2.3.1 Exchange Statistics#

Prompts

Why can particles in 3D only be bosons or fermions, while 2D allows any exchange phase?
What is the braid group, and how does it differ from the permutation group?
A charge-flux composite \((q, \Phi)\) is exchanged with an identical copy. Walk through how the Aharonov-Bohm effect produces the exchange phase \(\theta_{\mathrm{ex}} = q\Phi/2\hbar\).
How does the charge-flux model unify braiding statistics, exchange statistics, and the spin-statistics relation into a single formula?

Lecture Notes#

Overview#

In 3D, identical particles must be bosons or fermions — these are the only two options allowed by quantum mechanics. But in 2D, the topology of particle worldlines is richer: exchanging two particles twice is not topologically trivial, so any exchange phase is allowed. This lesson explores why 2D topology opens the door to anyons and develops the charge-flux composite as a concrete toy model for anyonic statistics.

Why 2D Is Special#

In 3D, exchanging two identical particles twice is the same as doing nothing — the particles can always “go around” each other and return to the original configuration. This forces the exchange phase to satisfy \(\mathrm{e}^{2\mathrm{i}\theta} = 1\), giving only two possibilities:

Exchange Statistics in 3D

\[\text{Bosons: } \theta = 0 \quad (\text{no phase}), \qquad \text{Fermions: } \theta = \pi \quad (\text{phase } {-1})\]

These are the only options in three spatial dimensions.

In 2D, the situation changes. When one particle moves around another in a plane, it traces out a loop that cannot be shrunk to a point — the other particle is always “in the way.” Clockwise and counterclockwise exchanges are topologically distinct, so in general \(\hat{\mathcal{P}} \neq \hat{\mathcal{P}}^{-1}\). Exchanging twice is no longer trivial. The exchange phase \(\mathrm{e}^{\mathrm{i}\theta}\) can take any value, not just \(\pm 1\).

Braiding vs. Permutation

In 3D, particle exchange is described by the permutation group \(S_N\). Exchanging twice gives the identity: \(\hat{\mathcal{P}}_{12}^2 = I\).

In 2D, particle exchange is described by the braid group \(B_N\). Exchanging twice does not give the identity — the worldlines of the particles form a braid that cannot be untangled.

Think of particle worldlines as strings extending in time. In 2D+1D spacetime:

Two strings can wind around each other, forming knots and braids
The order in which you braid matters (for non-abelian anyons)
The number of windings is a topological invariant — it cannot be changed by smooth deformations

Definition: (Abelian) Anyons

Particles in 2D with exchange phase \(\theta \neq 0, \pi\) are called anyons (from “any-ons,” because \(\theta\) can be any value). Under a single counterclockwise exchange, the many-body state \(\vert\Psi\rangle\) acquires a phase:

\[\vert \Psi\rangle \to \mathrm{e}^{\mathrm{i}\theta} \vert \Psi\rangle\]

\(\theta = 0\): boson
\(\theta = \pi\): fermion
\(0 < \theta < \pi\): anyon (fractional statistics)

Abelian vs. Non-Abelian

If exchanging particles only produces a phase \(\mathrm{e}^{\mathrm{i}\theta}\), the anyons are abelian — different exchanges commute.

If exchanging particles acts as a unitary matrix on a degenerate ground-state space, i.e. \(\vert \Psi\rangle \to U \vert \Psi\rangle\), the anyons are non-abelian — the order of exchanges matters.

Charge-Flux Composite#

A useful toy model for abelian anyons is the charge-flux composite \((q, \Phi)\): a point charge \(q\) bound to a thin magnetic flux tube \(\Phi\). The Aharonov-Bohm effect then generates statistical phases.

Aharonov-Bohm Phase

When a charge \(q\) completes a closed loop encircling a magnetic flux \(\Phi\), it accumulates a phase:

(52)#\[ \theta_{\mathrm{AB}} = \frac{q\,\Phi}{\hbar} \]

This single formula unifies three kinds of statistics for charge-flux composites:

Three Kinds of Statistics

Braiding statistics. When one composite \((q_1, \Phi_1)\) moves in a full closed loop around another \((q_2, \Phi_2)\), the charge \(q_1\) encircles the flux \(\Phi_2\), accumulating an Aharonov-Bohm phase:

(53)#\[ \theta_{\mathrm{br}} = \frac{q_1 \Phi_2}{\hbar} \]
Exchange statistics. Exchanging two identical composites \((q, \Phi)\) is a half-braid, so the exchange phase is:

(54)#\[ \theta_{\mathrm{ex}} = \frac{\theta_{\mathrm{br}}}{2} = \frac{q\,\Phi}{2\hbar} \]
Spin-statistics relation. A \(2\pi\) self-rotation of a composite produces the same phase as an exchange. The spin quantum number is:

(55)#\[ s = \frac{\theta_{\mathrm{ex}}}{2\pi} = \frac{q\,\Phi}{4\pi\hbar} \]

“Charge around flux” vs “flux around charge”

When composite A encircles composite B, one can equivalently view this as charge \(q_1\) moving around flux \(\Phi_2\), or flux \(\Phi_1\) moving around charge \(q_2\). These are two descriptions of the same topological interaction (the same linking of worldlines), not two separate contributions to be summed. The braiding phase is \(q_1\Phi_2/\hbar\), counted once.

Recovering Bosons and Fermions

Bosons: \(q\Phi/(2\hbar) = 0 \pmod{2\pi}\), so \(\theta_{\mathrm{ex}} = 0\) and \(s \in \mathbb{Z}\) (integer spin).
Fermions: \(q\Phi/(2\hbar) = \pi \pmod{2\pi}\), so \(\theta_{\mathrm{ex}} = \pi\) and \(s \in \mathbb{Z} + \tfrac{1}{2}\) (half-integer spin).
Anyons: any other value gives fractional statistics and fractional spin.

Discussion: Why Does Spin = Exchange?

The spin-statistics relation \(\theta_s = \theta_{\mathrm{ex}}\) follows naturally in the charge-flux picture: both self-rotation and exchange involve a charge sweeping past a flux in a half-braid topology. But in 3D, the spin-statistics theorem is proven from relativistic quantum field theory (CPT symmetry). Is the charge-flux argument a “derivation” of spin-statistics, or merely a consistency check? Can you think of a topological reason why self-rotation and exchange must give the same phase in 2D?

Poll: Boson exchange phase

A boson state \(\vert\psi\rangle\) satisfies \(\hat{P}_{12}\vert\psi\rangle = \vert\psi\rangle\) under particle exchange \(\hat{P}_{12}\). What does this tell you about the wavefunction \(\psi(1, 2)\) relative to \(\psi(2, 1)\)?

(A) \(\psi(2, 1) = \psi(1, 2)\) (symmetric wavefunction).

(B) \(\psi(2, 1) = -\psi(1, 2)\) (antisymmetric wavefunction).

(D) The wavefunction is undefined for identical particles.

Summary#

3D topology forces bosons or fermions: In three spatial dimensions, exchanging two particles twice is topologically equivalent to the identity, so \(\hat{\mathcal{P}}^2 = I\). This forces the exchange phase to satisfy \(\mathrm{e}^{2\mathrm{i}\theta} = 1\), giving only \(\theta = 0\) (bosons) or \(\theta = \pi\) (fermions).
2D allows anyons: In two dimensions, the braid group \(B_N\) (not permutation group) governs particle exchange. One particle looping around another cannot be continuously shrunk to a point, so \(\hat{\mathcal{P}}^2 \neq I\) in general. The exchange phase \(\theta\) can be any value in \([0, 2\pi)\).
Aharonov-Bohm phase: When a charge \(q\) completes a loop encircling flux \(\Phi\), it acquires phase \(\theta_\mathrm{AB} = q\Phi / \hbar\). This is the mechanism for anyonic statistics.
Charge-flux unification: For composites \((q, \Phi)\), braiding gives \(\theta_\mathrm{br} = q_1\Phi_2 / \hbar\), exchange is half-braid \(\theta_\mathrm{ex} = q\Phi / (2\hbar)\), and spin equals exchange phase: \(s = \theta_\mathrm{ex} / (2\pi)\). This recovers bosons, fermions, and interpolates to anyons.

Homework#

1. Exchange topology in 3D. In three spatial dimensions, exchanging two identical particles twice is topologically equivalent to the identity.

(a) Explain why the path of one particle looping around the other can be continuously shrunk to a point in 3D (the loop can “slip off” through the third dimension).

(b) Conclude that \(\hat{\mathcal{P}}^2 = I\), so the exchange phase satisfies \(\mathrm{e}^{2\mathrm{i}\theta} = 1\). What are the only two solutions?

2. Exchange topology in 2D. In two spatial dimensions, one particle looping around another traces a path that cannot be shrunk to a point.

(a) Explain (with a sketch or in words) why the second particle blocks the contraction of the loop — there is no third dimension to “slip through.”

(b) Why does this mean \(\hat{\mathcal{P}}^2 \neq I\) in general, allowing the exchange phase \(\theta\) to take any value in \([0, 2\pi)\)?

3. Exchange phases. State the exchange phase \(\mathrm{e}^{\mathrm{i}\theta}\) acquired under a single counterclockwise exchange for:

(a) bosons (\(\theta = 0\)),

(b) fermions (\(\theta = \pi\)),

For each case, compute the phase acquired after exchanging the same pair twice (a full braid). In which case(s) does a double exchange return the system to its original state?

4. Permutation vs braid group. In 3D, particle exchange is governed by the permutation group \(S_N\); in 2D, by the braid group \(B_N\).

(a) How many distinct elements does \(S_2\) have? How many does \(B_2\) have?

(b) Explain the key difference: \(S_2\) has \(\hat{\mathcal{P}}^2 = I\) (finite order) while \(B_2\) has elements \(\hat{\mathcal{P}}^n\) for all integers \(n\) (infinite group). What physical operations correspond to \(\hat{\mathcal{P}}^n\) for \(n > 2\)?

5. Charge-flux composite. A charge-flux composite \((q, \Phi)\) consists of a point charge \(q\) bound to a magnetic flux tube \(\Phi\). The Aharonov-Bohm phase for a charge encircling a flux is \(\theta_\text{AB} = q\Phi/\hbar\).

(a) Composite A with \((q_1, \Phi_1)\) performs a full loop around composite B with \((q_2, \Phi_2)\). Show that the braiding phase is \(\theta_\text{br} = q_1\Phi_2/\hbar\).

(b) Explain why the exchange phase (half-braid) for two identical composites \((q, \Phi)\) is \(\theta_\text{ex} = q\Phi/(2\hbar)\).

(c) What values of \(q\Phi/\hbar\) give bosonic exchange (\(\theta_\text{ex} = 0\))? Fermionic exchange (\(\theta_\text{ex} = \pi\))?

6. Anyon accumulation. An anyon with statistical angle \(\theta = 2\pi/5\) is exchanged with an identical anyon \(n\) times in succession.

(a) What is the total accumulated phase after \(n\) exchanges? Express it as \(n\theta\).

(b) After how many exchanges is the system first returned to its original state (phase equal to a multiple of \(2\pi\))?

7. Spin-statistics relation. The charge-flux model predicts that a \(2\pi\) self-rotation of a composite \((q, \Phi)\) produces the same phase as an exchange, giving spin \(s = \theta_\text{ex}/(2\pi) = q\Phi/(4\pi\hbar)\).

(a) For a boson (\(\theta_\text{ex} = 0\)), show that \(s\) is an integer. For a fermion (\(\theta_\text{ex} = \pi\)), show that \(s\) is a half-integer.

(b) For an anyon with \(\theta_\text{ex} = \pi/3\), what is \(s\)? Is it integer, half-integer, or neither?

(c) Explain why anyons cannot exist as fundamental particles in 3D but can appear as quasiparticle excitations in 2D materials. What role does the dimensionality of space play?