Anyonic Page Curve in Fusion Spaces

• Anyonic Page curve is the average anyonic entanglement entropy in fusion-constrained Hilbert spaces, characterized by a volume law dictated by the quantum dimension of the anyon species.
• It employs Haar-random pure states in fixed total charge sectors, using anyonic partial traces and quantum traces to yield exact closed-form entropy averages.
• The framework demonstrates strong typicality and reveals a distinctive topological asymmetry between abelian and non-abelian sectors, absent in conventional Page curves.

The anyonic Page curve is the average anyonic entanglement entropy (AEE) of Haar-random pure states in a fixed total topological-charge sector of a one-dimensional anyon chain. In this setting, the Hilbert space is not a tensor product of local factors but a fusion space determined by the fusion rules of a unitary pre-modular category, or in modular cases a UMTC. The resulting Page curve is therefore a Page-type entanglement benchmark for topological many-body systems rather than for unconstrained spin chains. Its central features are a volume law with coefficient set by the quantum dimension of the physical anyon species, the absence of the universal $O(\sqrt{L})$ or $O(1)$ symmetry-type corrections familiar from Lie-group symmetry-resolved Page curves, and a distinctly topological asymmetry governed by the quantum dimension $d_J$ of the total charge sector $J$ (Yauk et al., 26 Mar 2026).

1. Fusion-constrained Hilbert spaces and categorical setting

The formal setting is an anyon chain on a disk with $L$ identical anyons of type $\mathfrak j$. The anyon types are the simple objects $a\in\mathcal S$ of a unitary pre-modular category, with fusion algebra

$$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$

where $N_{ab}^c\in \mathbb N$ are fusion multiplicities. The analysis is restricted to the multiplicity-free case, $N_{ab}^c\in\{0,1\}$. The full Hilbert space decomposes into fixed-total-charge sectors,

$O(1)$0

where $O(1)$1 is the fusion space for total topological charge $O(1)$2, and $O(1)$3 counts admissible fusion paths ending in $O(1)$4 (Yauk et al., 26 Mar 2026).

This structure differs fundamentally from unconstrained tensor-product systems. Basis states are fusion trees rather than independent local basis labels, and the observable algebra respects superselection: $O(1)$5 Different total-charge sectors therefore cannot be coherently mixed by physical observables. The framework is described as a quantum-group generalization of non-abelian symmetry-resolved entanglement. For $O(1)$6-invariant spin chains, one has Schur–Weyl-type decompositions into symmetry irreps and multiplicity spaces; for $O(1)$7 anyons, the analogous multiplicity spaces arise after semisimplifying the representation theory of $O(1)$8 at root of unity. In that sense, $O(1)$9 is the $d_J$0-deformed multiplicity space, and the anyonic Page curve extends Page-curve analysis beyond Lie-group symmetries (Yauk et al., 26 Mar 2026).

2. Bipartition and anyonic entanglement entropy

To define the curve, the chain is bipartitioned into a left block $d_J$1 of $d_J$2 anyons and a right block $d_J$3 of $d_J$4 anyons, with subsystem fraction

$d_J$5

Inside a fixed total-charge sector $d_J$6, the fusion space decomposes as

$d_J$7

where $d_J$8 and $d_J$9 are subsystem total charges constrained by $J$0. The Page curve is the average entanglement as a function of $J$1 (Yauk et al., 26 Mar 2026).

The entropy used here is not the ordinary von Neumann entropy but the anyonic entanglement entropy, defined with the quantum trace. For a pure state in sector $J$2,

$J$3

the reduced state on $J$4 is defined by the anyonic partial trace, and the AEE is

$J$5

The quantum trace differs from the ordinary trace by quantum-dimension weights: $J$6 This is the trace compatible with isotopy invariance and tensor products of copies (Yauk et al., 26 Mar 2026).

For a generic state, the reduced state decomposes into subsystem-charge sectors,

$J$7

where $J$8 is the probability that subsystem $J$9 carries topological charge $L$0. Writing $L$1 for the eigenvalues of the corresponding normalized sector matrix $L$2, the AEE of a single state takes the form

$L$3

Relative to the ordinary von Neumann entropy, the additional term is the topological contribution $L$4, weighted by the charge distribution (Yauk et al., 26 Mar 2026).

3. Exact Haar average and large-$L$5 asymptotics

The random-state ensemble is Haar-uniform on the fixed fusion space $L$6. Under this measure, the sector weights $L$7 have a Dirichlet distribution determined by the dimensions of the $L$8 sectors, while each $L$9 has a fixed-trace Wishart–Laguerre distribution. This yields an exact closed-form average: $\mathfrak j$0 with

$\mathfrak j$1

Here $\mathfrak j$2, $\mathfrak j$3, and $\mathfrak j$4 (Yauk et al., 26 Mar 2026).

For UMTCs, the large-$\mathfrak j$5 behavior of the fusion-space dimensions is controlled by the modular $\mathfrak j$6-matrix through the Verlinde formula. For identical anyons $\mathfrak j$7, the dominant eigenvalue of the fusion matrix $\mathfrak j$8 is

$\mathfrak j$9

so the sector dimensions scale exponentially,

$a\in\mathcal S$0

The leading entanglement coefficient is therefore $a\in\mathcal S$1, which acts as an effective local Hilbert-space dimension in the Page-curve asymptotics (Yauk et al., 26 Mar 2026).

A key cancellation occurs when these asymptotics are substituted into the exact average formula. The $a\in\mathcal S$2-dependence of $a\in\mathcal S$3 disappears at leading and constant order, and the large-$a\in\mathcal S$4 anyonic Page curve becomes

$a\in\mathcal S$5

This result is notably close to the ordinary Page curve, but it carries a topological asymmetry when $a\in\mathcal S$6 is non-abelian (Yauk et al., 26 Mar 2026).

4. Absence of symmetry-type corrections and topological asymmetry

The leading term is a volume law, but its subleading structure differs sharply from the Lie-group case. In symmetry-resolved Page curves for $a\in\mathcal S$7, $a\in\mathcal S$8, or more general Lie-group symmetries, Hilbert-space dimensions typically scale like $a\in\mathcal S$9, and the polynomial prefactor generates nontrivial $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$0 or $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$1 corrections. In the anyonic setting, the fusion-category Hilbert-space dimensions scale purely exponentially, without such polynomial factors, and the use of AEE with quantum trace removes the would-be constant terms. Consequently there are no universal $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$2 or $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$3 symmetry-type corrections, except for a subleading topological correction term (Yauk et al., 26 Mar 2026).

What remains is controlled by the global topological charge sector $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$4. If $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$5 is abelian, then $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$6, and the curve is symmetric up to the usual half-cut Page dip. If $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$7 is non-abelian, then $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$8, and the curve is asymmetric under $$a\otimes b=\bigoplus_{c\in\mathcal S} N_{ab}^c\, c,$$9. Away from the midpoint,

$N_{ab}^c\in \mathbb N$0

equivalently,

$N_{ab}^c\in \mathbb N$1

up to finite-size smoothing near $N_{ab}^c\in \mathbb N$2. The correction depends on the quantum dimension of the total charge, not directly on the subsystem charge after averaging. It is therefore a boundary or superselection effect of the total charge sector (Yauk et al., 26 Mar 2026).

The asymmetry is tied to the failure of ordinary complementary-region symmetry for anyonic reduced states. In the unconstrained tensor-product Page curve, pure states give exact symmetry under $N_{ab}^c\in \mathbb N$3. Here that symmetry fails because for non-abelian total charge the reduced states inherit a superselection asymmetry, expressed by the relation

$N_{ab}^c\in \mathbb N$4

The apparent discontinuity at $N_{ab}^c\in \mathbb N$5 is a large-$N_{ab}^c\in \mathbb N$6 artifact. A double-scaling analysis with

$N_{ab}^c\in \mathbb N$7

shows that the relevant crossover occurs at $N_{ab}^c\in \mathbb N$8, smoothing the half-cut Page correction into an exponential crossover and recovering the $N_{ab}^c\in \mathbb N$9 jump continuously (Yauk et al., 26 Mar 2026).

5. Typicality, model examples, and chaotic benchmarks

The average curve is also typical. The exact variance of the AEE can be written in closed form, and its leading asymptotics are

$N_{ab}^c\in\{0,1\}$0

The variance therefore vanishes exponentially with system size, establishing strong typicality: Haar-random pure states in a fixed anyonic charge sector have AEE sharply concentrated around the average anyonic Page curve (Yauk et al., 26 Mar 2026).

Several model classes illustrate the general theory. For $N_{ab}^c\in\{0,1\}$1, the modular $N_{ab}^c\in\{0,1\}$2-matrix is

$N_{ab}^c\in\{0,1\}$3

and the quantum dimensions are $N_{ab}^c\in\{0,1\}$4. For Fibonacci anyons, $N_{ab}^c\in\{0,1\}$5 with fusion rule

$N_{ab}^c\in\{0,1\}$6

and

$N_{ab}^c\in\{0,1\}$7

The asymptotic Page curve specializes to

$N_{ab}^c\in\{0,1\}$8

For $N_{ab}^c\in\{0,1\}$9, the curve is symmetric; for $O(1)$00, the asymmetry is $O(1)$01 (Yauk et al., 26 Mar 2026).

The numerical benchmark is the golden chain, the standard nearest-neighbor Fibonacci anyon chain, with Hamiltonian

$O(1)$02

At $O(1)$03 the model is integrable; at $O(1)$04 it enters a quantum-chaotic regime identified through level-spacing ratios, with agreement with GOE statistics, while $O(1)$05 follows Poissonian statistics. For $O(1)$06, $O(1)$07, parity $O(1)$08, the average AEE over the central 2000 eigenstates in the chaotic regime closely follows the analytical Page curve, including the exact finite-$O(1)$09 formula and its large-$O(1)$10 asymptote. In the integrable chain, the entanglement is visibly submaximal for extensive subsystems. For non-abelian total charge, the numerics reproduce the predicted asymmetry, and the measured difference between complementary subsystem entropies approaches $O(1)$11 (Yauk et al., 26 Mar 2026).

6. Terminological scope and relation to Page–Wootters constructions

The expression “anyonic Page curve” admits a potential ambiguity because “Page” can refer either to the entanglement Page curve of random pure states or to the Page–Wootters relational-time proposal. In the entanglement sense, the term denotes the AEE curve just described: a Haar-average entanglement benchmark for fusion-constrained Hilbert spaces with topological asymmetry controlled by $O(1)$12 (Yauk et al., 26 Mar 2026).

By contrast, “Relational time in anyonic systems” studies a Page–Wootters construction implemented with anyons in Chern–Simons theories and explicitly states that it is not about a black-hole Page curve, nor about entropy versus time for evaporating black holes. There, the relevant notion of “Page” is relational time, encoded through correlations between clock and system subsystems, with non-universal anyon models such as $O(1)$13 yielding discrete relational time because only finitely many inequivalent clock POVMs are physically accessible (Nikolova et al., 2017).

This distinction matters conceptually. The entanglement anyonic Page curve concerns typical bipartite AEE in fixed fusion sectors and serves as a chaotic benchmark for topological many-body systems. The Page–Wootters anyonic construction concerns clock-system conditionalization in Hamiltonian-free topological theories. The shared use of “Page” does not indicate a shared observable or a shared curve. Instead, it marks two separate lines of inquiry: one about typical entanglement in fusion spaces, the other about relational time in anyonic models (Yauk et al., 26 Mar 2026, Nikolova et al., 2017).