Topological Order

Intrinsic topological order is the universal long-distance structure of a gapped, long-range-entangled many-body phase whose bulk supports nontrivial superselection sectors and, in two dimensions, anyonic fusion and braiding. It is not defined by breaking an ordinary global symmetry and does not become trivial merely because an optional symmetry is removed.

The phrase names a phase, not one observable. In the standard two-dimensional setting, the same universal structure appears in several forms:

• a locally indistinguishable ground-state manifold on nontrivial space;
• Wilson loops and flux sectors associated with noncontractible cycles;
• anyon types with fusion, braiding, and topological spin;
• modular transformations acting on the ground-state manifold;
• quantized electromagnetic or thermal response when the needed probes exist;
• universal subleading entanglement data.

The aim here is to calculate and compare the universal data of two-dimensional phases: their fusion rules, modular transformations, ground-state counts, charge response and heat transport. Two phases can agree on several of these quantities and still have different quasiparticle statistics. The toric code and double-semion theory below provide an explicit example.

Topological Order Preview develops local indistinguishability and derives the toric-code ground space. Anyons and Braiding explains fusion spaces and the operations that exchange quasiparticles. This page builds on those ideas without repeating their microscopic derivations.

Required background. Topological Order Preview supplies local indistinguishability and the toric-code diagnostic baseline, Anyons and Braiding supplies fusion and exchange data, and Entanglement Entropy supplies the reduced-state diagnostic language.

Helpful background. Fractional Quantum Hall Effect supplies the principal electronic realization and the interpretation of its measurements.

Long-range entanglement and intrinsic topological order

Unless a qualification is stated, the discussion assumes:

• two spatial dimensions and one time dimension;
• a local or sufficiently short-range Hamiltonian;
• a stable bulk gap in the thermodynamic limit;
• finite correlation length away from edges and quasiparticles;
• a closed orientable surface for ground-state topology;
• finitely many pointlike superselection sectors;
• nondegenerate braiding for the bosonic modular-category statements.

The vacuum sector is labeled $0$. Anyon labels are $a,b,c,\ldots$, with conjugate $\bar a$. Fusion multiplicities are $N_{ab}^{c}$. Quantum dimensions are $d_a$, and

$$\mathcal D = \sqrt{ \sum_a d_a^2 }$$

is the total quantum dimension.

The modular matrices use

$$S_{0a} = \frac{d_a}{\mathcal D},$$

and

$$T_{ab} = \delta_{ab} \theta_a \exp\!\left( -\frac{2\pi i}{24}c_- \right),$$

where $\theta_a$ is the topological spin and $c_-$ is the chiral central charge. Authors sometimes remove the common framing phase from $T$ and call the resulting diagonal matrix $U$. Always check the convention before comparing tables.

For Abelian Chern–Simons formulas, $K$ is a symmetric, integral, nondegenerate matrix and $\mathbf t$ is the conserved-charge vector. Orientation fixes the signs of braiding and chiral response. The explicit convention below gives positive winding phase $2\pi\boldsymbol\ell^TK^{-1}\boldsymbol\ell'$. Electronic Hall-sign conventions are discussed in Fractional Quantum Hall Effect.

Local correlations and global distinctions

A local bulk measurement cannot distinguish the ground sectors of an ordinary intrinsic topological phase, up to corrections that vanish with system size. This is stronger than having several nearly degenerate states: branches of a symmetry-breaking phase can be distinguished by a local order parameter. The many-body introduction derives this difference in the toric code.

Long-range entanglement does not require power-law local correlations. A gapped topological phase can have exponentially decaying connected correlations while retaining noncontractible loop operators and nontrivial braiding. A finite-depth local circuit cannot remove these structures. Conversely, a generic symmetric gapped path produces quasi-local continuation; exact finite-depth circuits describe special representatives or approximations, not every microscopic ground state.

An SPT phase becomes trivial after its optional protecting symmetry is relaxed. Intrinsic anyons survive that operation. Symmetry can also permute or fractionalize on anyons, producing symmetry-enriched topological order. A chiral invertible phase such as the bosonic $E_8$ state is different again: it has no nontrivial anyons, yet its thermal chirality cannot be removed by a local circuit.

During a uniformly gapped local deformation, microscopic energies and correlation lengths may change while the anyon theory stays fixed. At finite size the ground levels need not be exactly degenerate. Gap stability results require locality and appropriate topological-order hypotheses, not merely a guessed low-energy level count (Bravyi, Hastings and Michalakis; Hastings and Wen).

Ground-State Manifolds and Topology

Let $\Sigma_g$ be a closed orientable surface of genus $g$. A two-dimensional topological order assigns a finite-dimensional ground-state space

$$\mathcal H(\Sigma_g).$$

For a unitary modular tensor category, its dimension is

$$\dim\mathcal H(\Sigma_g) = \sum_a \left( S_{0a} \right)^{2-2g}.$$

Using $S_{0a}=d_a/\mathcal D$,

$$\dim\mathcal H(\Sigma_g) = \mathcal D^{2g-2} \sum_a d_a^{2-2g}.$$

Three checks are immediate:

$$\dim\mathcal H(S^2)=1,$$ $$\dim\mathcal H(T^2) = \#\{\text{anyon types}\},$$

and higher genus introduces more independent noncontractible cycles. In a topological-field-theory construction, cutting a surface into three-holed pieces introduces intermediate anyon labels. Gluing sums over compatible labels; the modular matrix diagonalizes these fusion constraints, giving the displayed power of $S_{0a}$. The formula counts topological states, not oscillator excitations or microscopic spin configurations.

Degeneracy means a low-energy band

At a fixed point, ground states can be exactly degenerate. In a generic finite sample, virtual anyon tunneling around a noncontractible cycle produces a splitting

$$\delta E_{\mathrm{top}} \sim e^{-L/\xi_a},$$

where $\xi_a$ depends on the lightest process that changes the flux sector. The defining scaling is

$$\frac{ \delta E_{\mathrm{top}}(L) }{ \Delta_{\mathrm{bulk}}(L) } \longrightarrow 0$$

as $L\to\infty$, with $\Delta_{\mathrm{bulk}}\to\Delta_\infty>0$.

Counting several low-energy states on one small torus is therefore not enough. Symmetry breaking, center-of-mass degeneracy, accidental levels, and edge states can all imitate the count.

Mapping-class-group action

Large diffeomorphisms of the torus act on $\mathcal H(T^2)$. Two generators are:

• $S$, which exchanges the two primitive cycles up to orientation;
• $T$, a Dehn twist that shears one cycle around the other.

With the framing phase included in the $T$ defined above, one can choose the lift so that

$$S^2 = C, \qquad (ST)^3 = C,$$

where $C_{ab}=\delta_{a,\bar b}$ is charge conjugation. If $U=\operatorname{diag}(\theta_a)$ omits that framing phase, the corresponding relation is $(SU)^3=e^{2\pi i c_-/8}C$. Confusing $U$ with $T$ makes this consistency check fail. The ground-space geometry permits extraction of braiding data without explicitly moving localized quasiparticles.

Minimally entangled states

For a cylindrical bipartition of a torus, special ground-state combinations diagonalize the anyon flux through the cut. These minimally entangled states provide a flux-sector basis. For non-Abelian theories, different definite-flux states need not have equal entropy: the sector quantum dimension contributes to the constant. The name does not mean that every flux sector attains one common global minimum. Overlaps between bases adapted to inequivalent cuts can yield modular matrices (Zhang and colleagues; Cincio and Vidal).

This procedure requires:

• a converged basis for the full ground-state manifold;
• sufficiently large circumference and length;
• careful phase and permutation fixing;
• control of lattice symmetries and cut conventions;
• checks that the low-energy states belong to one phase.

An arbitrary basis returned by diagonalization does not carry canonical anyon labels.

Fusion, braiding and modular data

An anyon theory must say more than “Abelian” or “non-Abelian.” Its universal data include:

1. simple sectors $a$ and conjugates $\bar a$;
2. fusion multiplicities $N_{ab}^{c}$;
3. associativity transformations $F$;
4. braiding transformations $R$;
5. topological spins $\theta_a$;
6. quantum dimensions $d_a$;
7. chiral central charge $c_-$, including information not fixed by bulk anyons alone.

Anyons and Braiding develops the fusion-space bases and consistency equations. Here their consequences constrain possible phases.

Fusion and quantum dimension

Fusion is

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

Quantum dimensions form a positive solution of

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

For an Abelian anyon, $d_a=1$. A non-Abelian anyon has $d_a>1$, and the fusion-space dimension for many such anyons grows asymptotically like a power of $d_a$.

Modular S and T

The entry $S_{ab}$ encodes a normalized mutual-braiding amplitude. The first row gives quantum dimensions:

$$d_a = \mathcal D S_{0a}.$$

The diagonal matrix $T$ records topological spins together with a common framing phase. Fusion can be recovered from $S$ through the Verlinde formula:

$$N_{ab}^{c} = \sum_x \frac{ S_{ax}S_{bx}S_{cx}^* }{ S_{0x} }.$$

Central charge from Gauss and Milgram

The Gauss–Milgram relation connects bulk spins to the chiral central charge:

$$\frac{1}{\mathcal D} \sum_a d_a^2\theta_a = \exp\!\left( \frac{2\pi i}{8}c_- \right).$$

For a bosonic topological order, the anyon data therefore determine $c_-$ modulo $8$. Stacking with the bosonic $E_8$ invertible phase changes $c_-$ by $8$ without changing anyon sectors.

Modular data are powerful but not complete

$S$ and $T$ determine fusion through Verlinde and capture spins and mutual braiding. They are not, in full mathematical generality, a complete invariant of a modular tensor category: inequivalent categories can share the same modular data. Complete specification can require $F$ and $R$ data or equivalent higher link and punctured-surface invariants.

For many low-rank physical examples, $S$, $T$, and $c_-$ are highly discriminating. Agreement of those quantities can distinguish a specified family of candidates, but matching two matrices is not a general proof of equivalence (Mignard and Schauenburg).

Bosonic and fermionic orders

For a bosonic phase with nondegenerate braiding, the low-energy category is modular: the only sector that braids trivially with every sector is the vacuum.

In an electronic system, the physical electron is a local transparent fermion. It is not treated as a nontrivial emergent anyon, yet it cannot be identified with the bosonic vacuum. Fermionic topological orders require spin-sensitive or super-modular refinements. Applying a bosonic modular formula without tracking the local fermion can double-count sectors or assign the wrong central-charge equivalence.

Topological Field Theory

At energies well below the bulk gap and distances much larger than the correlation length, local propagating bulk modes disappear. The remaining universal amplitudes depend on topology, links, and background fields. A topological quantum field theory assigns:

$$\Sigma \longmapsto \mathcal H(\Sigma)$$

for a closed spatial surface and

$$(M;\text{links}) \longmapsto Z(M;\text{links})$$

for a spacetime with anyon worldlines. Gluing spacetimes corresponds to composing linear maps. This is why ground-space dimension, braiding, and knot invariants belong to one structure.

TQFT is an infrared description. It does not determine the microscopic gap, correlation length, quasiparticle mass, edge velocity, disorder sensitivity, or which Hamiltonian realizes the phase.

Chern–Simons descriptions

Chern–Simons theories express this infrared structure using gauge fields. Wilson lines represent quasiparticle trajectories, and linked trajectories acquire braiding amplitudes. Canonical quantization on a closed surface produces the topological ground space. Non-Abelian examples require fusion spaces as well as phases; their line representations and allowed levels depend on the gauge group and global structure (Witten; Kitaev).

A TQFT does not determine the microscopic gap, edge velocity or which material realizes a phase. The Abelian matrix description below is a particularly calculable family, not a formula for every topological order.

Abelian K-Matrix Theory

Set $\hbar=1$ in the action and use an oriented spacetime. An Abelian theory with $N$ compact gauge fields $a_I$ can be written

$$S[a,A] = -\frac{1}{4\pi} \int K_{IJ}\, a_I\wedge da_J + \frac{q}{2\pi} \int t_I\, A\wedge da_I.$$

Here $A$ is a background electromagnetic field and $q>0$ is the unit-charge magnitude. A line labeled by $\boldsymbol\ell$ inserts $\exp(i\ell_I\oint a_I)$. With the displayed minus sign in the internal action, a positively oriented loop linking that source acquires the positive winding phase used below. The charge current is $j^\mu=\delta S/\delta A_\mu$. Reversing orientation reverses the signed statistics and Hall response. This is a choice of conventions for the Wen–Zee matrix description.

Anyon labels and local particles

An anyon is labeled by an integer vector

$$\boldsymbol\ell \in \mathbb Z^N.$$

Vectors differing by a local excitation are equivalent:

$$\boldsymbol\ell \sim \boldsymbol\ell + K\boldsymbol\Lambda, \qquad \boldsymbol\Lambda\in\mathbb Z^N.$$

Therefore the number of Abelian anyon types is

$$\left| \det K \right|.$$

Fusion is vector addition modulo the $K$ lattice.

Charge and statistics

The conserved charge is

$$Q_{\boldsymbol\ell} = q\, \mathbf t^{\mathsf T} K^{-1} \boldsymbol\ell.$$

The exchange angle is

$$\Theta_{\boldsymbol\ell} = \pi \boldsymbol\ell^{\mathsf T} K^{-1} \boldsymbol\ell \pmod{2\pi},$$

and the full winding phase of $\boldsymbol\ell$ around $\boldsymbol\ell'$ is

$$\Phi_{\boldsymbol\ell,\boldsymbol\ell'} = 2\pi \boldsymbol\ell^{\mathsf T} K^{-1} \boldsymbol\ell' \pmod{2\pi}.$$

Thus

$$\theta_{\boldsymbol\ell} = e^{i\Theta_{\boldsymbol\ell}}.$$

Response and ground-space dimension

Completing the square with $\widetilde a=a-qK^{-1}\mathbf t A$ leaves a background term $q^2(\mathbf t^TK^{-1}\mathbf t)\int A\wedge dA/(4\pi)$. In the corresponding Hall orientation, and restoring $h$,

$$\sigma_{xy}=\frac{q^2}{h}\mathbf t^TK^{-1}\mathbf t.$$

On genus $g$,

$$\dim\mathcal H(\Sigma_g) = \left| \det K \right|^g.$$

The edge has net chiral central charge

$$c_- = \operatorname{signature}(K)$$

for this $K$-matrix theory and its chosen edge orientation. Invertible layers already included as blocks of $K$ contribute to this signature even when they add no anyons.

Basis changes and stable equivalence

An integral change of gauge-field basis,

$$W \in GL(N,\mathbb Z), \qquad \det W=\pm1,$$

acts as

$$K \longmapsto W^{\mathsf T}KW, \qquad \mathbf t \longmapsto W^{\mathsf T}\mathbf t.$$

With $a=Wa'$, a line label transforms as $\boldsymbol\ell'=W^T\boldsymbol\ell$. Charges, winding phases and the signature are unchanged. Adding a physically trivial unimodular block is also allowed in stable equivalence. Adding an arbitrary invertible block is different: an $E_8$ block changes $c_-$ by eight and thermal transport, although it introduces no new anyons. If conserved-charge response is being compared, an invertible charged layer must likewise be retained. The equivalence relation must say explicitly which layers are being ignored.

For a bosonic microscopic system, diagonal entries of $K$ are even in the usual local-boson convention. Fermionic systems allow odd diagonal entries and require the local transparent fermion to be tracked.

Toric-code and double-semion statistics

Topological Order Preview derives the toric-code Hamiltonian and loops. Here its four sectors give a first check on the general formulas.

Order the sectors as

$$(0,e,m,\epsilon).$$

Their fusion is

$$e\times e = m\times m = \epsilon\times\epsilon = 0,$$

and

$$e\times m = \epsilon.$$

All quantum dimensions are one:

$$d_0=d_e=d_m=d_\epsilon=1.$$

Thus $\mathcal D=2$.

The modular matrices, with the common framing phase omitted, are

$$S = \frac12 \begin{pmatrix} 1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1 \end{pmatrix},$$ $$T = \operatorname{diag} (1,1,1,-1).$$

The minus sign for $\epsilon$ says it is a fermion. The entries $S_{em}=S_{me}=-1/2$ encode the mutual $\pi$ phase between $e$ and $m$ after normalization.

An Abelian Chern–Simons representation is

$$K_{\mathrm{TC}} = \begin{pmatrix} 0&2\\ 2&0 \end{pmatrix}.$$

Since

$$\left| \det K_{\mathrm{TC}} \right| = 4,$$

there are four anyon types and four torus ground states. The signature is zero, so $c_-=0$. Gauss–Milgram gives

$$\frac{1}{2} \left( 1+1+1-1 \right) = 1,$$

consistent with $c_-=0\pmod8$.

Same state counts, different statistics

Now compare the double-semion theory, described by

$$K_{\mathrm{DS}}=\begin{pmatrix}2&0\\0&-2\end{pmatrix}.$$

Choose representatives $0=(0,0)$, $s_+=(1,0)$, $s_-=(0,1)$ and $b=(1,1)$. Each semion is its own fusion antiparticle: $s_+\times s_+=s_-\times s_-=0$, while $s_+\times s_-=b$. The signs label opposite exchange chirality. Their spins are

$$\begin{aligned} \theta_0&=\theta_b=1,\\ \theta_{s_+}&=i,\qquad \theta_{s_-}=-i. \end{aligned}$$

The toric code and double-semion theory both have four Abelian sectors, $\mathcal D=2$, genus-$g$ dimension $4^g$, vacuum entropy constant $\ln2$ and $c_-=0$. Yet their spins differ. Winding $e$ once around another $e$ in the toric code gives phase $+1$; winding $s_+$ once around another $s_+$ gives $-1$. No relabeling turns the toric-code fermion and two bosons into two semions and a boson.

Each entry is the full winding phase $M_{ab}$ for the row anyon around the column anyon. White cells have phase $+1$, gray cells $-1$. For these Abelian theories $S_{ab}=M_{ab}/2$. Both have four torus sectors and the same entropy constant; the different diagonal winding phases and topological spins distinguish their order. These are calculated universal data, not a microscopic material spectrum.

This pair also shows why adding an $E_8$ layer and changing the anyon theory are separate operations. The former changes thermal chirality without changing these winding matrices; the latter can change the matrices while leaving thermal chirality zero.

Fractional Quantum Hall Order

The fractional quantum Hall effect is the canonical equilibrium material realization because transport, quasiparticle charge, edge structure, and topological order are accessible in one platform.

Laughlin order as a one-component K matrix

For a Laughlin state at filling magnitude $\nu=1/m$, take

$$K=(m), \qquad \mathbf t=(1),$$

with odd $m$ for a one-component electronic state. Anyons are labeled by

$$\ell = 0,1,\ldots,m-1.$$

Their universal data are

$$Q_\ell = q\frac{\ell}{m},$$ $$\Theta_\ell = \pi\frac{\ell^2}{m} \pmod{2\pi},$$

For odd $m$, replacing $\ell$ by $\ell+m$ attaches a local electron. Mutual braiding is unchanged, but the exchange angle shifts by $\pi$. The displayed $\Theta_\ell$ therefore refers to the chosen representatives $0\leq\ell<m$; a complete fermionic description also retains the transparent electron and spin structure.

$$\left| \sigma_{xy} \right| = \frac{q^2}{mh},$$

and

$$\dim\mathcal H(\Sigma_g) = m^g.$$

Every sector is Abelian, so

$$d_\ell=1, \qquad \mathcal D=\sqrt m.$$

The vacuum-sector topological entanglement entropy is therefore

$$\gamma = \ln\mathcal D = \frac12\ln m.$$

These data describe the Laughlin order after identifying sectors that differ by a local electron, with chosen representatives for exchange phases. They do not derive its wavefunction or select which interaction stabilizes it. Fractional Quantum Hall Effect explains those physical questions and the interpretation of charge, interference and transport experiments.

Larger $K$ matrices can have the same $\mathbf t^TK^{-1}\mathbf t$ while differing in neutral sectors, spins or chiral central charge. Non-Abelian phases additionally require fusion spaces; an ordinary Abelian matrix cannot encode them. A filling fraction or Hall plateau alone is therefore an incomplete phase label.

Topological Entanglement Entropy

For a smooth simply connected region $A$ in the vacuum sector of a suitable two-dimensional gapped topological phase,

$$S(A) = \alpha \frac{ \lvert\partial A\rvert }{a} - \gamma +o(1),$$

where $L$ scales the region and $a$ is a short-distance cutoff. Here

$$\gamma = \ln\mathcal D.$$

If a definite anyon $a$ lies in the region, the universal constant changes to

$$\gamma_a = \ln \left( \frac{\mathcal D}{d_a} \right).$$

Thus a non-Abelian anyon with $d_a>1$ raises the universal entropy by $\ln d_a$ relative to the vacuum sector. This assumes a specified topological charge, an appropriate spatial factorization and a cut far from quasiparticles. Smooth geometric corrections can decay as inverse powers of $L$; a finite correlation length does not make every entropy correction exponentially small. Sector superpositions and cuts with multiple boundary components require different constants.

Why one fitted intercept is insufficient

The coefficient $\alpha$ is nonuniversal. Corners, finite correlation length, Goldstone modes, symmetry breaking, finite bond dimension, and cut geometry can all contribute constants. Reliable schemes combine several regions so boundary terms cancel.

Topological Entanglement Entropy Preview develops the Kitaev–Preskill and Levin–Wen subtraction geometries and their numerical limitations.

Even a reliable $\gamma$ gives only $\mathcal D$. Distinct anyon theories can share the same total quantum dimension. It must be joined with sector counting, modular data, response, or excitation information.

Chiral Central Charge and Thermal Response

The chiral central charge is

$$c_- = c_R-c_L,$$

the difference between right- and left-moving edge central charges. In a well-equilibrated chiral edge, the thermal Hall conductance obeys

$$\frac{\kappa_{xy}}{T} = c_- \frac{\pi^2k_B^2}{3h}.$$

This response detects information invisible to electrical Hall conductance, including neutral modes. It also sees invertible chiral layers that carry no nontrivial anyons.

Real measurements require caution:

• edge modes may not equilibrate over the device length;
• phonons and metallic channels carry heat;
• contacts can mix modes;
• upstream and downstream sectors can exchange energy;
• the measured plateau may be a crossover rather than the asymptotic limit.

The Gauss–Milgram relation fixes $c_-$ only modulo $8$ for a bosonic modular category. Thermal response can distinguish phases that share anyon data but differ by an $E_8$ layer.

What measurements can distinguish

A useful comparison starts with candidate phases and asks which predicted quantities differ. The toric-code/double-semion example needs statistics-sensitive information; ground-state counting and the vacuum entropy alone cannot decide it. Even full modular data have known mathematical ambiguities, so conclusions remain tied to the candidate family and measured precision.

Quantity	Possible measurement	Main limitation
Bulk gap	Spectroscopy or size scaling	Spectral versus mobility gap
Ground sectors	Torus or cylinder calculation	Competing ordinary order
Quantum dimensions	Entropy or fusion counting	Sector and size dependence
Spins and winding	Modular transformations or interference	Geometry and phase offsets
Fractional charge	Charge sensing or shot noise	Nonequilibrium bunching
Thermal chirality	Heat transport	Mode equilibration

For a numerical model, first converge the gap, correlation length, energy and truncation error. Then compare the low-energy states across sizes and geometries, test local indistinguishability and competing order, and extract flux, entropy or modular data. The internal width of the ground-state band must collapse relative to a nonzero excitation gap. An arbitrary small-system degeneracy is not enough.

In a material, charge and thermal transport must be separated from parallel channels, contacts and bulk heat leakage. Interference also requires control of dynamical and electrostatic phases. In a simulator, preparing a few states or applying programmed string operations does not by itself demonstrate a stable equilibrium phase. The Hamiltonian, preparation error and excitation dynamics determine which conclusion is supported.

Boundaries, Condensation, and Interfaces

A topological bulk does not uniquely determine one boundary spectrum. Boundary conditions can condense selected bosonic anyons. A gapped boundary is possible only when the condensed set has mutually compatible statistics and is large enough to remove the boundary anomaly.

At an interface between phases $A$ and $B$, the relevant order is the folded product

$$A\boxtimes\overline B.$$

Anyon condensation can identify, confine, or split sectors across the interface. Chiral central charge constrains whether a fully gapped interface is possible:

$$c_-^{A} - c_-^{B} = 0$$

is a necessary thermal condition for the actual fixed interface, but is not sufficient. Equality modulo eight is inadequate: an $E_8$ mismatch leaves eight units of chiral central charge and cannot have a fully gapped interface to the unchanged other phase. Adding an inverse chiral layer changes the physical interface problem.

For a bosonic Abelian order, a fully gapped boundary requires a maximal collection of mutually trivial bosons that can consistently condense, called a Lagrangian subgroup. In the toric code, $\{0,e\}$ or $\{0,m\}$ is such a choice; in double-semion order, $\{0,b\}$ is a choice. Their sizes are $\mathcal D=2$. Additional symmetries may forbid a proposed condensate. For non-Abelian orders the corresponding object is a condensable algebra, not just a list of sectors (Levin).

Edge and Surface States discusses boundary spectra and measurements; Bulk–Boundary Correspondence develops relative bulk constraints. A topological bulk allows a set of boundary realizations, not one unique edge dispersion.

Beyond gapped two-dimensional phases

The modular-category framework is tailored to gapped pointlike excitations in two dimensions.

Three-dimensional topological order

In three spatial dimensions, excitations can include particles, loops, and membranes. Universal processes include particle–loop and loop–loop braiding. Ground-state structure and higher-form symmetries replace parts of the two-dimensional anyon description. A two-dimensional $S$ matrix is not a complete classifier.

Fracton phases

Fracton phases can have excitations with restricted mobility and degeneracy that scales with system size or foliation structure rather than only manifold topology. They require subsystem-symmetry, tensor-gauge, or foliated-equivalence language.

Gapless spin liquids

Fractionalized excitations can occur in a gapless phase. Emergent photons, spinon Fermi surfaces, and algebraic correlations do not define a fully gapped TQFT. Calling every fractionalized state “topologically ordered” without specifying the gap convention obscures this difference.

Nonunitary effective theories

Unitary topological order in a stable Hermitian quantum system requires positive-definite Hilbert spaces and unitary braiding. Nonunitary conformal blocks may be useful trial-state constructions but do not automatically define a stable gapped phase.

Common pitfalls

Counting states without testing locality. Ordinary symmetry breaking and accidental levels can imitate a topological degeneracy. Combine scaling with local matrix elements and extended operators.

Identifying a phase from one scalar. Equal Hall response, total quantum dimension or thermal chirality does not imply equal anyon statistics. Compare quantities that distinguish the actual alternatives.

Dropping a local fermion or chiral layer. Attaching an electron changes exchange spin, and stacking an invertible layer changes thermal response. Neither effect can be discarded silently.

Demanding an exact microscopic edge spectrum. Termination, reconstruction and interactions change dispersions. Condensation rules and net chirality constrain what survives.

Exercises

1. A non-Abelian genus count

An Ising theory has sectors $0,\psi,\sigma$, dimensions $1,1,\sqrt2$ and spins $1,-1,e^{i\pi/8}$. Find the genus-$g$ ground-space dimension and check $g=0,1,2$. What does Gauss–Milgram give for $c_-$?

Solution

Here $\mathcal D=2$. Thus

$$\begin{aligned} \dim\mathcal H(\Sigma_g) &=2^{2g-2}(2+2^{1-g})\\ &=2^{g-1}(2^g+1). \end{aligned}$$

The first three dimensions are $1,3,10$. Unlike an Abelian theory, three torus sectors do not imply $3^g$ states on higher genus. The weighted spin sum is

$$\frac{1-1+2e^{i\pi/8}}{2}=e^{i\pi/8},$$

so $c_-=1/2\pmod8$. A microscopic realization can carry an additional invertible chiral layer; the anyon data alone do not fix that multiple of eight.

2. Distinguishing two four-sector phases

Use $K_{\mathrm{TC}}$ and $K_{\mathrm{DS}}$ to calculate the full winding phase of two identical nontrivial anyons. Why does the answer distinguish these phases even though both have $\gamma=\ln2$?

Solution

For the toric code,

$$K_{\mathrm{TC}}^{-1}=\begin{pmatrix}0&1/2\\1/2&0\end{pmatrix}.$$

For each representative $\ell=(1,0),(0,1),(1,1)$, $\exp(2\pi i\ell^TK^{-1}\ell)=1$. The fermion has exchange phase $-1$ but full self-winding phase $+1$.

For double-semion order,

$$K_{\mathrm{DS}}^{-1}=\operatorname{diag}(1/2,-1/2).$$

Both semions have full self-winding phase $-1$; their bound state $b$ has $+1$. The diagonal entries of the winding matrices cannot agree under any permutation. Equal total quantum dimension specifies a shared entropy constant, not equal statistics.

3. A two-component electronic Hall order

Let

$$K=\begin{pmatrix}3&1\\1&3\end{pmatrix},\qquad \mathbf t=\begin{pmatrix}1\\1\end{pmatrix}.$$

Find the torus count, Hall response, minimal positive fractional charge, spin of $\ell=(1,0)^T$ and $c_-$. Show that this $\ell$ generates all sectors. What changes when a local electron is attached?

Solution

The determinant is eight and

$$K^{-1}=\frac18\begin{pmatrix}3&-1\\-1&3\end{pmatrix}.$$

Consequently

$$\sigma_{xy}=\frac{q^2}{2h},\qquad Q_\ell=\frac q4,$$ $$\theta_\ell=e^{3\pi i/8},\qquad c_-=2.$$

There are eight torus sectors. Every charge is an integer multiple of $q/4$, and this representative achieves that minimum. The vector $nK^{-1}(1,0)^T=(3n,-n)^T/8$ is integral first at $n=8$, so fusion by $\ell$ generates a cyclic group of order eight.

Attaching the local particle $K(1,0)^T$ changes the charge by $q$ and the exchange phase by $e^{i\pi K_{11}}=-1$. Mutual winding phases are unchanged because their exponent shifts by an integer multiple of $2\pi$. Spin modulo $2\pi$ therefore belongs to a chosen fermionic representative, not to the quotient sector alone.

4. Changing a matrix basis

Take $a=Wa'$ with integral unimodular $W$. Show that charge, full winding and the Hall coefficient remain unchanged under the transformed $K$, $\mathbf t$ and line labels.

Solution

Substitution into the action and line integral gives

$$K'=W^TKW,\qquad \mathbf t'=W^T\mathbf t,$$ $$\ell'=W^T\ell.$$

Define $x=K^{-1}\ell$. The primed equation $K'x'=\ell'$ is solved by $x'=W^{-1}x$. Therefore

$$(\mathbf t')^Tx'=\mathbf t^Tx,$$

so charge is unchanged. Replacing $\mathbf t$ by a second line label proves winding invariance. Replacing $\ell$ by $\mathbf t$ proves invariance of the Hall coefficient. Unimodularity preserves the integer lattice, and Sylvester’s law preserves the signature. A real invertible change of basis alone is not enough: it can change which lines are local integer excitations.

5. Equal anyons but a chiral interface

One sample realizes toric-code order; another is the same order stacked with a bosonic $E_8$ phase. Which of the anyon sectors, vacuum entropy constant and thermal Hall coefficient differ? Can the interface be fully gapped without changing either bulk?

Solution

The anyon sectors and $\gamma=\ln2$ agree. The additional invertible layer has a one-dimensional closed-surface ground space and no nontrivial anyons, so it changes neither count. It changes $c_-$ by eight, hence

$$\Delta(\kappa_{xy}/T)=8\frac{\pi^2k_B^2}{3h}.$$

Matching anyons across the interface can remove their nonchiral sector, but cannot remove the remaining net thermal chirality. A fully gapped interface between these fixed bulks is impossible. Stacking an inverse $E_8$ layer would cancel the mismatch by changing the physical system.

Continue studying

Fractionalization and Quantum Spin Liquids connect these ideas to magnetic excitations and distinguish gapped from gapless phases. Emergent Gauge Fields explains how gauge descriptions arise from local constraints.

Topological Entanglement Entropy and Entanglement Spectrum develop numerical extraction methods. Topological Codes and Surface Code explain active error correction; Topological Quantum Computation develops operations in finite anyon spaces.

References

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Further reading

• S. H. Simon, Topological Quantum, Oxford University Press (2023). A detailed introduction to fusion, braiding and topological field theory.
• X.-G. Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press (2004), Chapters 4–10.