Entanglement entropy of two disjoint intervals in conformal field theory II (1011.5482v1)

Abstract: We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We compute Tr\rho_A^n for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form.

• The paper extends entanglement entropy methods by employing Riemann-Siegel theta functions to calculate moments of the reduced density matrix for two disjoint intervals.
• It develops a systematic asymptotic expansion for the four-point ratio scaling function in the short interval regime, improving analytical precision.
• Analytical continuation of these expansions yields von Neumann entropy expressions that align with established results in conformal field theory.

Entanglement Entropy of Two Disjoint Intervals in Conformal Field Theory

The paper "Entanglement entropy of two disjoint intervals in conformal field theory II" by Calabrese, Cardy, and Tonni advances the exploration of entanglement entropy (EE) for systems described by conformal field theories (CFTs). Herein, the authors extend previous analyses, specifically addressing the calculation of the moments of the reduced density matrix, $\rho_A^n$, for two disjoint intervals in Ising universality class CFTs, further refining the theoretical framework initially proposed in a 2009 publication.

The core contribution of the paper involves tackling the intricate geometry of two disjoint intervals. The authors systematically develop methods to examine the asymptotic expansion of the scaling function for the four-point ratio, particularly in regimes where the intervals are short. This expansion provides deeper insights into the rich operator content inherent in CFTs. Central to this approach is the use of Riemann-Siegel theta functions, which elegantly encapsulate the complex nature of the system under paper.

Key Findings and Methods

1. Riemann-Siegel Theta Functions: The work extends the use of Riemann-Siegel theta functions to express predictions about $\rho_A^n$ for integer $n$. This framework allows for connection and comparison with existing numerical data, bolstering the theoretical underpinnings of CFTs.
2. Systematic Method for Asymptotic Expansion: A notable advancement is a robust methodology for obtaining the complete asymptotic expansion of the scaling function for small values of the four-point ratio, $[u_1,v_1]\cup[u_2,v_2]$. The methodology holds potential applicability across various CFTs beyond the specific Ising universality class studied.
3. Analytical Continuation and Entropy Scaling: The authors meticulously perform analytic continuation for these expansions to derive expressions for the von Neumann entropy. Such derivations are crucial for understanding the general behavior and subtleties of entanglement across different parameter spaces, offering a clearer picture of the entropic scaling laws within CFT frameworks.
4. Comparison with Known Results: The paper effectively integrates comparisons with known results from free compactified boson models and analytically characterizes Ising models, further validating the presented methods. For instance, the authors derive expressions that are consistent with known theoretical values for compactified bosons, thereby demonstrating the generality and robust nature of their analytical techniques.

Implications and Future Work

The implications of these findings are manifold, both in enhancing our understanding of the theoretical structure of CFTs and in providing groundwork for further explorations of quantum entanglement in more complex and varied systems. The rigorous approach to the small interval limit expansion paves the way for future studies in more convoluted or higher-dimensional CFTs, as well as possible extensions into non-conformal field theories using similar methodological frameworks.

Additionally, the analytical techniques described could be instrumental in exploring emergent phenomena in systems with non-trivial topologies or boundary conditions, offering new angles on long-standing challenges in theoretical physics related to integrable models and quantum fields.

Speculations on Future Developments

While this paper specifically addresses CFTs, the methodologies employed may inspire broader applications within quantum information theory, particularly those concerning multipartite entanglement and information dynamics in holographic models. Moreover, as AI and machine learning techniques continue to develop, these advancements could potentiate new computational approaches to managing the complex datasets and simulations necessary for larger-scale empirical verification of theoretical predictions in quantum systems.

In conclusion, Calabrese, Cardy, and Tonni's work enhances the quantitative tools available for analyzing entanglement properties in critical systems modeled by CFTs. This paper not only strengthens the theoretical underpinnings of entanglement entropy calculations but also broadens the horizon for future research in both fundamental physics and applied quantum computing domains.