Form factors of branch-point twist fields in quantum integrable models and entanglement entropy (0706.3384v2)

Abstract: In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the "replica trick" whereby the entropy is obtained as the derivative with respect to n of the trace of the n-th power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the n-th power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.

• The paper introduces branch-point twist fields to compute leading corrections to bipartite entanglement entropy in integrable quantum field models.
• It employs the replica trick and form factor program to analyze two-point correlations, supported by case studies on the Ising and sinh-Gordon models.
• Robust numerical results validate a universal entropy correction term, paving the way for applications in complex quantum systems.

An Analysis of Branch-Point Twist Fields and Entanglement Entropy in Integrable Quantum Field Models

The paper explores the calculation of the leading corrections to bipartite entanglement entropy in two-dimensional integrable quantum field theories (IQFT) characterized by diagonal scattering matrices. This paper involves the innovative introduction of branch-point twist fields, which serve as tools to analyze the scaling limit of the entanglement entropy through the "replica trick." The primary aim is to understand the entropic behavior of a quantum system, where the existence of entanglement is a quintessential feature distinguishing it from classical systems.

Core Concepts and Methodologies

1. Branch-Point Twist Fields:
   • Branch-point twist fields are introduced as a novel type of twist fields in IQFTs. These fields are linked with the geometry of an $n$-sheeted Riemann surface, effectively enabling the computation of the entanglement entropy through their form factors.
2. Replica Trick:
   • The entanglement entropy is computed using the replica trick, which involves evaluating the trace of the $n^{\text{th}}$ power of the subsystem's reduced density matrix. This approach is rendered feasible by introducing twist fields that implement branch points such that their correlation functions are equivalent to partition functions on multi-sheeted Riemann surfaces.
3. Form Factor Program:
   • The form factors of the twist fields provide the key to understanding the large-distance behavior of the two-point correlation functions. Despite being a familiar method within integrable quantum field theory, adapting this framework to incorporate branch points necessitates modifications to traditional form factor equations.
4. Case Studies - Ising and sinh-Gordon Models:
   • Detailed analyses of the Ising and sinh-Gordon models are presented to substantiate theoretical findings. The Ising model serves as an archetype due to its simplicity and integrability, while the sinh-Gordon model offers a rich structure with a single particle spectrum devoid of bound states, facilitating the exploration of the variables involving branch-point twist fields.

Numerical Results and Practical Implications

• The paper unveils a remarkable universality in the leading correction term for the entanglement entropy, indicating its independence from specific scattering matrices and reliance solely on the particle mass spectrum. This revelation holds profound implications for the application of IQFT techniques in evaluating quantum entropy.
• Strong numerical results validate the theoretical predictions, for instance, confirming the expected scaling dimensions in both the Ising and sinh-Gordon models using the $\Delta$-sum rule. These results emphasize the robustness of the form factor approach in calculating the entanglement entropy for various IQFTs.
• The research delineates a pathway for pursuing entropy calculations in complex systems with multiple particles and bound states, thereby expanding the applicability of the proposed framework.

Theoretical Developments and Future Directions

This paper not only advances the understanding of entanglement entropy in IQFTs via branch-point twist fields but also enriches the methodological repertoire available to quantum physics researchers. Theoretical implications abound, offering novel insights into the universal properties of quantum entanglement that transcend intricate model details. Future research could further extend this work by addressing non-diagonal theories and exploring twist fields in models with non-abelian symmetries. In addition, the findings might find applications in quantum computing and information theory, where understanding entanglement plays a pivotal role.

For researchers exploring the intersection of statistical mechanics, quantum field theory, and information theory, this paper offers a foundational framework upon which further theoretical and applied investigations can be constructed. The potential for discovering even more unified principles across diverse quantum phenomena remains an enticing prospect for continued exploration.