When the Lyapunov exponent λ_L in a quantum chaotic system saturates the bound λ_L ≤ 2πk_BT , it is proposed that this system has a holographic dual described by a gravity theory. In particular, the butterfly effect as a prominent phenomenon of chaos can ubiquitously exist in a black hole system characterized by a shockwave solution near the horizon. In this paper we propose that the butterfly velocity can be used to diagnose quantum phase transition (QPT) in holographic theories. We provide evidences for this proposal with an anisotropic holographic model exhibiting metal-insulator transitions (MIT), in which the derivatives of the butterfly velocity with respect to system parameters characterizes quantum critical points (QCP) with local extremes in zero temperature limit. We also point out that this proposal can be tested by experiments in the light of recent progress on the measurement of out-of-time-order correlation function (OTOC).

In this paper, we analyze the light rays around a static hairy black hole in Horndeski gravity with the use of ray-tracing procedure. We find that a stronger Horndeski hairy parameter corresponds to larger photon sphere as well as critical impact parameter, and wider ranges of photon ring and lensed ring emissions. These influences can be robustly interpreted from the shape of the effective potential of the photon’s radial motion. Based on the distribution of the light rays, we then investigate the optical appearances of the Horndeski hairy black hole surrounded by various thin accretions. First, we consider that the Horndeski hairy black hole is illuminated by the optically and geometrically thin accretion disk. We carefully clarify the contributions from the direct, lensed ring and photon ring intensities to the total observed intensity via the transfer function, which is rarely discussed in this scenario. We find that the Horndeski hair has significant influences on both shadow size and distributions of direct, lensed ring and photon ring brightness in three standard emission profiles. As a result, the rings and images of Horndeski hairy black hole and the origination of their brightness differentiate from those of Schwarzschild black hole (SBH) when viewed in face-on orientation. Then, when the Horndeski hairy black hole is illuminated by thin spherical accretions, the hairy black hole’s shadow surrounded by a bright ring is larger than that of SBH, but the brightness of ring is fainter. Similar to that of SBH, the size of hairy black hole shadow does not change as the radial moving of the spherical accretion, and the brightness for the infalling accretion is fainter than that for the static accretion due to the Doppler effect. Therefore, we argue that the black hole image consisting of the shadow and accretion construction could, in theory, reflect the observational differences between the Horndeski hairy black hole and SBH.

We investigate the holographic DC and Hall conductivity in massive Einstein-Maxwell-Dilaton (EMD) gravity. Two special EMD backgrounds are considered explicitly. One is dyonic Reissner-Nordström-AdS (RN-AdS) geometry and the other one is hyperscaling violation AdS (HV-AdS) geometry. We find that the linear-T resistivity and quadratic-T inverse Hall angle can be simultaneously achieved in HV-AdS models, providing a hint to construct holographic models confronting with the experimental data of strange metal in future.

The entanglement exhibits extremal or singular behavior near quantum critical points (QCPs) in many condensed matter models. These intriguing phenomena, however, still call for a widely accepted understanding. In this paper we study this issue in holographic framework. We investigate the connection between the holographic entanglement entropy (HEE) and the quantum phase transition (QPT) in a lattice-deformed Einstein-Maxwell-Dilaton theory. Novel backgrounds exhibiting metal-insulator transitions (MIT) have been constructed in which both metallic phase and insulating phase have vanishing entropy density in zero temperature limit. We find that the first order derivative of HEE with respect to lattice parameters exhibits extremal behavior near QCPs. We propose that it would be a universal feature that HEE or its derivatives with respect to system parameters can characterize QPT in a generic holographic system. Our work opens a window for understanding the relation between entanglement and the QPT from a holographic perspective.

A hairy Schwarzschild black hole describes the deformation of a Schwarzschild black hole due to including additional sources. It is found that depending on the hairy parameters, the photons’ configurations around this black hole can be classified into two cases, corresponding to the hairy Schwarzschild black hole with a single photon sphere and double photon spheres, respectively. We focus on the shadows and images of the hairy Schwarzschild black hole under two types of static thin illuminations conditions: disk accretion and spherical accretion, respectively. Under both illuminations, the two hairy parameters ($\alpha$ and $l_o$) have competitive effects on the shadow and optical appearance image of the hairy Schwarzschild black hole with a single photon sphere. This means that even though the parameters have significant influences on the rings and shadows, its images with certain groups of $\alpha$ and $l_o$ could be indistinguishable to that of Schwarzschild black hole, namely, the degeneracies of images raise between the hairy Schwarzschild black hole and Schwarzschild black hole. Moreover, the optical appearance image of the hairy Schwarzschild black hole with double photon spheres will exhibit new additional rings and accretion features, which are not present in the images of a (hairy) Schwarzschild black hole with a single photon sphere. Our theoretical studies on the rings and shadows provide a potential tool to differentiate the hairy Schwarzschild black hole with double photon spheres from the Schwarzschild black hole, but they are not always helpful for the cases with a single photon sphere due to the degeneracies.

In this note we remark that the butterfly effect can be used to diagnose the phase transition of superconductivity in a holographic framework. Specifically, we compute the butterfly velocity in a charged black hole background as well as anisotropic backgrounds with Q-lattice structure. In both cases we find its derivative to the temperature is discontinuous at critical points. We also propose that the butterfly velocity can signalize the occurrence of thermal phase transition in general holographic models.Comment: 10 pages, 4 figures; minor revision, one figure added, journal reference adde