Abstract
We investigate the properties of pole-skipping of the sound channel in which the translational symmetry is broken explicitly or spontaneously. For this purpose, we analyze, in detail, not only the holographic axion model, but also the magnetically charged black holes with two methods: the near-horizon analysis and quasi-normal mode computations. We find that the pole-skipping points are related with the chaotic properties, Lyapunov exponent (λL) and butterfly velocity (vB), independently of the symmetry breaking patterns. We show that the diffusion constant (D) is bounded by Dge {v}_B^2/{lambda}_L , where D is the energy diffusion (crystal diffusion) bound for explicit (spontaneous) symmetry breaking. We confirm that the lower bound is obtained by the pole-skipping analysis in the low temperature limit.
Highlights
Imposing the ingoing boundary condition at the horizon
We investigate the properties of pole-skipping of the sound channel in which the translational symmetry is broken explicitly or spontaneously
We have studied the properties of pole-skipping in the sound channel with the breaking of translational invariance
Summary
We will introduce the Einstein-Maxwell-Dilaton with Axion model (EMDAxion model) and show that this model can describe the EXB or the SSB of translational invariance depending on the couplings. We will study the pole-skipping of the sound channel in these models using the near horizon analysis. 2.1 Einstein equations and pole-skipping We consider the following (3+1) dimensional Einstein gravity. Where δΦ represents general matter field perturbations that couple to gravitational perturbations. The pole-skipping phenomena is related to the near-horizon properties of Einstein equations. Using (2.11), for generic value of ω, the equation (2.9) imposes the constraints between the horizon values of metric components: δgv(0v) , δgv(0x) , δgx(0x) and δgy(0y). For the generic wave number k, this equation gives δgv(0v) = 0, when (2.12). Assuming the non-trivial identity (2.11), equation (2.9) at the following special point (2.14) is identically satisfied and we cannot impose constraints between the horizon metric components, leading to pole-skipping in the two-point functions. 4πT C2H DH BH which is consistent with the one by shock-wave analysis [55]
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