Kavli Affiliate: Huajia Wang
| First 5 Authors: Guan-Cheng Lu, Huajia Wang, , ,
| Summary:
Modular flows probe important aspects of the entanglement structures,
especially those of QFTs, in a dynamical framework. Despite the expected
non-local nature in the general cases, the majority of explicitly understood
examples feature local space-time trajectories under modular flows. In this
work, we study a particular class of non-local modular flows. They are
associated with the relativistic vacuum state and sub-regions whose boundaries
lie on a planar null-surface. They satisfy a remarkable algebraic property
known as the half-sided modular inclusion, and as a result the modular
Hamiltonians are exactly known in terms of the stress tensor operators. To be
explicit, we focus on the simplest QFT of a massive or massless free scalar in
$2+1$ dimensions. We obtain explicit expressions for the generators. They can
be separated into a sum of local and non-local terms showing certain universal
pattern. The preservation of von-Neumann algebra under modular flow works in a
subtle way for the non-local terms. We derive a differential-integral equation
for the finite modular flow, which can be analyzed in perturbation theory of
small distance deviating from the entanglement boundary, and re-summation can
be performed in appropriate limits. Comparison with the general expectation of
modular flows in such limits are discussed.
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