[Paper Review] Rényi Entropy for the $\sun1$ WZW model on the torus

Howard J. Schnitzer|arXiv (Cornell University)|Oct 20, 2015

Quantum many-body systems28 references3 citations

TL;DR

This paper computes the Rényi entropy for the $ {SU}(N)_1$ Wess-Zumino-Witten (WZW) model on a torus at finite temperature using a n-sheeted branched Riemann surface. It derives the Rényi entropy via the replica trick, showing universal low-temperature and interval-size limits, though analytic continuation to $n\to1$ fails due to Riemann-Siegel theta functions, preventing entanglement entropy extraction.

ABSTRACT

The $\sun1$ WZW model is constructed on a n-sheeted branched torus, which allows the investigation of the Rényi entropy for a single interval at finite temperature. The small and large interval limits, as well as the low temperature expansion are presented for this theory.

Motivation & Objective

• To compute the Rényi entropy for the $ {SU}(N)_1$ WZW model on a torus at finite temperature.
• To investigate the behavior of Rényi entropy in the small and large interval limits and in the low-temperature regime.
• To explore the challenges in analytic continuation to $n\to1$ for entanglement entropy due to Riemann-Siegel theta functions.
• To lay the groundwork for future study of Bose-Fermi duality and topological holographic duals of Rényi entropy in this model.

Proposed method

• Construct the $ {SU}(N)_1$ WZW model on a n-sheeted branched torus using the replica trick, with the partition function $Z_n$ defined on the n-sheeted Riemann surface.
• Use the bosonic formulation of the WZW model based on the weight lattice and Cartan matrix, with action involving metric $g_{ij}$ and anti-symmetric tensor $b_{ij}$.
• Compute the classical and quantum contributions to $Z_n = Z_{n,\text{classical}} Z_{n,\text{quantum}}$, where $Z_{n,\text{classical}} \propto |\Theta(0|i\Omega)|^2$.
• Derive the small and large interval limits of Rényi entropy using asymptotic expansions of theta functions.
• Perform a low-temperature expansion of $Z_n$ in powers of $e^{-\pi\beta}$, with $\beta$ the inverse temperature.
• Extract Rényi entropy via $S_n = -\frac{1}{n-1}\log\left(\frac{Z_n}{Z_1^n}\right)$, focusing on leading-order terms in the limits.

Experimental results

Research questions

• RQ1 What is the behavior of Rényi entropy for the $ {SU}(N)_1$ WZW model on a torus in the small and large interval limits?
• RQ2 How does the Rényi entropy behave in the low-temperature regime for this model?
• RQ3 Why is analytic continuation to $n\to1$ obstructed in this theory, and what are the implications for entanglement entropy?
• RQ4 What is the role of Riemann-Siegel theta functions in the partition function and their impact on the replica trick?
• RQ5 How does the Rényi entropy of $ {SU}(N)_1$ WZW model compare to universal CFT predictions in the low-temperature limit?

Key findings

• The Rényi entropy in the small interval limit is consistent with universal CFT behavior predicted by Cardy and Herzog, with leading correction $\propto e^{-\pi\beta}$.
• In the large interval limit, the Rényi entropy scales as $S_n \sim \tilde{c}_n - \frac{1}{n-1} \cdot 4n(N-1) \left( \frac{\sin \pi(l/L)}{n \sin(\pi/n)(l/L)} - 1 \right) e^{-\pi\beta} + \cdots$.
• The low-temperature expansion of $Z_n$ is dominated by the vacuum sector, with leading correction term proportional to $e^{-\pi\beta}$ and $N-1$ dependence.
• The classical partition function $Z_{n,\text{classical}}$ depends on $|\Theta(0|i\Omega)|^2$, which prevents analytic continuation to $n\to1$, thus entanglement entropy cannot be extracted.
• The Rényi entropy exhibits a universal $\sim \frac{c}{12} n(1 - 1/n^2)$ scaling in the small interval limit, matching CFT expectations.
• The fermionic formulation and Bose-Fermi duality of Rényi entropy are identified as open problems for future work, despite the bosonic formulation being used here.