Entanglement and Pseudo Entanglement Dynamics versus Fusion in CFT

TL;DR

This work analyzes how fusion rules and OPE coefficients in RCFTs influence entanglement dynamics under local operator quenches. By studying linear and refined linear combinations of primaries, it demonstrates that EE preferentially detects the heaviest operators while pseudo entropy and its replica-derived generalizations capture information from all primaries, revealing pseudo entropy amplification. Using Schmidt decomposition and the replica trick, the authors derive late time formulas that connect fusion data, quantum dimensions, and OPE coefficients, and they establish a direct relation between fusion numbers and OPE data. The refined operator construction restores light operator information in EE, and the quasiparticle picture remains valid with a block diagonal structure emerging in the reduced density matrix at late times. These results illuminate deep links between algebraic data in RCFTs and quantum information measures, and point to extensions to more general CFTs and holographic settings.

Abstract

The fusion rules and operator product expansion (OPE) serve as crucial tools in the study of operator algebras within conformal field theory (CFT). Building upon the vision of using entanglement to explore the connections between fusion coefficients and OPE coefficients, we employ the replica method and Schmidt decomposition method to investigate the time evolution of entanglement entropy (EE) and pseudo entropy (PE) for linear combinations of operators in rational conformal field theory (RCFT). We obtain a formula that links fusion coefficients, quantum dimensions, and OPE coefficients. We also identify two definition schemes for linear combination operators. Under one scheme, the EE captures information solely for the heaviest operators, while the PE retains information for all operators, reflecting the phenomenon of pseudo entropy amplification. Irrespective of the scheme employed, the EE demonstrates a step-like evolution, illustrating the effectiveness of the quasiparticle propagation picture for the general superposition of locally excited states in RCFT. From the perspective of quasiparticle propagation, we observe spontaneous block-diagonalization of the reduced density matrix of a subsystem when quasiparticles enter the subsystem.

Figures (4)

• Figure 1: Fusion transformations to obtain $\lim\limits_{t\to\infty}\langle\mathcal{O}_1(w_1)\mathcal{O}^\dagger_2(w_2)\mathcal{O}_3(w_3)\mathcal{O}^\dagger_4(w_4)\mathcal{O}(w_5)\mathcal{O}^\dagger_6(w_6)\rangle_{\Sigma_3}$
• Figure 2: The variation of the late-time second REE (in panels (a) and (b)) and the late-time second PREE (in panels (c) and (d)) with respect to the combination coefficient $C_{\sigma}$ in the critical Ising model. The solid lines are plotted using the late-time formula \ref{['resultsfinalapp']}, while the hollow circles denote numerical data computed using the well-known four-point functions in the critical Ising model. The vertical dotted lines in panels (c) and (d), plotted in terms of eqn. \ref{['Cstar-secton2-PREE']}, correspond to the maximum values of the second PREE. All four panels are symmetric along $C_\sigma=0$ since the late-time formula \ref{['resultsfinalapp']} depends solely on the magnitude of the combination coefficients. Note that the UV regulator $\epsilon$, while a sensitive parameter in REE calculations, does not impact PREE computations and can be safely set to 0.
• Figure 3: The variation of the late-time second REE with respect to the combination coefficients in the critical Ising model. Panels (a) and (b): A mixture of spin $\sigma$, energy density $\varepsilon$, and identity $\mathbb{I}$. Panels (c) and (d) correspond to linear combinations of $\varepsilon$ with $\sigma$ and $\varepsilon$ with $\mathbb{I}$, respectively. Panel (b) as well as the hollow circles in panels (c) and (d) represent numerical data obtained from known correlation functions in the critical Ising model. Panel (a) as well as the solid lines in panels (c) and (d) are plotted according to eqn. \ref{['normal-latetime']}. Note that the maximum value of the second REE is $\log(\sqrt{2}+2)$, as the quantum dimensions of three primary are $d_\sigma=\sqrt{2}$, $d_{\varepsilon}=d_{\mathbb{I}}=1$.
• Figure 4: The complete time evolution of EE, which exhibits a step-like behavior.