[Paper Review] Pseudo-entropy for descendant operators in two-dimensional conformal field theories

Song He, Jie Yang+2 more|arXiv (Cornell University)|Jan 12, 2023

Quantum many-body systems4 citations

TL;DR

This paper investigates the late-time behavior of pseudo-Rényi entropy in two-dimensional rational conformal field theories (RCFTs) for locally excited states created by descendant operators. It shows that when two descendant operators arise from the same Virasoro generator acting on a primary operator, the late-time excess of pseudo-Rényi entropy matches the logarithm of the primary operator's quantum dimension—mirroring entanglement entropy. However, for linear combinations of generators, an additional contribution emerges due to holonorphic-antiholomorphic mixing, expressible as the pseudo-Rényi entropy of an effective finite-dimensional transition matrix.

ABSTRACT

We study the late-time behaviors of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories (RCFTs). To construct the transition matrix, we utilize two non-orthogonal locally excited states that are created by the application of different descendant operators to the vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudo-entropy and pseudo-Rényi entropy corresponds to the logarithmic of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(Rényi) entropy compared to that for (Rényi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(Rényi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(Rényi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.

Motivation & Objective

• To understand the late-time behavior of pseudo-Rényi entropy in locally excited states within rational conformal field theories (RCFTs).
• To analyze how the structure of descendant operators—generated by Virasoro generators—affects the transition matrix and resulting pseudo-entropy.
• To compare the late-time excess of pseudo-Rényi entropy with that of entanglement entropy, especially in cases involving mixed holomorphic and antiholomorphic generator contributions.
• To identify and characterize additional contributions to pseudo-Rényi entropy arising from non-orthogonal superpositions of descendant states.

Proposed method

• Construct a transition matrix $ \mathcal{T}^{\psi|\varphi} = \frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle} $ from two non-orthogonal locally excited states created by descendant operators.
• Compute the reduced transition matrix $ \mathcal{T}_A^{\psi|\varphi} = \text{tr}_B[\mathcal{T}^{\psi|\varphi}] $ on a subsystem $ A $, which is non-Hermitian and requires careful logarithmic treatment.
• Use pseudo-Rényi entropy $ S_A^{(n)} = \frac{1}{1-n}\log\text{tr}[(\mathcal{T}_A^{\psi|\varphi})^n] $ for $ n \geq 2 $ to avoid branch cut issues in infinite-dimensional Hilbert spaces.
• Analyze the asymptotic behavior of correlation functions involving descendant operators in the late-time limit using conformal field theory techniques.
• Derive the leading-order behavior of four-point functions involving $ \mathcal{O}^{(-n)} $ and $ \mathcal{O}^{(-m)\dagger} $, mapping them to the transition matrix spectrum.
• Express the additional late-time contribution in the case of mixed generators as the pseudo-Rényi entropy of an effective finite-dimensional transition matrix.

Experimental results

Research questions

• RQ1 How does the late-time pseudo-Rényi entropy behave for locally excited states created by descendant operators in 2D RCFTs?
• RQ2 Does the late-time excess of pseudo-Rényi entropy match the logarithm of the quantum dimension of the primary operator when both descendants arise from the same Virasoro generator?
• RQ3 What happens to the late-time pseudo-Rényi entropy when the two excited states are generated by a linear combination of Virasoro generators?
• RQ4 Can the additional contribution to pseudo-Rényi entropy in the mixed generator case be systematically captured and expressed in terms of an effective transition matrix?
• RQ5 How does the mixing of holomorphic and antiholomorphic generators affect the entanglement structure in the late-time regime?

Key findings

• When two descendant operators are generated by the same Virasoro generator acting on a primary operator, the late-time excess of pseudo-Rényi entropy is $ \log d_p $, where $ d_p $ is the quantum dimension of the primary operator, matching the entanglement entropy case.
• For linear combinations of Virasoro generators, the late-time excess of pseudo-Rényi entropy deviates from the entanglement entropy formula, indicating a distinct behavior in the non-orthogonal state setup.
• The additional contribution to pseudo-Rényi entropy in the mixed generator case arises due to enhanced entanglement from holomorphic-antiholomorphic mixing.
• This extra contribution is quantitatively expressed as the pseudo-Rényi entropy of an effective transition matrix in a finite-dimensional Hilbert space.
• The leading-order asymptotic behavior of the four-point function $ \langle \mathcal{O}^{(-n)}(1)\mathcal{O}^{(-m)\dagger}(2)\mathcal{O}^{(-n)}(3)\mathcal{O}^{(-m)\dagger}(4) \rangle_{\Sigma_2} $ at late times is derived, showing explicit dependence on $ \Delta $, $ c $, and the operator weights.
• The full late-time excess formula includes terms proportional to $ \Delta $, $ \Gamma(m+n) $, and $ (-1)^n $, with higher-order corrections involving $ \Gamma(m-1) $, $ \Gamma(n-1) $, and $ \Gamma(m+n) $, reflecting the non-trivial structure of the transition matrix spectrum.