Anyons and Braiding

An anyon is a pointlike topological excitation in two spatial dimensions whose adiabatic exchanges realize a unitary representation of the braid group rather than only the bosonic or fermionic representations of the permutation group. For Abelian anyons, a braid multiplies the state by a phase. For non-Abelian anyons, braids act as matrices on a protected fusion space, and different braid sequences can fail to commute.

The word anyon describes exchange topology and superselection data, not a new elementary-particle species in vacuum. Intrinsic anyons are deconfined excitations of a two-dimensional topologically ordered phase. Externally imposed defects can carry analogous protected fusion spaces and projective braid operations, but they are not automatically deconfined particles of that phase. Engineered code Hamiltonians provide another setting in which these operations can be studied.

We first distinguish an exchange from a closed winding, then build the fusion space on which a braid acts. A four-Ising-anyon example shows how changing the order of two braids can change a measured fusion probability. Fractional Quantum Hall Effect develops a material realization; Topological Quantum Computation explains how fusion spaces encode and process quantum information.

Required background. Symmetrization Postulate supplies the boson–fermion exchange baseline, and Topological Order Preview supplies the phase, excitation, and superselection framework.

Helpful background. Symmetric Group supplies the permutation comparison, Fractional Quantum Hall Effect supplies a material anyon realization, and the Adiabatic Theorem supplies the controlled-transport conditions used in braiding arguments.

Statistics in Two Dimensions

Ordered and unordered configuration spaces

For $N$ point particles moving on a spatial manifold $M$, let $D\subset M^N$ be the configurations in which at least two particles coincide. Remove these collisions to obtain the ordered configuration space:

$$\operatorname{Conf}_N(M)=M^N\setminus D.$$

For identical particles, configurations related by relabeling represent the same physical arrangement. The unordered configuration space is

$$\mathcal C_N(M) = \frac{\operatorname{Conf}_N(M)} {S_N}.$$

A closed path in $\mathcal C_N(M)$ can exchange particles even though it returns to the same unordered configuration. Paths that can be continuously deformed into one another without collisions belong to the same homotopy class. Quantum transport around such loops may act on the state by a unitary representation of

$$\pi_1\left( \mathcal C_N(M) \right).$$

For the plane,

$$\pi_1\left( \mathcal C_N(\mathbb R^2) \right) \simeq B_N,$$

the braid group. The ordered space instead gives the pure braid group, in which every labeled worldline returns to its original endpoint.

Why the plane is special

In simply connected three-dimensional space, one worldline can pass around another using the extra direction, and an exchange followed by the same exchange can be contracted. For ordinary point particles this reduces the topology to the permutation group. In two dimensions, clockwise and counterclockwise exchanges are distinct, and a double exchange winds one particle around another.

The braid group $B_N$ is generated by $\sigma_i$, the counterclockwise exchange of adjacent particles $i$ and $i+1$, with

$$\sigma_i\sigma_j = \sigma_j\sigma_i, \qquad \lvert i-j\rvert\ge2,$$

and

$$\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}.$$

The first relation says that well-separated exchanges commute. The second says that two ways of moving three adjacent strands produce the same braid. Crucially, $B_N$ does not impose

$$\sigma_i^2=1.$$

Products act on states from right to left: $B_1B_2$ means perform $B_2$ first. Adding that relation turns the braid group into the symmetric group $S_N$. Anyonic statistics live in precisely the information discarded by that quotient.

One spatial dimension is different again: point particles cannot pass without collision unless an interaction, boundary condition, internal path, or lattice process supplies additional structure. Calling every one-dimensional exchange phase “anyon statistics” without specifying that structure is too loose.

Statistics as holonomy

Suppose a gapped family of Hamiltonians depends smoothly on quasiparticle positions $\mathbf R=(\mathbf R_1,\ldots,\mathbf R_N)$ and has a low-energy subspace spanned by $\lvert\psi_\alpha(\mathbf R)\rangle$. After removing the common dynamical phase, transport in an exactly degenerate subspace gives the Wilczek–Zee geometric operator

$$W_\gamma = S_\gamma\,\mathcal P \exp\left( i \oint_\gamma \mathcal A \right),$$

with matrix-valued connection

$$\mathcal A_{\alpha\beta} = i \left\langle \psi_\alpha(\mathbf R) \middle| \nabla_{\mathbf R} \psi_\beta(\mathbf R) \right\rangle \cdot d\mathbf R.$$

The endpoint matrix $S_\gamma$ compares the final basis with the initial one, including the relabeling of identical particles. It equals the identity only in a single-valued frame. Omitting it can miss a braid phase even when the local Berry connection vanishes. With a residual matrix-valued splitting, the projected Hamiltonian must also be included in the time-ordered evolution.

In a topological limit, the statistical part depends on the braid class $[\gamma]$, while electromagnetic and other geometric phases may depend on the path. A multidimensional subspace permits matrix-valued holonomy; it does not by itself guarantee noncommuting holonomies. Holonomy explains the general geometric construction (Wilczek and Zee, 1984).

Topological Charge and Fusion

An anyon type $a$ labels a topological superselection sector. A local operator cannot change the total topological charge in a region unless compensating charge crosses the boundary or another excitation is created. The vacuum sector is $1$, and every $a$ has an antiparticle $\bar a$ with

$$a\times\bar a \supset 1.$$

When anyons $a$ and $b$ are brought together, their total charge can lie in several channels:

$$a\times b = \sum_c N_{ab}^{\ \ c}\,c.$$

The nonnegative integer $N_{ab}^{\ \ c}$ is the dimension of the elementary fusion space

$$\mathcal V_{ab}^{c}.$$

For more anyons, a fusion tree chooses an order in which charges are combined. Different trees are bases of the same physical space, related by unitary $F$-moves. For three incoming charges fusing to $d$,

$$\bigoplus_e \mathcal V_{ab}^{e} \otimes \mathcal V_{ec}^{d} \quad \overset{F_d^{abc}}{\longrightarrow} \quad \bigoplus_f \mathcal V_{af}^{d} \otimes \mathcal V_{bc}^{f}.$$

The intermediate labels $e$ and $f$ are not generally local observables. They encode nonlocal fusion information shared by separated anyons.

The quantum dimensions satisfy

$$d_a d_b = \sum_c N_{ab}^{\ \ c}d_c.$$

For $n$ identical anyons of type $a$, the fusion-space dimension grows asymptotically like $d_a^n$ on the particle counts compatible with the fixed total charge. The dimension at each finite $n$ is an integer; $d_a$ is the asymptotic growth factor, not the number of internal states of one isolated anyon. Abelian anyons have $d_a=1$. A non-Abelian anyon in a unitary topological phase has

$$d_a>1.$$

A merely accidental degeneracy does not make an anyon non-Abelian. The fusion space must be topologically organized and braid operations must act on it noncommutatively.

Abelian Anyons

A one-dimensional unitary representation of $B_N$ maps every generator to the same phase:

$$\rho(\sigma_i) = e^{i\vartheta}.$$

The braid relation forces the phases for neighboring generators to agree. Bosons and fermions are the special cases

$$\vartheta=0 \quad\text{and}\quad \vartheta=\pi \pmod{2\pi}.$$

Other values describe Abelian anyons. “Abelian” means every braid acts by a scalar on the relevant state space; it does not mean the phase is small or physically trivial.

Exchange, full braid, and spin

Keep three operations separate:

1. $R_c^{ab}$ exchanges $a$ and $b$ in fusion channel $c$.
2. $R_c^{ba}R_c^{ab}$ takes one anyon fully around the other.
3. $\Theta_a$ rotates one anyon by $2\pi$ and is its topological spin phase.

On each fusion channel they obey the ribbon identity; when fusion multiplicities occur, the right-hand side multiplies the identity on that channel:

$$R_c^{ba}R_c^{ab} = \frac{\Theta_c} {\Theta_a\Theta_b}.$$

For identical Abelian anyons in a unique fusion channel, a full braid is the square of the exchange amplitude. For distinguishable anyon types, a single exchange swaps the ordered labels and need not be a scalar operation on the original basis; the closed full braid is usually the cleaner observable.

Topological spin is related to exchange statistics but is not ordinary microscopic spin angular momentum. A fractional Hall quasiparticle can have a fractional topological spin even though the underlying electrons are spin-$1/2$ fermions.

Toric-code mutual statistics

The toric-code sectors obey

$$\begin{aligned} e\times e&=m\times m=\epsilon\times\epsilon=1,\\ e\times m&=\epsilon. \end{aligned}$$

The $e$ and $m$ anyons each have bosonic self-statistics in the conventional model, yet a full braid of $e$ around $m$ contributes

$$M_{em} = -1.$$

Their mutual statistics are nontrivial even though every sector is Abelian. The composite $\epsilon=e\times m$ is a fermion. Topological Order Preview derives these statements from crossing string operators.

Non-Abelian Anyons

For non-Abelian anyons, braiding gives a matrix representation

$$\rho: B_N \longrightarrow U(\mathcal V),$$

where $\mathcal V$ is a multi-anyon fusion space. Two braid words can obey

$$\rho(\mathcal B_1) \rho(\mathcal B_2) \ne \rho(\mathcal B_2) \rho(\mathcal B_1).$$

This is the meaning of non-Abelian statistics. It has nothing to do with a non-Abelian microscopic spin-rotation or gauge symmetry.

Ising anyons

The Ising theory has charges

$$1, \qquad \sigma, \qquad \psi$$

with fusion

$$\sigma\times\sigma = 1+\psi,$$ $$\sigma\times\psi = \sigma, \qquad \psi\times\psi = 1.$$

Because two $\sigma$ anyons have two possible fusion outcomes,

$$d_\sigma = \sqrt2.$$

Four $\sigma$ anyons constrained to total charge $1$ span a two-dimensional space. In the basis where the first pair fuses to $x\in\{1,\psi\}$, a common gauge choice is (Nayak et al., 2008)

$$R^{\sigma\sigma} = \begin{pmatrix} e^{-i\pi/8} & 0\\ 0 & e^{3i\pi/8} \end{pmatrix},$$

and

$$F_{\sigma}^{\sigma\sigma\sigma} = \frac{1}{\sqrt2} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}.$$

The first adjacent braid is diagonal:

$$B_1 = R^{\sigma\sigma}.$$

To braid a pair that is not diagonal in this fusion basis, change basis, apply $R$, and change back:

$$B_2 = F^{-1} R^{\sigma\sigma} F.$$

Then $B_1B_2\ne B_2B_1$. Individual matrix entries depend on fusion-tree and gauge conventions; probabilities for a complete preparation, braid, and fusion-measurement protocol do not.

Turning braid order into a measured probability

Write $|0\rangle$ for first-pair fusion to $1$ and $|1\rangle$ for fusion to $\psi$. Prepare $|0\rangle$ by creating two vacuum pairs, keeping total charge fixed to $1$. Compare $B_1B_2$ with $B_2B_1$. A direct measurement in this basis gives probabilities $(1/2,1/2)$ in both cases: noncommuting matrices need not be distinguished by every measurement.

Apply the same analyzing braid $B_2^{-1}$ before measuring in both cases:

$$\begin{aligned} |A\rangle&=B_2^{-1}B_1B_2|0\rangle,\\ |B\rangle&=B_2^{-1}B_2B_1|0\rangle. \end{aligned}$$

Direct multiplication gives

$$\begin{aligned} P_A(1)&=\tfrac12,\qquad P_A(\psi)=\tfrac12,\\ P_B(1)&=1,\qquad P_B(\psi)=0. \end{aligned}$$

The preparation, analyzer and measurement are identical. The different probabilities are therefore sensitive to the order of the middle two braids. They are predictions of the ideal Ising model, not a claim about a particular device.

Top: one generator exchanges the endpoints; two same-sign exchanges return them while retaining a winding. Time runs upward, and the gaps distinguish over- and undercrossings in the projected worldlines. Bottom: the two Ising protocols above have different fusion probabilities after the same analyzing braid. These probabilities, unlike individual matrix entries, are unchanged by a consistent change of fusion basis.

Fibonacci anyons

The Fibonacci theory contains $1$ and $\tau$ with

$$\tau\times\tau = 1+\tau.$$

Its nontrivial quantum dimension is the golden ratio,

$$d_\tau = \varphi = \frac{1+\sqrt5}{2}.$$

Let $a_n$ and $b_n$ count fusion paths of $n$ $\tau$ anyons with total charge $1$ and $\tau$. Adding one $\tau$ gives

$$a_{n+1}=b_n,\qquad b_{n+1}=a_n+b_n.$$

Number of anyons	Total charge $1$	Total charge $\tau$
2	1	1
3	1	2
4	2	3
5	3	5
6	5	8

The recursion matrix has largest eigenvalue $\varphi$, explaining the noninteger quantum dimension through integer state counts.

Ising versus Fibonacci braid universality

Fibonacci braid representations can approximate arbitrary unitaries on suitable encoded spaces. By contrast, braiding ordinary Ising anyons generates protected Clifford operations but not a universal gate set by itself. Non-Abelian does not automatically mean computationally universal.

Braiding Operations

A physical braid protocol

An ideal adiabatic protocol has four stages:

1. prepare a definite total topological charge and a separated set of anyons;
2. move them along worldlines without collisions or uncontrolled pair creation;
3. return to the original spatial arrangement after a chosen braid word;
4. fuse or interferometrically measure a basis-sensitive outcome.

Let $\Delta_{\min}$ be the minimum gap from the transported fusion space to any unwanted excitation, including localized core states, $\delta$ the residual splitting within the nominal fusion space, $\tau_{\mathrm{qp}}$ a quasiparticle-poisoning time, and $\tau_\phi$ a coherence time. A necessary scale separation is

$$\frac{\hbar}{\Delta_{\min}}\ll\tau_{\rm braid}, \qquad \tau_{\rm braid}\ll\frac{\hbar}{\delta},$$

with the braid also short compared with poisoning and coherence times. The first inequality alone is not a sufficient adiabatic condition. For a low-energy state $a$ and an unwanted state $n$, the local parameter

$$\eta_{na}(t)= \frac{\hbar|\langle n|\dot H|a\rangle|}{|E_n-E_a|^2}$$

must be controlled along the path. Integrated errors also depend on path length, smoothness, endpoint switching and possible coherent accumulation; there is no universal leakage formula determined by one bulk gap and one total duration (Jansen, Ruskai and Seiler, 2007).

Finite-separation splitting contributes relative dynamical phases of order $\delta\tau_{\rm braid}/\hbar$. Making a braid slower reduces leakage but can increase that error. Exponentially small splitting with separation is a useful local, gapped-model limit, not a guarantee for arbitrary long-range couplings, gapless environments or defects.

Path independence and its limits

Two trajectories implement the same topological operation when their worldlines can be smoothly deformed into one another while:

• preserving endpoints and braid orientation;
• avoiding quasiparticle collisions;
• avoiding boundaries or defects that change the topology;
• keeping the bulk gap open;
• preserving the set of topological charges.

Path independence does not mean spatial control can be crude. Moving too quickly causes nonadiabatic leakage; moving too close creates exponentially larger splittings; moving around an unintended quasiparticle changes the braid class; and long-range electromagnetic phases can retain geometric dependence.

Creating and measuring anyons

Local operations create total-vacuum combinations such as $a,\bar a$ pairs. Fusion measurement brings charges together or couples their total channel to an interferometer, resonator, charge sensor, parity detector, or boundary. A measured fusion probability depends on preparation, braid, readout model, and detector fidelity.

Thus an observable prediction includes a preparation, an ordered braid word and a readout, as in the Ising example. A common phase can disappear from fusion probabilities yet remain measurable relative to a reference path in an interferometer.

Quantum Hall Examples

Laughlin anyons

For the electron Laughlin phase at $\nu=1/m$, $q=-e$, and $B>0$, use physical counterclockwise braids in the $xy$ plane as on Fractional Quantum Hall Effect. Topological sectors can be labeled by

$$l \in \mathbb Z_m.$$

Fusion is addition modulo $m$:

$$l\times l' = l+l' \pmod m.$$

For chosen integer representatives $0\leq l<m$, the charge, physical counterclockwise exchange phase, and physical full mutual braid are

$$\frac{q_l}{e} = \frac{l}{m},$$ $$R^{ll} = \exp\left( -\frac{i\pi l^2}{m} \right),$$

and

$$M_{ll'} = \exp\left( -\frac{i2\pi ll'}{m} \right),$$

The familiar positive exponents instead describe counterclockwise paths in the reflected analytic plane $z=x-iy$, which are clockwise physical paths. This translation is specific to the displayed Hall realization; the abstract braid group and the independent Ising examples retain their own declared data.

For odd $m$, replacing $l$ by $l\pm m$ attaches a local electron or electron hole. The charge changes by an integer multiple of $e$, and the exchange phase changes by $-1$, although every mutual braid is unchanged. The displayed self-exchange phase therefore depends on the chosen representative, with the transparent local fermion retained in a complete fermionic description. Every sector has quantum dimension one, so these anyons are Abelian. The fractional Hall page derives fractional charge and explains the corresponding measurements.

From an anyon model to a physical system

The Moore–Read construction contains Ising-type sectors together with a charge sector; its quasiparticle braid phases are not obtained by dropping the charged part of the theory. Read–Rezayi constructions give other non-Abelian examples. A candidate wavefunction and its braid matrices do not establish that a particular material realizes that order.

Likewise, a programmed toric-code string experiment tests operations in an encoded model. It does not identify deconfined non-Abelian excitations of the processor’s underlying superconducting material. Anyon Research and Fractional Hall Frontiers discuss the experimental questions. Topological Quantum Computation develops encodings, readout and the extra operations required beyond Ising braids.

Local perturbations may shift the entire fusion space by a common energy. Protection means that their ability to distinguish or mix its channels is small when the anyons are well separated under the appropriate gap and locality assumptions. It does not make every matrix element of a local operator small. Thermal excitations, finite separation, leakage and measurement errors still require control.

Exercises

1. From braids to permutations

Show that imposing $\sigma_i^2=1$ on the braid-group presentation produces the adjacent-transposition presentation of $S_N$. What topological information is lost?

Solution

The adjacent transpositions $s_i=(i,i+1)$ obey

$$s_i^2=1,$$ $$s_i s_j=s_j s_i, \qquad \lvert i-j\rvert\ge2,$$

and

$$s_i s_{i+1}s_i = s_{i+1}s_i s_{i+1}.$$

These are the braid relations plus $s_i^2=1$. The quotient map sends $\sigma_i$ to $s_i$ and remembers only the endpoint permutation. It discards winding, exchange orientation, and the distinction between two braids with the same final permutation.

2. Mutual statistics in the toric code

Use the crossing-string algebra

$$W_e W_m = -W_m W_e$$

to explain why winding $e$ around $m$ gives a minus sign even though each has bosonic self-statistics.

Solution

A closed $e$ string implementing the winding intersects an $m$ string ending on the enclosed $m$ anyon once. Their operators anticommute, so reversing their order contributes

$$-1.$$

The closed process therefore multiplies the state by $-1$. Self-statistics concerns exchanging two $e$ anyons or two $m$ anyons; mutual statistics concerns winding one species around the other. The two statements are independent.

3. Check the Ising readout

Calculate the probabilities in the two protocols $A$ and $B$ above. Why must a change of fusion basis also transform the prepared state and the measurement projectors?

Solution

Discarding a common phase for the calculation,

$$B_1\sim\begin{pmatrix}1&0\\0&i\end{pmatrix},$$ $$B_2\sim\frac12 \begin{pmatrix}1+i&1-i\\1-i&1+i\end{pmatrix}.$$

Protocol $B$ reduces to $B_1|0\rangle$, so vacuum fusion has probability one. Protocol $A$ has amplitudes of equal magnitude $1/\sqrt2$, hence probabilities $1/2$ in each channel. Under a unitary basis change $G$, transform states by $G$, operators by $GUG^\dagger$ and projectors by $G\Pi G^\dagger$. Then $\langle\psi|U^\dagger\Pi U|\psi\rangle$ is unchanged. Transforming only the braid matrices changes the experiment.

4. Integer dimensions and a noninteger growth factor

Use the Fibonacci fusion recursion to count seven $\tau$ anyons with total vacuum. Contrast this with six Ising $\sigma$ anyons with total vacuum.

Solution

The preceding Fibonacci row is $(a_6,b_6)=(5,8)$, so $(a_7,b_7)=(8,13)$. The fixed-vacuum dimension is eight. For six Ising anyons, fuse three pairs independently to $1$ or $\psi$. Even total $\psi$ parity leaves four of the eight assignments. More generally, $2n$ Ising anyons with total vacuum have dimension $2^{n-1}$. Two anyons with total vacuum have only one state; two possible pair charges do not form a qubit when the total charge has already been fixed.

5. A duration is not an adiabatic guarantee

Consider the two-level Hamiltonian

$$H(t)=\frac{\Delta}{2} [\cos(\omega t)\sigma_z+\sin(\omega t)\sigma_x].$$

Its gap is always $\Delta$. If the field makes $w$ full turns in duration $T$, estimate $\eta_{eg}$. Why does $T\gg\hbar/\Delta$ not guarantee small leakage when $w$ is unrestricted?

Solution

The off-diagonal derivative has magnitude $\Delta|\omega|/2$, giving

$$\eta_{eg}=\frac{\hbar|\omega|}{2\Delta} =\frac{\pi\hbar |w|}{\Delta T}.$$

At $T=100\hbar/\Delta$, one turn gives about $0.0314$, whereas one hundred turns gives about $3.14$. The same gap and duration can therefore describe very different drives. Endpoint smoothness and the full evolution are also needed to bound the final leakage; this local estimate is not a universal error probability.

Further Reading

• C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian Anyons and Topological Quantum Computation,” Reviews of Modern Physics 80, 1083–1159 (2008), doi:10.1103/RevModPhys.80.1083.
• J. K. Pachos, Introduction to Topological Quantum Computation, Cambridge University Press, 2012.
• A. Stern, “Anyons and the Quantum Hall Effect: A Pedagogical Review,” Annals of Physics 323, 204–249 (2008), doi:10.1016/j.aop.2007.10.008.

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