Resolving modular flow: a toolkit for free fermions

Abstract

Modular flow is a symmetry of the algebra of observables associated to spacetime regions. Being closely related to entanglement, it has played a key role in recent connections between information theory, QFT and gravity. However, little is known about its action beyond highly symmetric cases. The key idea of this work is to introduce a new formula for modular flows for free chiral fermions in $1+1$ dimensions, working directly from the \textit{resolvent}, a standard technique in complex analysis. We present novel results -- not fixed by conformal symmetry -- for disjoint regions on the plane, cylinder and torus. Depending on temperature and boundary conditions, these display different behaviour ranging from purely local to non-local in relation to the mixing of operators at spacelike separation. We find the modular two-point function, whose analytic structure is in precise agreement with the KMS condition that governs modular evolution. Our ready-to-use formulae may provide new ingredients to explore the connection between spacetime and entanglement.

arXiv comment33 pages, 10 figures, v2: minor corrections made

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Cited by in corpus (10)

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• Moving mirrors, Page curves and bulk entropies in AdS${}_2$
• Symmetry-resolved modular correlation functions in free fermionic theories
• Modular Hamiltonian for de Sitter diamonds
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• Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT${}_2$
• Rényi entropy of single-character CFTs on the torus