Abstract
In this work we study the Tomita-Takesaki construction for a family of excited states that, in a strongly coupled CFT - at large $N$-, correspond to coherent states in an asymptotically AdS spacetime geometry. We compute the modular flow and modular Hamiltonian associated to these excited states in the Rindler wedge and for a ball shaped entangling surface. Using holography, one can compute the bulk modular flow and construct the Tomita-Takesaki theory for these cases. We also discuss generalizations of the entanglement regions in the bulk and how to estimate the modular Hamiltonian in a large N approximation. Finally we present a holographic formula, based on the BDHM prescription, to compute the modular evolution of operators in the corresponding CFT algebra.
Highlights
Modular Hamiltonians are unbounded and Hermitian operators properly defined in the context of axiomatic quantum field theories [1,2], and in particular in the framework of the Tomita-Takesaki (TT) theorem [3]
In this paper we studied the modular Hamiltonian and modular flow of a family of excited states whose holographic description is precise in both sides of the AdS=CFT duality and are related to bulk coherent states at large N [30,37]
By using thermofield dynamics (TFD) and Schwinger-Keldysh techniques, we manage to frame our excited system as a Tomita-Takesaki theory, allowing us to find the correct Δ and J operator of our excited system, matching the expressions derived via path integral methods
Summary
Modular Hamiltonians ( called entanglement Hamiltonians in the condensed matter community) are unbounded and Hermitian operators properly defined in the context of axiomatic quantum field theories [1,2], and in particular in the framework of the Tomita-Takesaki (TT) theorem [3]. The modular Hamiltonian KA is a self-adjoint operator that belongs to the algebra of operators, defined on the region A It has a very precise definition in the AQFT setup in the context of the TT modular theory [3]. In the present work we will study the modular Hamiltonian, modular operator and its modular flow for a family of excited states in equipartite Hilbert spaces in the context of holographic CFTs. Some related works in the field theory context are [26–28] (see [29] for an axiomatic approach).
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