The complex life of hydrodynamic modes

Abstract

We study analytic properties of the dispersion relations in classical hydrodynamics by treating them as Puiseux series in complex momentum. The radii of convergence of the series are determined by the critical points of the associated complex spectral curves. For theories that admit a dual gravitational description through holography, the critical points correspond to level-crossings in the quasinormal spectrum of the dual black hole. We illustrate these methods in N = 4 supersymmetric Yang-Mills theory in 3+1 dimensions, in a holographic model with broken translation symmetry in 2+1 dimensions, and in conformal field theory in 1+1 dimensions. We comment on the pole-skipping phenomenon in thermal correlation functions, and show that it is not specific to energy density correlations. 4 The connection between the pole-skipping values (q 2 * , ω * ) and the growth of the out-of-time-ordered fourpoint correlation functions (OTOC) of local operators in quantum field theory has been studied in refs. [29][30][31][32].

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

“…However, there are yet another ways to explore the relation between hydrodynamics and quantum chaos. Following [19] and in the language of [66], one can investigate whether the chaos points given in (2.30) are located on the analytically continued dispersion relation of sound waves. To proceed, one has to firstly find the spectral curve corresponding to the sound channel.…”

“…The latter gives spectral curve of sound channel Z(0; ω, k 2 , kB) = 0 and E z (0; ω, k 2 , kB) = 0. Then by using the method developed in [66], one can find the dispersion relation of sound modes in an expansion over both k and B. If it behaves like the non-chiral case [19], luckily, the first few orders in the expansion will be sufficient to observe that chaos point is on the analytic continuation of this spectral curve.…”

Higher curvature corrections to pole-skipping

“…In the absence of any higher curvature correction, it was argued in [20] that two important parameters of chaos, i.e. λ L and v B , can be recovered, irrespectively of the channel of metric perturbations, as…”

“…This indicates that pole-skipping may not always be directly related to quantum chaos, but could be a consequence of a more general feature of near horizon bulk equations. Relevant discussions can also be found in [18,19,20].where A and B are coefficients in the asymptotic expansion of the scalar field near the boundary φ → Ar ∆−4 + Br −∆ .(2.5) 2 In this paper, the AdS radius is always set to unity for convenience. 3 One may well consider the equivalent form ∇µ∇ µ ϕ − m 2 ϕ = 0.…”

“…This indicates that pole-skipping may not always be directly related to quantum chaos, but could be a consequence of a more general feature of near horizon bulk equations. Relevant discussions can also be found in [18,19,20].…”