Abstract
We consider a chiral fermion at non-zero temperature on a circle (i.e., on a torus in the Euclidean formalism) and compute the modular Hamiltonian corresponding to a subregion of the circle. We do this by a very simple procedure based on the method of images, which is presumably generalizable to other situations. Our result is non-local even for a single interval, and even for Neveu-Schwarz boundary conditions. To the best of our knowledge, there are no previous examples of a modular Hamiltonian with this behavior.
Highlights
The study of entanglement and its measures has proven to be very useful in unveiling some of the deepest properties of quantum field theory (QFT)
Entanglement measures are based on the reduced density matrix, or equivalently on its logarithm, the modular Hamiltonian
The knowledge of modular Hamiltonians was essential for the proof of the averaged null energy condition [1], the derivation of quantum energy inequalities [2,3], and the formulation of a well-defined version of the Bekenstein bound [4]
Summary
The study of entanglement and its measures has proven to be very useful in unveiling some of the deepest properties of quantum field theory (QFT). The result is universal and local for the vacuum of any QFT reduced to Rindler space [9,10], and from this result one can derive CFT expressions for the vacuum reduced to a ball in the plane [11], for a thermal state reduced to an interval in the plane in 1 þ 1 dimensions [12], and for the vacuum reduced to an interval in the cylinder in 1 þ 1 dimensions [13] In these cases, the modular Hamiltonian turns out to be local, but nonlocal contributions are expected to appear in general. Our result is nonlocal even for a single interval, even for Neveu-Schwarz (antiperiodic) boundary conditions
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