Resolving modular flow: a toolkit for free fermions

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.

Moving mirrors, Page curves and bulk entropies in AdS$_2$

“…, N labels the interval. These are similar -but not identical -to those already encountered in [16,21,29,30], and includes the local solution y = x. For opposite chiralities i = −j, we have a novel set of solutions which we call y − to indicate that they are associated to a change in chirality.…”

“…This section contains the main technical tools of this work, following the approach of [19][20][21] . Given an operator G of bounded spectrum and a function f (λ) holomorphic in the interior of a contour γ enclosing the spectrum -in the case of fermions, the interval [0, 1] -Cauchy's integral formula defines the function of an operator by…”

“…where f = − log λ −1 − 1 , and we have abbreviated q i = q i (x) and t = Z(x)−Z(y). Contour integrals of this form have been evaluated in [19,21]. The result is:…”

Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT2

“…A natural question that arises is whether one can extend this construction of algebra and its representation beyond the vacuum in CFT for at least some class of excited states. 13 Perhaps a more tractable direction to pursue would be to find the extension of such algebras for disconnected multi-interval cases where analytic expression of modular Hamiltonian are known [43,44]. 14 We hope to return to some of these questions in the near future.…”

Modular Hamiltonians for the massless Dirac field in the presence of a boundary

“…We remark that we consider anti-periodic boundary conditions. Non local terms can occur in the modular Hamiltonian if other boundary conditions are imposed [31,32,74]. In this case, which has been already studied in [8,9], the weight function β 0 (x) and the corresponding w(x) read respectively…”