Horizon constraints on holographic Green’s functions

Abstract

We explore a new class of general properties of thermal holographic Green's functions that can be deduced from the near-horizon behaviour of classical perturbations in asymptotically anti-de Sitter spacetimes. We show that at negative imaginary Matsubara frequencies and appropriate complex values of the wavenumber the retarded Green's functions of generic operators are not uniquely defined, due to the lack of a unique ingoing solution for the bulk perturbations. From a boundary perspective these 'pole-skipping' points correspond to locations in the complex frequency and momentum planes at which a line of poles of the retarded Green's function intersects with a line of zeroes. As a consequence the dispersion relations of collective modes in the boundary theory at energy scales ω ∼ T are directly constrained by the bulk dynamics near the black-brane horizon. For the case of conserved U (1) current and energy-momentum tensor operators we give examples where the dispersion relations of hydrodynamic modes pass through a succession of pole-skipping points as real wavenumber is increased. We discuss implications of our results for transport, hydrodynamics and quantum chaos in holographic systems. arXiv:1904.12883v3 [hep-th] 13 Jan 2020 45 E.3 Transverse metric perturbations 48 E.4 Longitudinal metric perturbations 49 E.5 Transverse metric perturbations in a charged black hole 52 -1 -1 v B is the 'butterfly velocity' [15][16][17] of the dual quantum field theory. 2 Pole-skipping in G R εε at (ω * , k * ) was also shown to hold in holographic theories dual to higher derivative gravity in [20].3 The symbol ω n is more conventionally used to refer to the real set of Matsubara frequencies ω E = 2πT n at which the Euclidean Green's function is defined. Here we are interested in studying the real time correlator G R (ω, k) and will abuse the more common notation somewhat by using the symbol ω n to refer to the special pure imaginary frequencies ω = ω n = −i2πT n. These are referred to as the 'negative imaginary Matsubara frequencies' for obvious reasons.We are grateful to Nejc Ceplak, Saso Grozdanov, Hong Liu, and Andrei Starinets for helpful discussions. M. B. received support from the Office of High Energy Physics of U.S.

Quantum chaos, pole-skipping and hydrodynamics in a holographic system with chiral anomaly

“…It is clear that at these 2 points, the Green's function of the boundary operator, dual to the bulk scalar field Φ, is multi-valued. According to the explanations given in Introduction, based on [37], one concludes that to every Matsubara frequency ω = −i2πT , pole-skipping points of the dual boundary operator correspond. This simply shows that how near horizon dynamics can strictly constrain the Green's function of a generic boundary operator, even beyond the regime of hydrodynamics, namely at frequencies ω ∼ T .…”

“…In another way, recently it has been shown that the lack of information to uniquely define a correlation function is not specific to energy density correlation functions at chaos point; Green's functions of generic operators have also the same feature but at negative Matsubara frequencies and some appropriate complex values of wavenumber [37] 7 . The presence of such set of pole-skipping points in the lower half of complex Fourier plane shows that the dispersion relations of collective modes in boundary theory at energy scales ω ∼ T are directly constrained by the near horizon dynamics of bulk fields.…”

“…In the absence of chiral effects, these four are represented by the red points at Im ω 2πT = −2 in Figs.1 and 2. It is shown that the wave number of red points satisfy the following relations[37]…”

Higher curvature corrections to pole-skipping

“…Of course, the physics will remain the same. Here we simply follow the convention used in [17] for the sake of comparison.11 Note that at λGB = 1/4, N 2 GB = 1/2, the shear viscosity vanishes, and the theory exhibits unusual properties in many aspects, such as quasinormal modes and thermodynamics, see [54,59,60] for detailed discussions. Since this value lies far outside of the causality range (4.3), we will not consider it in the following.…”

“…Of course, the physics will remain the same. Here we simply follow the convention used in [17] for the sake of comparison.…”