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Typical entanglement in anyon chains: Page curves beyond Lie group symmetries

Yale Yauk, Lucas Hackl, Alexander Hahn

Abstract

We study bipartite entanglement statistics in one-dimensional anyon chains, whose Hilbert spaces are constrained by fusion rules of unitary pre-modular categories. Our setup generalizes previous frameworks on symmetry-resolved entanglement entropy for non-abelian Lie group symmetries to the setting of quantum groups. We derive analytical expressions for the average anyonic entanglement entropy and its variance. Surprisingly, despite the constrained Hilbert space structure, the large $L$ expansion has no universal $O(\sqrt{L})$ or $O(1)$ symmetry-type corrections except for a subleading topological correction term that produces a Page curve asymmetry. We further show that the variance decays exponentially with system size, establishing the typicality. Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions, supporting the use of eigenstate entanglement as a diagnostic of quantum chaos. Our results establish the anyonic Page curve as an appropriate chaotic benchmark in topological many-body systems and connect anyonic entanglement to Page-type universality in quantum many-body physics.

Typical entanglement in anyon chains: Page curves beyond Lie group symmetries

Abstract

We study bipartite entanglement statistics in one-dimensional anyon chains, whose Hilbert spaces are constrained by fusion rules of unitary pre-modular categories. Our setup generalizes previous frameworks on symmetry-resolved entanglement entropy for non-abelian Lie group symmetries to the setting of quantum groups. We derive analytical expressions for the average anyonic entanglement entropy and its variance. Surprisingly, despite the constrained Hilbert space structure, the large expansion has no universal or symmetry-type corrections except for a subleading topological correction term that produces a Page curve asymmetry. We further show that the variance decays exponentially with system size, establishing the typicality. Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions, supporting the use of eigenstate entanglement as a diagnostic of quantum chaos. Our results establish the anyonic Page curve as an appropriate chaotic benchmark in topological many-body systems and connect anyonic entanglement to Page-type universality in quantum many-body physics.

Paper Structure

This paper contains 32 sections, 1 theorem, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Anyonic quantum information
  3. Setup and conventions
  4. Examples
  5. Abelian anyons
  6. $\mathrm{SU}(2)_k$ anyons
  7. Fibonacci anyons
  8. Anyonic entanglement entropy
  9. Relation to symmetry-resolved entanglement
  10. Anyonic entanglement: average and variance
  11. Average anyonic entanglement entropy
  12. Large $L$ behavior and examples
  13. Abelian anyon models
  14. $\mathrm{SU}(2)_k$ anyons
  15. Fibonacci anyons
  16. ...and 17 more sections

Key Result

Lemma 1

Take a fixed $J$ sector, say $J_0$, and define the vector $\ket{\psi}=\ket{\vb{x},J_0;0}$ in $\mathcal{H}^0$ and corresponding vector state $\omega\in\mathscr{S}(\mathcal{M}_0)$ by $\omega(O)=\langle\psi, O\psi\rangle$, $O\in\mathcal{M}_0$. Restrict $\omega$ to $\mathcal{M}_\mathrm{sub}$ by defining

Figures (3)

  • Figure 1: Distribution of level-spacing ratios $r_m$ of the $L=26$ golden chain Hamiltonian \ref{['eq:golden_chain_OBC']} with $\lambda=0.9\, [0]$ in the sector $J=0$, $\mathop{\mathrm{parity}}\nolimits=+1$. The dashed chaotic GOE [integrable Poissonian] prediction $P_{\mathrm{GOE}}(r)=\frac{27}{4}(r+r^2)/(1+r+r^2)^{5/2}$ [$P_\mathrm{P}(r)=2/(1+r^2)$] is overlayed. Inset: the average level-spacing ratio $\overline{r}$ as a function of $\lambda$, compared to GOE $\overline{r}_{\mathrm{GOE}}\approx0.5307$ [Poissonian $\overline{r}_\mathrm{P}\approx0.3867$] predictions. The blue [red] dot corresponds to the value $\lambda=0.9\, [0]$ chosen in the main plot.
  • Figure 2: Normalized Page curve for the $L=26$ golden chain Hamiltonian \ref{['eq:golden_chain_OBC']} in the sector $J=0$, $\mathop{\mathrm{parity}}\nolimits=+1$. We plot the AEE of integrable (orange) and quantum-chaotic (blue) eigenstates as a function of subsystem fraction $f=L_A/L$. We also show the exact analytical prediction \ref{['eq:exact_AEE']} (red) and its leading-order \ref{['eq:Fib_AEE']} (black). The Page curve for $J=0$ is symmetric across $f=0.5$. Inset: an attempt at finite-size scaling for the half-chain AEE of quantum-chaotic eigenstates. The dashed line is a fit $1+a/L+b/L^2$ to the three largest chain sizes, $L=22,24,26$. The poor scaling results might indicate that the chain lengths used are not yet in the asymptotic regime.
  • Figure 3: Asymmetric normalized Page curve for the $L=24$ golden chain Hamiltonian \ref{['eq:golden_chain_OBC']} with $\lambda=0.9$ in the sector $J=1$, $\mathop{\mathrm{parity}}\nolimits=+1$, c.f. Figure \ref{['fig:OBC_AEE']}. We additionally plot the resolved curve Eq. \ref{['eq:resolved_delta']} at $s=1$. Inset: we plot the asymmetry $\Delta(f)=\abs{\overline{\tilde{S}_A}(f)-\overline{\tilde{S}_A}(1-f)}$ against the subsystem ratio $f$. Away from $f=1/2$ (finite-size effects), the asymmetry is $\log(\phi)$ as expected.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • proof