Entanglement Temperature and Entanglement Entropy of Excited States (1305.3291v2)

Abstract: We derive a general relation between the ground state entanglement Hamiltonian and the physical stress tensor within the path integral formalism. For spherical entangling surfaces in a CFT, we reproduce the \emph{local} ground state entanglement Hamiltonian derived by Casini, Huerta and Myers. The resulting reduced density matrix can be characterized by a spatially varying "entanglement temperature." Using the entanglement Hamiltonian, we calculate the first order change in the entanglement entropy due to changes in conserved charges of the ground state, and find a local first law-like relation for the entanglement entropy. Our approach provides a field theory derivation and generalization of recent results obtained by holographic techniques. However, we note a discrepancy between our field theoretically derived results for the entanglement entropy of excited states with a non-uniform energy density and current holographic results in the literature. Finally, we give a CFT derivation of a set of constraint equations obeyed by the entanglement entropy of excited states in any dimension. Previously, these equations were derived in the context of holography.

• The paper derives a universal entanglement Hamiltonian by linking the stress tensor with path integral methods in CFT.
• It introduces a spatially varying entanglement temperature that simplifies entropy calculations in regions with non-uniform energy density.
• It establishes a local first law-like relation for entanglement entropy while highlighting discrepancies with holographic predictions for excited states.

Entanglement Temperature and Entanglement Entropy of Excited States

The paper "Entanglement Temperature and Entanglement Entropy of Excited States" addresses the theoretical connections between the entanglement Hamiltonian and the physical stress tensor in a Conformal Field Theory (CFT) framework. Using the path integral method, the authors derive a relation that stands universally for ground state entanglement Hamiltonians across a variety of quantum states. The implications of this paper are profound for understanding the entanglement properties of quantum fields, specifically in calculating the variations in entanglement entropy in response to changes in a system's energy and conserved charges.

Key Contributions and Results

1. Derivation of the Entanglement Hamiltonian: The paper starts by establishing a connection between the ground state entanglement Hamiltonian and the stress tensor using the path integral formalism. For spherical entangling surfaces in a CFT, the paper successfully reproduces the local form of the entanglement Hamiltonian posited by previous works, notably that of Casini, Huerta, and Myers.
2. Entanglement Temperature: A distinctive contribution is the introduction of the notion of an "entanglement temperature", which varies spatially. This approach simplifies the calculation of entanglement entropy, particularly in cases with varying energy densities. The entanglement Hamiltonian can thus be understood as generating thermal-like states within the region, described by a spatially dependent temperature that characterizes the reduced density matrix.
3. First Law-like Relation: The research derives a local first law-like relation for the entanglement entropy changes due to variations in energy. Remarkably, this first law-like relation aligns with expectations from holographic techiques but is derived using purely field-theoretic approaches, underscoring its versatility beyond holographic systems.
4. Discrepancies with Holographic Results: An important point raised is the discrepancy between field-theoretic calculations and holographic results for excited states with non-uniform energy densities. These contradictions highlight potential limitations or areas for refinement in holographic calculations and advocate for a deeper analysis of approximate metrics and their implications in holography.
5. Constraint Equations for Entanglement Entropy: The paper further provides a CFT-based derivation of constraint equations for the entanglement entropy, originally discussed in holographic contexts. This serves to generalize findings across different spacetime dimensions and confirms consistency with prior holographic models.

Implications and Future Directions

The findings of this paper have significant theoretical implications. Not only do they provide new methods to compute entanglement entropy using field theory, but they also offer critical insights that can refine our understanding of entanglement in both high-energy physics and condensed matter systems. The introduction of entanglement temperature has potential applications in identifying effective thermal states in quantum many-body systems, providing a bridge between microscopic interactions and macroscopic thermodynamic properties.

From a future research perspective, the discrepancies noted between field-theoretic results and holographic models suggest a fertile ground for exploration. Investigating these deviations could yield new theoretical insights and contribute to resolving outstanding queries in the AdS/CFT correspondence. Additionally, extending this framework to non-conformal settings or integrating disorder and finite-size effects could enrich the current understanding of quantum entanglement across diverse systems.

Conclusion

The paper expertly navigates the complex interactions between entanglement Hamiltonians, localized charges, and the geometric peculiarity of quantum fields in the CFT regime. By demonstrating firm analytical control and offering perceptive critiques of existing methodologies, it experimentally validates theoretical concepts. With solid analytical frameworks and results open to various substantial applications, this paper forms a fundamental contribution to the ongoing discourse on quantum entanglement and field theory.