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

Abstract

It is well-known that chiral anomaly can be macroscopically detected through the energy and charge transport, due to the chiral magnetic effect. On the other hand, in a holographic many body system, the chaotic modes might be only associated with the energy conservation. This suggests that, perhaps, one can detect microscopic anomalies through the diagnosis of quantum chaos in such systems. To investigate this idea, we consider a magnetized brane in AdS space time with a Chern-Simons coupling in the bulk. By studying the shock wave geometry in this background, we first compute the corresponding butterfly velocities, in the presence of an external magnetic field $B$, in $μ\ll T$ and $ B \ll T^2$ limit. We find that the butterfly propagation in the direction of $B$ has a different velocity than in the opposite direction. The splitting of butterfly velocities confirms the idea that chiral anomaly can be macroscopically manifested via quantum chaos. We then show that the pole-skipping points of energy density Green's function of the boundary theory coincide precisely with the chaos points. This might be regarded as the hydrodynamic origin of quantum chaos in an anomalous system. Additionally, by studying the near horizon dynamics of a scalar field on the above background, we find the spectrum of pole-skipping points associated with the two-point function of dual boundary operator. We find that the sum of wavenumbers corresponding to pole-skipping points at a specific Matsubara frequency is a universal quantity, which is independent of the scaling dimension of the dual boundary operator. We then show that this quantity follows from a closed formula and can be regarded as another macroscopic manifestation of the chiral anomaly.

Figures (4)

• Figure 1: Spectrum of chaos points (in the upper half plane) together with the pole-skipping points of a boundary operator dual to a scalar field with mass $m=0$ in the bulk (in the lower half plane). Orange and red points, which are symmetric with respect to the vertical axis, are related to non-chiral matter at $\nu=0$ and $b=0$. Blue and green points, however, correspond to a chiral matter at $\nu=0.5$ and $b=0.5$. Due to the chiral anomaly, these points are located asymmetrically with respect to vertical axis. The vertical gray dashed lines at $\boldsymbol{k}=\pm\sqrt{3/2}$ show that at the special case when $m=0$, accidentally, the chaos points and the lowest frequency pole-skipping points have the same wavenumbers at $b=0$ and $\nu=0$.
• Figure 2: Spectrum of chaos points (in the upper half plane) together with the pole-skipping points of a boundary operator dual to a scalar filed with mass $m=2$ in the bulk (in the lower half plane). Orange and red points are related to non-chiral matter at $\nu=0$ and $b=0$. Blue and green points correspond to a chiral matter at $\nu=0.5$ and $b=0.5$. Orange and red points are symmetric with respect to the Im $\omega$-axis, however, due to the chiral anomaly, the blue and Green ones are located asymmetrically with respect to the same axis.
• Figure 3: Asymmetric analytic structure of $\tilde{h}$ in the presence of magnetic field and chiral anomaly. Black points, which are symmetric with respect to the real axis, show the location of poles in the absence of magnetic filed.
• Figure 4: Spectrum of chaos points together with the pole-skipping points of a boundary operator dual to a scalar filed, in the transverse channel. In the left panel, the mass of scalar field in the bulk is $m=0$ and in the right panel $m=2$. Orange and red points are related to non-chiral matter at $\nu=0$ and $b=0$. Blue and green points correspond to a chiral matter at $\nu=.5$ and $b=.5$.