Abstract
Modular flow is a symmetry of the algebra of observables associated to space-time 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 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.
Highlights
Despite the many contexts in which modular flow appears, there are very few cases where its action is explicitly known
Modular flow is a symmetry of the algebra of observables associated to spacetime regions
In the general context of quantum field theory (QFT), the vacuum modular flow in a Rindler wedge is fixed by Poincaré symmetry alone [42], while conformal symmetry fixes it for diamond shaped geometries [43]
Summary
Before we specialize to the free fermion, let us first recall some basic notions of In other words: if we only consider operators in R, the vector |Ω behaves like a thermal state ρR = Z−1e−K This coincides with the definition of modular flow in terms of “reduced density matrices” [8]. We can use the antiperiodicity (2.10) to rewrite this purely in terms of the function on the lower strip, Gmod(x, y; t − i0+) + Gmod(x, y; t − i + i0+) = Σt(x, y) This relation is important because it relates the analytic structure of the modular correlator to the locality properties of the modular flow, via the Kernel Σt of the operator flow. The salient example where analyticity fails is a single interval on the periodic vacuum on the cylinder, where the flow is completely non-local and Gmod has branch cuts
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