[Paper Review] Symmetric Orbifolds and Entanglement Entropy for Primary Excitations in Two Dimensional CFT

Amir Esmaeil Mosaffa|arXiv (Cornell University)|Aug 15, 2012

Black Holes and Theoretical Physics18 references5 citations

TL;DR

This paper computes the entanglement entropy (EE) of a single interval in a 2D CFT excited by a primary operator using symmetric orbifolding techniques. By mapping the replica trick to a covering space via $S_n$-twist fields, it derives a closed-form expression for the Rényi entropy as a $2n$-point function, yielding the EE in the $n \to 1$ limit and confirming agreement with prior results in the small-angle limit.

ABSTRACT

We use the techniques in symmetric orbifolding to calculate the Entanglement Entropy of a single interval in a two dimensional conformal field theory on a circle which is excited to a pure highest weight state. This is achieved by calculating the Reney Entropy which is found in terms of a 2n-point function of primary operators, n being the replica number.

Motivation & Objective

• To compute the entanglement entropy (EE) of a single interval in a 2D conformal field theory (CFT) excited by a primary operator.
• To extend the symmetric orbifolding technique—previously used in ground states—to excited pure states with highest weight primary operators.
• To provide a field-theoretic derivation of Rényi entropy using twist operators on a covering space, enabling holographic interpretation.
• To establish consistency with existing results from Alcaraz et al. and Berganza et al. in the small-angle ($\theta \ll 2\pi$) limit.
• To lay the groundwork for a holographic dual description of EE in excited CFT states via bulk field configurations.

Proposed method

• Applies the replica trick to compute Rényi entropy $S_A^{(n)} = \frac{1}{1-n} \ln \text{Tr}(\rho_A^n)$, mapping the problem to a $2n$-point function of primary operators on a covering space.
• Uses symmetric orbifolding to resolve the singular geometry of the replica manifold $\mathcal{R}_n$ by introducing $n$ copies of the target space fields and enforcing $S_n$-twist boundary conditions via twist operators.
• Constructs a coordinate transformation from the original $t$-plane to a covering space $\mathcal{M}_C$ with a conformal map that smooths the branch points, enabling calculation of the path integral on a smooth manifold.
• Evaluates the $2n$-point function $\langle \prod_{k=0}^{n-1} \mathcal{O}(t_k) \tilde{\mathcal{O}}(t_k') \rangle_t$ in terms of the primary operator's conformal dimensions $h, \bar{h}$ and its OPE coefficients.
• Derives the Rényi entropy as $\mathcal{F}^{(n)}_{\mathcal{O}}(\theta) = \left( \delta^n \frac{2^{n-1}}{\hat{\delta}} \frac{\sin^{n-1}(\theta/2)}{n^{n+1}} \right)^{2(h+\bar{h})} \times \frac{\langle \prod \mathcal{O}(t_k) \tilde{\mathcal{O}}(t_k') \rangle_t}{\langle \mathcal{O}(0) \tilde{\mathcal{O}}(\infty) \rangle_z^n}$, valid for general $n$.
• Takes the $n \to 1$ limit to extract the entanglement entropy $S_{\mathcal{O}}(\theta) = S_{GS}(\theta) - \partial_n \mathcal{F}^{(n)}_{\mathcal{O}}(\theta) \big|_{n=1}$, with $\text{Tr}\, \rho_{\mathcal{O}}(\theta) = 1$.

Experimental results

Research questions

• RQ1 How does the entanglement entropy of a single interval in a 2D CFT change when the system is excited by a primary operator?
• RQ2 Can symmetric orbifolding techniques, previously applied to ground states, be generalized to compute Rényi and entanglement entropy in excited pure states?
• RQ3 What is the precise field-theoretic expression for the Rényi entropy in terms of a $2n$-point function of twist fields and primary operators?
• RQ4 How does the resulting Rényi entropy behave in the small-angle limit ($\theta \ll 2\pi$), and does it match known results from other methods?
• RQ5 What is the holographic interpretation of the field-theoretic result, particularly in terms of bulk geometry and dual fields corresponding to the excited state?

Key findings

• The Rényi entropy for a single interval in a 2D CFT excited by a primary operator is derived as a $2n$-point function on a covering space, with the result $\mathcal{F}^{(n)}_{\mathcal{O}}(\theta) = \left( \delta^n \frac{2^{n-1}}{\hat{\delta}} \frac{\sin^{n-1}(\theta/2)}{n^{n+1}} \right)^{2(h+\bar{h})} \times \frac{\langle \prod_{k=0}^{n-1} \mathcal{O}(t_k) \tilde{\mathcal{O}}(t_k') \rangle_t}{\langle \mathcal{O}(0) \tilde{\mathcal{O}}(\infty) \rangle_z^n}$.
• In the small-angle limit ($\theta \ll 2\pi$), the Rényi entropy simplifies to $\mathcal{F}^{(n)}_{\mathcal{O}}(\theta) = 1 + \frac{h+\bar{h}}{3}\left(\frac{1}{n} - n\right)\left(\frac{\theta}{2}\right)^2 + O(\theta^{\Delta+\bar{\Delta}})$, matching the result of Alcaraz et al. (2011).
• The entanglement entropy is obtained as $S_{\mathcal{O}}(\theta) = S_{GS}(\theta) - \partial_n \mathcal{F}^{(n)}_{\mathcal{O}}(\theta) \big|_{n=1}$, with $\text{Tr}\, \rho_{\mathcal{O}}(\theta) = 1$, showing that the excited state contributes a correction to the ground state EE.
• The result is re-expressed in cylindrical coordinates via $s$-mapping, yielding $\mathcal{F}^{(n)}_{\mathcal{O}}(\theta) = n^{-2n(h+\bar{h})} \frac{\langle \prod_{k=0}^{n-1} \mathcal{O}(\frac{\theta+2\pi k}{n}) \tilde{\mathcal{O}}(\frac{2\pi k}{n}) \rangle_{cy}}{\langle \mathcal{O}(\theta) \tilde{\mathcal{O}}(0) \rangle_{cy}^n}$, confirming agreement with Alcaraz and Berganza.
• The method provides a field-theoretic framework for the holographic dual of excited-state EE, suggesting that bulk fields in AdS correspond to the excitation of the CFT primary state.
• The symmetric orbifolding technique successfully generalizes to excited states, offering a new route to compute EE in non-ground states and enabling future exploration of bulk geometry duals.