Abstract
The time evolution of entanglement tracks how information propagates in interacting quantum systems. We study entanglement entropy in ${\mathrm{CFT}}_{2}$ with a time-dependent Hamiltonian. We perturb by operators with time-dependent source functions and use the replica trick to calculate higher-order corrections to entanglement entropy. At first order, we compute the correction due to a metric perturbation in ${\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}$ and find agreement on both sides of the duality. Past first order, we find evidence of a universal structure of entanglement propagation to all orders. The central feature is that interactions entangle unentangled excitations. Entanglement propagates according to ``entanglement diagrams,'' proposed structures that are motivated by accessory spacetime diagrams for real-time perturbation theory. To illustrate the mechanisms involved, we compute higher-order corrections to free fermion entanglement entropy. We identify an unentangled operator, one which does not change the entanglement entropy to any order. Then, we introduce an interaction and find it changes entanglement entropy by entangling the unentangled excitations. The entanglement propagates in line with our conjecture. We compute several entanglement diagrams. We provide tools to simplify the computation of loop entanglement diagrams, which probe UV effects in entanglement propagation in CFT and holography.
Highlights
Entanglement is a fundamental feature of quantum field theory
We compute the change in entanglement entropy using the replica trick in the conformal field theories (CFTs) and find that the correction depends only on the metric perturbation at the interval’s endpoints, which is expected as perturbation by the trace of the stress tensor preserves conformal symmetry wherever ωðxÞ 1⁄4 0
While entanglement entropy is not itself a physical observable, it is determined by ρA, and all observable properties of ρA are fixed according to the expectation values of operators localized to DA, which themselves change according to standard real-time perturbation theory
Summary
Entanglement is a fundamental feature of quantum field theory. The program of studying entanglement and other information-theoretic quantities in field theory has recently led to new understanding of the well-known holographic correspondence between gravitational theories in anti–de Sitter spacetimes (AdS) and large-N strongly-coupled conformal field theories (CFTs). In [33], the first law was used to calculate the change in entanglement entropy to first order in J due to the Hamiltonian perturbation Jðx; tÞOðxÞ for operators O in the free scalar and fermion theories in various spacetime dimensions. Their CFT results agreed with that of the Hubeny-Rangamani-Takayanagi prescription (HRT) in the bulk, the covariant generalization. IV, we omit overall numerical factors that play no role in our results
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