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Twist operators and pseudo entropies in two-dimensional momentum space

Giancarlo Camilo, Andrea Prudenziati

TL;DR

This work defines momentum-space branch-point twist operators in a two-dimensional conformal field theory via a momentum-space replica construction and computes their two-point function. The resulting correlator reduces to a momentum-constrained, delta-function-regulated quantity that, in the $\Delta\to 0$ limit, yields a simple universal function $F(\lambda,0)$ for collinear momenta; the authors propose interpreting the derivative at $n=1$ as a pseudo-Rényi entropy in Nakata’s sense. The analysis reveals constraints from collinearity, mass-shell considerations, and rotation regularization, and discusses how this momentum-space object differs from ordinary entanglement measures while potentially offering new probes of correlations in momentum space and holographic cosmology contexts. Nevertheless, the interpretation relies on nontrivial regularization choices and raises questions about the precise states involved and the broader relevance to momentum-space entanglement literature.

Abstract

We use a replica trick construction to propose a definition of branch-point twist operators in two dimensional momentum space and compute their two-point function. The result is then tentatively interpreted as a pseudo Rényi entropy for momentum modes.

Twist operators and pseudo entropies in two-dimensional momentum space

TL;DR

This work defines momentum-space branch-point twist operators in a two-dimensional conformal field theory via a momentum-space replica construction and computes their two-point function. The resulting correlator reduces to a momentum-constrained, delta-function-regulated quantity that, in the limit, yields a simple universal function for collinear momenta; the authors propose interpreting the derivative at as a pseudo-Rényi entropy in Nakata’s sense. The analysis reveals constraints from collinearity, mass-shell considerations, and rotation regularization, and discusses how this momentum-space object differs from ordinary entanglement measures while potentially offering new probes of correlations in momentum space and holographic cosmology contexts. Nevertheless, the interpretation relies on nontrivial regularization choices and raises questions about the precise states involved and the broader relevance to momentum-space entanglement literature.

Abstract

We use a replica trick construction to propose a definition of branch-point twist operators in two dimensional momentum space and compute their two-point function. The result is then tentatively interpreted as a pseudo Rényi entropy for momentum modes.

Paper Structure

This paper contains 13 sections, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Twist operators and the replica trick in position space
  3. Momentum space twist operators
  4. Two-point function
  5. Physical interpretation
  6. Pseudo-Renyi entropy and Twist operator locations
  7. Discussion on the sign
  8. Symmetry condition for Entanglement Entropy under the exchange $k_1\leftrightarrow k_2$
  9. Behaviour under momentum rotation $k_1\rightarrow e^{i\theta}k_1$
  10. Conclusions
  11. Integration in polar coordinates for $\lambda>0$
  12. Integration in spherical coordinates for $\lambda>0$
  13. $\lambda<0$

Figures (1)

  • Figure 1: Integration region $\mathcal{R}(\lambda)$ for different values of $\lambda$. Top: $\lambda=-100,-1,-0.01$, respectively. Bottom: $\lambda=0.01,2,100$, respectively. For negative $\lambda$ the region is $\mathbb{R}^3$ with a solid hyperboloid removed (for $\lambda<-1$ the hyperboloid becomes two-sheeted). For positive $\lambda$ the region is the interior of an ellipsoid which is oblate for $0<\lambda<1$ and prolate for $\lambda>1$.