Entanglement hamiltonians in two-dimensional conformal field theory

Abstract: We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O(1) terms in the entanglement entropy.

• The paper demonstrates that entanglement Hamiltonians in 2d CFT can be locally expressed as integrals of the energy-momentum tensor under conformal mappings, clarifying the universal entanglement spectrum.
• The paper reveals that boundary conditions at the entangling surface significantly influence modular Hamiltonians and contribute universal O(1) terms in entanglement entropy.
• The paper introduces methodologies for dynamic quantum quenches, establishing conditions for local integral representations and techniques to regularize UV divergences.

Entanglement Hamiltonians in Two-Dimensional Conformal Field Theory

The paper authored by John Cardy and Erik Tonni presents a meticulous examination of entanglement Hamiltonians in the context of two-dimensional conformal field theory (2d CFT). The research focuses on the entanglement or modular Hamiltonian, defined as the logarithm of the reduced density matrix for a given bipartition of a quantum system. Specifically, it investigates the conditions under which this Hamiltonian can be expressed as an integral involving the energy-momentum tensor with a local weight.

Summary of Key Findings

1. Examples of Entanglement Hamiltonians: The paper lists cases where the entanglement Hamiltonian, denoted as $K_A$, can be represented as a local integral over the energy-momentum tensor. These include known examples and novel cases associated with time-dependent scenarios induced by global and local quantum quenches.
2. Role of Boundary Conditions: The research underscores the importance of boundary conditions at the entangling surface. It shows how these conditions affect the modular Hamiltonian and influence universal $O(1)$ terms in the entanglement entropy, such as boundary entropies.
3. Entanglement Spectrum: Across various examples, the entanglement spectrum corresponds to that of an appropriate boundary CFT. This indicates that the universal properties are dictated by boundary scaling dimensions of the CFT.
4. UV Divergences and Regularization: The discussion points out that in quantum field theories, expressions for both Rényi entropies and entanglement entropy exhibit UV divergences. These need to be regularized, particularly in field theories, by projecting onto a common eigenstate or by spatially smearing local operators.
5. Time-Dependent Scenarios: The paper introduces new methodologies for dynamic cases, delineating the behavior of entanglement Hamiltonians under global and local quenches. In such setups, $K_A$ can also depend on the momentum density.
6. Parameters and Universal Features: The authors derive explicit conditions under which $K_A$ is expressed as a local integral. The key condition involves conformal equivalence of the euclidean space-time region to an annulus, which determines the universality in the entanglement spectrum.

Implications and Future Work

This study has important implications both theoretically and practically. Theoretically, it provides a deeper understanding of the modular Hamiltonian in 2d CFT and reveals complex interactions between geometry, topology, and entanglement. Practically, these insights can be relevant for quantum computing and simulations, where control over entanglement is crucial.

The researchers suggest several avenues for future exploration. These include extending the analysis to higher-dimensional CFTs and investigating further the effects of different boundary conditions. Also, exploring the applicability of these concepts to non-conformal theories remains an open area for investigation.

This work is pivotal for those involved in the study of quantum entanglement in field theories and may catalyze further developments in the field. The elucidation of the entanglement spectrum and its relation to boundary conditions opens new pathways to understanding quantum many-body systems, potentially impacting fields ranging from condensed matter physics to quantum information science.