Chaos near to the critical point: butterfly effect and pole-skipping

Abstract

AbstractWe study the butterfly effect and pole-skipping phenomenon for the 1RCBH model which enjoys a critical point in its phase diagram. Using the holographic idea, we compute the butterfly velocity and interestingly find that this velocity can probe the critical behavior of this model. We calculate the dynamical exponent of this quantity near the critical point and find a perfect agreement with the value of the other quantity’s dynamical exponent near this critical point. We also find that at special points, namely the $$(\omega _\star = i \lambda _L, \, k_\star = i\lambda _L/v_B),$$ ( ω ⋆ = i λ L , k ⋆ = i λ L / v B ) , where $$\lambda _L$$ λ L and $$v_B$$ v B are Lyapunov exponent and butterfly velocity respectively, the phenomenon of pole-skipping appears which is a sign of a multivalued retarded Green’s function. Furthermore, we observe that $$v_B^2 \ge c_s^2$$ v B 2 ≥ c s 2 at each point of parameter space of the 1RCBH model where $$c_s$$ c s is the speed of sound wave propagation.