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
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:
\(\theta = 0\): 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. \(|\Psi\rangle \to U |\Psi\rangle\), 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:
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:
(55)#\[ \theta_{\mathrm{br}} = \frac{q_1 \Phi_2}{\hbar} \]Exchange statistics. Exchanging two identical composites \((q, \Phi)\) is a half-braid, so the exchange phase is:
(56)#\[ \theta_{\mathrm{ex}} = \frac{\theta_{\mathrm{br}}}{2} = \frac{q\,\Phi}{2\hbar} \]Spin-statistics relation. A \(2\pi\) self-rotation of a composite produces the same phase as an exchange. The spin quantum number is:
(57)#\[ s = \frac{\theta_{\mathrm{ex}}}{2\pi} = \frac{q\,\Phi}{4\pi\hbar} \]
“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?
Summary#
Dimension |
Exchange group |
Allowed statistics |
Exchange squared |
|---|---|---|---|
3D |
Permutation \(S_N\) |
Bosons (\(\theta=0\)) or fermions (\(\theta=\pi\)) |
\(\hat{\mathcal{P}}^2 = I\) |
2D |
Braid \(B_N\) |
Any \(\theta \in [0, 2\pi)\) |
\(\hat{\mathcal{P}}^2 \neq I\) in general |
Discussion: Can Anyons Exist in 3D?
In 3D, the topology of configuration space forces \(\hat{\mathcal{P}}^2 = I\), seemingly ruling out anyons. But what about particles confined to a 2D layer embedded in 3D (like electrons in a quantum well)? Are these “truly 2D” for the purposes of exchange statistics? What physical conditions must be met for anyonic behavior to emerge?
Homework#
1. In 3D, exchanging two identical particles twice is topologically equivalent to the identity. Why? (Hint: think about whether the path of one particle around the other can be continuously shrunk to a point.)
2. In 2D, explain (with a picture or in words) why the path of one particle going around another cannot be shrunk to a point. What physical object blocks the contraction?
3. State the exchange phase for: (a) bosons, (b) fermions, (c) an anyon with statistical angle \(\theta = \pi/3\). For each case, compute the phase acquired after exchanging the same pair of particles twice.
4. What is the difference between the permutation group \(S_2\) and the braid group \(B_2\)? How many distinct elements does each have?
5. An anyon with statistical angle \(\theta = 2\pi/5\) is exchanged with an identical anyon 5 times in succession. What is the total accumulated phase? Is the system back to its original state?
6. Explain in one or two sentences why anyons cannot exist as fundamental particles in 3D, but can exist as quasiparticle excitations in 2D materials.