Entanglement entropy of two disjoint intervals in conformal field theory (0905.2069v2)

Abstract: We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr\rho_A^n for any integer n is calculated as the four-point function of a particular type of twist fields and the final result is expressed in a compact form in terms of the Riemann-Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.

• The paper introduces a novel replica trick approach with twist operators to calculate the Rényi entropy of two disjoint intervals in conformal field theory.
• It validates analytic predictions with numerical data from the Luttinger liquid and XXZ spin chain, confirming universal scaling behaviors.
• The study establishes a robust framework for exploring multipartite entanglement in quantum systems, paving the way for further theoretical and experimental advancements.

Entanglement Entropy of Two Disjoint Intervals in Conformal Field Theory

The paper "Entanglement Entropy of Two Disjoint Intervals in Conformal Field Theory" by Calabrese, Cardy, and Tonni explores the characterization of entanglement in systems described by conformal field theories (CFTs), particularly focusing on the Luttinger liquid model. Entanglement entropy serves as a crucial metric for understanding quantum correlations and scaling behaviors in these systems, especially near critical points.

Core Contributions

The authors concentrate on calculating the entanglement entropy of two disjoint intervals in a CFT framework, extending the understanding of single interval entanglement entropy. They meticulously evaluate the Rényi entropy for two intervals, $A = [u_1, v_1] \cup [u_2, v_2]$, using the replica trick method tailored for CFTs. This involves considering the $n$-th power of the reduced density matrix $\rho_A^n$ as a partition function defined on an $n$-sheeted Riemann surface composed of the intervals in question.

A central innovation in the approach is the representation of $\rho_A^n$ as a four-point function of twist operators on the Riemann surface, yielding results expressed via Riemann-Siegel theta functions. The analysis capably handles the transition between compactified and decompactified limits — the latter notably lacking explicit dependence on system parameters, thus showcasing universal properties.

Theoretical and Numerical Validation

The analytic results in the compactified regime are compared against existing numerical data, showing consistency with theoretical predictions up to specific scaling limits. For instance, in the decompactification limit, a clear analytic continuation is provided for all model parameters, which has been corroborated with numerics for the XXZ spin chain, a well-studied system within the same universality class as the Luttinger liquid.

Implications and Future Directions

This work significantly enhances the toolkit for analyzing entanglement entropy in more complex quantum systems and provides key insights into universal scaling laws. The results extend beyond theoretical curiosity, offering implications for experimental observations in one-dimensional Bose gases, Heisenberg spin chains, and similar systems where Luttinger liquids emerge as effective descriptions.

The methodology paves the way for examining more intricate configurations involving multiple intervals or going beyond one-dimensional systems. Furthermore, the difficulty in analytically continuing results to real $n$ highlights an area ripe for research, potentially involving insights from integrable models or leveraging advanced numerical techniques such as tensor networks or quantum Monte Carlo simulations.

The paper lays groundwork for future explorations, particularly in scenarios where the universality and correspondence with physical models can be directly tested in controlled quantum experiments. Such endeavors might also benefit from a cross-disciplinary approach, combining insights from condensed matter, high energy theory, and quantum information science.

In conclusion, the paper offers a robust analytical framework for understanding multipartite entanglement in critical one-dimensional systems and poses stimulating questions for further theoretical progress and empirical validation.