Tutorial 2 · Arovas–Schrieffer–Wilczek, 1984

The Phase
You Pick Up

Two years after Wilczek showed that two-dimensional particles may have any statistics, Arovas, Schrieffer and Wilczek computed what the statistics actually are in a real material. Four pages, and the method is a holonomy calculation.

What this is

A worksheet, not a result. The mathematics on this page is standard topological quantum computation, roughly 1984–2009. None of it is new. “It was simply new to my hands. I had read it for years and never computed with it, and those turn out to be different states of knowing.” These sheets exist to move a reader from the first state to the second, and they make no claim of their own.

The method is predict–then–check. Every input box, drill and gate here is built for one move: write down what you think the number is before you reveal it. Numbers are sourced to a paper where a paper says them, and marked unchecked where none does — a sheet whose whole method is check it yourself cannot ask you to take its own arithmetic on trust.

If you are coming from an operator formalism, you may want to ask of every quantity here what the declared observable is, and what mapping puts it into the domain before the algebra acts. That is a fair question, and these sheets sit upstream of it — the arithmetic of d² = d + 1 is the same arithmetic whichever way it is answered. They take no position on it. The full note is on the index.

Prerequisites: Section 3 genuinely needs the Berry phase and adiabatic transport — that is Derivation 5 in the paper course, and it is honestly labelled weeks, not hours. Sections 1, 4 and 5 stand on their own and need nothing beyond arithmetic.

Start from a number. At filling ν = 1/3 the exchange phase is θ = π/3 — sixty degrees, neither the boson's 0 nor the fermion's π. This tutorial is the four-page 1984 calculation that produced that number, and the thirty-six years it took to measure it. If you have not done the Berry phase, read Section 3 as a preview rather than a lesson and come back to it; nothing else here depends on it.

⛔ The one real prerequisite

This is the one tutorial in the series that genuinely needs Berry phase and adiabatic transport, and everything in Section 3 rests on them. The groundwork is Derivation 5 in Five Derivations: spin-½ in a rotating field → the Berry connection → Aharonov–Bohm → ASW. Four steps, honestly labelled weeks.

Section 1What was and wasn't known in 1984

Three things were already settled:

Leinaas & Myrheim (1977), and Wilczek (1982): in two dimensions, exchanging two identical particles is not necessarily ±1. The configuration space of two indistinguishable points in the plane is not simply connected, so the exchange phase can be any value e^iθ. Wilczek named the resulting particles anyons. This is a statement about what 2D permits.
Tsui, Störmer & Gossard (1982): the fractional quantum Hall effect exists — a plateau at filling ν = 1/3.
Laughlin (1983): a trial wavefunction for ν = 1/m, and the claim that its quasiparticle excitations carry fractional charge e/m.

What was not known: whether the FQHE quasiparticles are anyons, and if so, with what θ. Fractional charge does not imply fractional statistics — they are separate claims. ASW closed that gap by direct computation.

Section 2The construction

Laughlin's state at ν = 1/m, with m odd:

ψ_m = Π_i<j (z_i − z_j)^m · exp( −Σ_i |z_i|² / 4ℓ² )z = x + iy in units of the magnetic length ℓ. The exponent m is what makes this fractional.

A quasihole at position z₀ is made by multiplying in a factor that pushes electrons away from that point:

ψ_m^(z₀) = Π_i (z_i − z₀) · ψ_mEach electron coordinate acquires a simple zero at z₀ — a hole in the fluid.

The depleted charge is e/m — one m-th of an electron, missing from a region around z₀.

Fig. 1 — a hole worth one third of an electron

Multiplying the Laughlin state by ∏(z_i − z₀) gives every electron coordinate a simple zero at z₀ — the electrons are pushed away and a hole opens. Fractional charge, but that alone says nothing about statistics. Charge and statistics are separate claims; ASW's contribution was computing the second.

Section 3The calculation, which is a holonomy

Here is the move. Drag the quasihole slowly around a closed loop and ask what phase the wavefunction accumulates. Slowly means adiabatically, so the state stays in the ground-state manifold and the phase picked up is a Berry phase.

γ = i ∮ ⟨ψ(z₀) | ∂_z₀ | ψ(z₀) ⟩ dz₀The holonomy of the Berry connection around the loop. Geometry, not dynamics — it does not depend on how fast you go.

Step one — one quasihole, empty loop. Transporting charge e/m around a loop enclosing magnetic flux Φ gives the Aharonov–Bohm phase for that fractional charge:

γ₁ = −2π · (1/m) · (Φ / Φ₀)The 1/m is the fractional charge showing up directly in the phase. This alone re-derives Laughlin's charge.

Step two — put a second quasihole inside the loop. Now repeat the transport. The enclosed charge has changed: there is an additional deficit of e/m sitting inside the path. The phase changes by the amount corresponding to that extra enclosed charge:

Δγ = 2π / mThe extra phase from encircling one quasihole with another. A full loop = two exchanges.

A full circuit is two exchanges, so a single exchange carries half:

θ = π / mAt ν = 1/3: θ = π/3. Neither 0 (boson) nor π (fermion). Anyons, with a computed value.

Fig. 2 — the same trip, twice, with one difference

The difference between the two panels is the statistics. Both transports pick up an Aharonov–Bohm phase from the magnetic flux. The right-hand loop encloses an extra charge deficit of e/3, and that extra enclosed charge contributes an extra 2π/3. Nothing else changed — same path, same speed, same field. Subtract one from the other and what is left is the exchange statistics.

The statistical phase is a holonomy

The statistical phase is not a property of the particles considered individually. It is the holonomy of a connection over the configuration space of their positions — which is why the calculation in Section 3 is a loop integral and not a property of either quasihole on its own.

Fig. 3 — a full loop is two exchanges

Carrying one quasihole all the way around another is two exchanges, so a single exchange carries half the phase. At ν = 1/3 that gives θ = π/3 — a value between the boson's 0 and the fermion's π, which is what "anyon" means. Predicted 1984, confirmed by interferometry in 2020.

Section 4What the argument assumes

Worth refreshing alongside the result, because the caveats are where the later literature lives:

Adiabaticity. There must be a gap, and the transport must be slow relative to it. Fine deep in a plateau; not fine near a transition.
Well-separated quasiholes. The derivation treats the enclosed charge as a clean deficit. Overlapping quasiholes have no such clean accounting.
Berry phase versus monodromy. The phase splits into an explicit monodromy of the wavefunction and a Berry connection term. For Laughlin states these combine cleanly. For non-abelian candidates — Moore–Read at ν = 5/2 — the separation is subtle, and Kjønsberg & Leinaas showed the naive reading can mislead. Read & Moore (1992) is the paper where that harder case begins.
Abelian only. ASW gives a phase, an element of U(1). That is the abelian case. Non-abelian statistics needs a matrix acting on a degenerate ground-state manifold — a different and later object.

Where this lands in the classification

In tutorial 1's language, ASW's quasiholes have quantum dimension d = 1. They store a phase and cannot compute. The braid group image is a circle in U(1), abelian, and Freedman–Larsen–Wang density is nowhere in sight.

So ASW is the floor of the ladder rather than a step on it: it establishes that fractional statistics are real and computable in a laboratory system, which is what makes the harder rows worth asking about at all.

Section 5The 36-year gap

ASW predicted θ = π/3 in 1984. Direct experimental confirmation of anyonic braiding arrived in 2020 — Nakamura, Liang, Gardner & Manfra, using a Fabry–Pérot interferometer at ν = 1/3, with the phase jumps matching θ = 2π/3 per quasiparticle in the interference loop. Bartolomei et al. reported complementary evidence the same year from anyon collisions.

Sources

Checked against the paper itself: Arovas, Schrieffer & Wilczek (1984) for the calculation and for θ = π/m; Laughlin (1983) for the wavefunction and the fractional charge; Tsui, Störmer & Gossard (1982) for the ν = 1/3 plateau; Wilczek (1982) for anyons; Halperin (1984); Read & Moore (1992) for the non-abelian case; Nakamura, Liang, Gardner & Manfra (2020) for the interferometric confirmation.

Stated on reputation, not checked here: Leinaas & Myrheim (1977); Bartolomei et al. (2020) for the anyon-collision evidence; Kjønsberg & Leinaas on the monodromy/Berry split; Freedman–Larsen–Wang for density. The mathematics is standard; treat these attributions as the soft part and open them before quoting them.