the solenoid's actual A, while the field is finite-differenced to zero at every point the electron visits — plus a 40-site ring whose Peierls-phase spectrum (the Peierls phase is the lattice version of the same magnetic phase as in (13.7): the twist an electron's amplitude picks up hopping from one site to the next) is checked against the exact band formula at machine precision:

1// the closed counterclockwise loop is lower-forward + upper-reversed
2// (our "upper" parametrization runs theta from pi to 0, i.e.
3// clockwise — the first version of this referee compared the wrong
4// orientation to +Phi and failed by exactly 2*Phi_max)
5loop_err = loop_err.max(((lower - upper) - phi_flux).abs());
6// … inside curl_of, the central-difference differencer itself:
7let day_dx = (a(x + h, y).1 - a(x - h, y).1) / (2.0 * h);
8let dax_dy = (a(x, y + h).0 - a(x, y - h).0) / (2.0 * h);
9day_dx - dax_dy
10// …
11// Differentiate A at a few points on the electron's own path. Twice:
12// once on the solenoid's A (must read 0 — the electron flies through
13// field-free space) and once, at the same points with the same
14// differencer, on the symmetric gauge of a UNIFORM field B (must
15// read B). The second call is what makes the first one evidence.
16for &(x, y) in &[(0.0, 3.0), (2.1, 2.1), (0.0, -3.0)] {
17    let c = curl_of(|x, y| a_field(phi_flux, x, y), x, y).abs();
18    worst_curl = worst_curl.max(c);
19    curl_samples.push(c);
20    let probe = curl_of(|x, y| a_uniform(b, x, y), x, y);
21    curl_probe = curl_probe.max((probe - b).abs());
22}

1. (F) Verify gauge covariance: show $(\hat{\vec{p}}-q\vec{A}-q\nabla \chi )e^{iq\chi /\hslash }\psi =e^{iq\chi /\hslash }(\hat{\vec{p}}-q\vec{A})\psi$, and conclude that the two Hamiltonians have identical spectra and identical probability densities.
2. (F) Earn minimal coupling's first credential: from (13.2) and equation (5.4), compute $md⟨\vec{v}⟩/dt$ (where $\vec{v}=(\hat{\vec{p}}-q\vec{A})/m$) and recover the Lorentz force, operator-ordered as $\frac{q}{2}(\vec{v}\times \vec{B}-\vec{B}\times \vec{v})$.
3. (C) In the Landau fan, fix E_F and record the B values where ν jumps. Show they occur at B = E_F/(n+½) — evenly spaced in 1/B. That periodicity (Shubnikov–de Haas oscillations) is how experimentalists measure Fermi surfaces to this day.
4. (F) Derive the shelf capacity (13.6): in an L_x × L_y sample, k is quantized in steps 2π/L_y and the center x₀ = ħk/qB must lie inside [0, L_x]. Count.
5. (F, hard) The lowest Landau level in the symmetric gauge: show that every function ${\psi }_m\propto (x+iy{)}^m{e}^{-qB(x^2+y^2)/4\hslash }$ (m = 0, 1, 2, …) has energy exactly ħω_c/2 — an infinite holomorphic family. (Feed it to the symmetric-gauge radial operator, or better: build the two ladder operators of this problem and show one annihilates all of them.) This “holomorphic arena” is where the fractional quantum Hall effect lives.
6. (F, hard) Flux quantization in superconductors: the condensate wavefunction of paired electrons (charge 2e) must be single-valued around a superconducting ring, while deep inside the material the gauge-invariant momentum vanishes. Show the enclosed flux is forced to integer multiples of h/2e — half this chapter's Φ₀ — and explain why the measured value h/2e was smoking-gun evidence that superconducting carriers come in pairs (Chapter 16's subject).
7. (P, hard) The Hofstadter game: put the AB-ring's Peierls trick on a 2D square lattice with flux p/q per plaquette (use magnetic Bloch theory: the unit cell grows q-fold, giving a q×q matrix per momentum). Compute and plot the q sub-bands for q = 2, 3, 4, 5. You are sketching the wings of the Hofstadter butterfly — a fractal spectrum, and one of the first places topology entered condensed-matter physics.