We consider a composite system where AdS gravity is coupled to a flat heat bath and investigate the mutual information between two subregions on the intersection of the AdS and bath, referred to as the boundary mutual information (BMI). The corresponding entanglement entropy is captured via quantum extremal surfaces (QES), which holographically be computed by a surface optimization algorithm based on “Surface Evolver”. We focus on both connected and disconnected configurations of the quantum entanglement wedge (Q-EW) in the AdS bulk and analyze the finite corrections to the BMI. Our numerical results reveal a phase transition of the BMI as the separation between two subregions increases. Furthermore, we find that the BMI can naturally be decomposed into two distinct components: a geometric term arising from the areas of the quantum extremal surfaces, and a correction term resulting from bulk quantum fields within the Q-EW. Interestingly, the geometric contribution always exceeds the total BMI, indicating a negative correction from the bulk matter fields. This negativity can be understood as the result of subtracting a greater contribution from quantum fields in the connected Q-EW than in the disconnected one. We also reproduce the negative contribution of bulk quantum fields to BMI within a random tensor network (RTN) toy model of double holography. Modeling the bulk as a highly mixed state entangled with a large bath leads to a volume-law bulk entropy. In the large bond-dimension limit, the geometric part of the BMI remains non-negative, while the bulk entropy contribution becomes non-positive when the Q-EWs merge.

China

Yuxuan Liu

Yi Ling

Germany

Zhuo-Yu Xian

Boundary mutual information in double holography

We present a class of exact, dynamical, and fully analytic solutions describing regular black holes formed via the gravitational collapse of matter obeying a generalized polytropic equation of state. Starting from a Vaidya-type geometry with a radially dependent mass function, we demonstrate that regularization of the Kiselev solutions can be achieved through a physically motivated modification of the energy density profile. This procedure leads to nonsingular spacetimes with a de Sitter core and finite curvature invariants at the center. We show that the resulting matter content is naturally described by a generalized polytropic equation of state of the form $$P=\alpha \rho -\zeta \rho ^{\gamma }$$ , where the polytropic index $$\gamma $$ is uniquely determined by the regularization scheme. Within this framework, we obtain exact dynamical generalizations of several well-known regular black hole solutions, including the Hayward and Bardeen spacetimes, as particular cases corresponding to specific values of the polytropic parameters. Remarkably, the requirement that the equation of state remains coordinate independent imposes a universal constraint relating the regularization scale to the mass function, which in turn guarantees the existence of a regular de Sitter core with a curvature scale independent of the black hole mass. Our results provide a unified analytic description of Hayward-like and Bardeen-like black holes emerging from gravitational collapse, offering a consistent effective-matter interpretation rooted in generalized polytropic matter.

Russia

Dmitriy Kudryavcev

Azerbaijan

Vitalii Vertogradov

Exact dynamical regular black holes from generalized polytropic matter

We establish a direct connection between the interior curvature structure of nonsingular black holes (BHs) with a Minkowski core and their observable optical signatures. By classifying these spacetimes into three fundamental types, (Kretschmann scalar $$K_{\text {max}}$$ increasing with mass $\textit{M}$), (mass-independent $$K_{\text {max}}$$ ), and ( $$K_{\text {max}}$$ decreasing with $\textit{M}$), we demonstrate how subtle variations in the core geometry imprint distinguishable features on the BH shadow. A detailed analysis of photon dynamics reveals that the parameters $$\alpha $$ and $\textit{n}$, which control the deviation from Schwarzschild geometry and the radial decay of the regularizing factor, respectively, systematically alter the properties of the photon sphere. These intrinsic geometric differences propagate outward: for fixed parameters, BHs, with the most compact photon sphere, produce the smallest and brightest shadows, whereas BHs yield the largest and dimmest ones. Shadow computations under both static and infalling spherical accretion models confirm that the curvature-based classification directly corresponds to observable differences. Critically, BHs exhibit the strongest sensitivity to parameter variations, making them optimal probes for constraining the underlying spacetime geometry. Our work reveals that even among nonsingular BHs sharing the same asymptotic core, differences in internal curvature are reflected in the shadow morphology, thereby providing a new pathway to test quantum-gravity-inspired models using upcoming high-resolution observations.

Ming-Xin Li

Jin Pu

Guo-Ping Li

Linking interior curvature to observable shadows: a case study of nonsingular black holes

We investigate second-order gravitational perturbations in asymptotically AdS black branes, developing a gauge-invariant framework to compute the amplitude ratio between quadratic and linear quasi-normal modes. Our analysis reveals resonant divergences of this ratio when the summed frequencies of two source modes coincide with the frequency of a third mode. These divergences are shown to manifest as poles in three-point fully retarded correlators of the energy-momentum tensor in the holographically dual quantum field theory, establishing a concrete connection between bulk gravitational nonlinearities and observables in the dual boundary theory. Our findings contribute to the understanding of nonlinearity in quantum many-body systems while deepening the holographic dictionary between spacetime dynamics and quantum correlations.

Wen-Bin Pan

Zhangping Yu

The second-order quasi-normal modes for AdS black branes

We construct a holographic model to study the striped superconductor on ionic lattices. This model features a phase diagram with three distinct phases, namely the charge density wave (CDW) phase, ordinary superconducting phase (SC) and the striped superconducting phase (SSC). The effect of the ionic lattices on the phase diagram is investigated in detail. First, due to the periodic nature of the background, different types of CDW solutions can be found below the critical temperature. Furthermore, with the increase of the lattice amplitude these solutions are locked in different commensurate states. Second, we find that the critical temperature of CDW phase decreases with the increase of the lattice amplitude, while that of the SC phase increases. Additionally, the background solutions are obtained for different phases, and it is verified that the SSC phase has the lowest free energy among all three phases.

Kai Li

Peng Liu

Meng-He Wu

Holographic striped superconductor with ionic lattice

We investigate the perturbation of the scalar field as well as the electromagnetic field over a sort of regular black holes which are characterized by the sub-Planckian curvature and the Minkowskian core. Specifically, we compute the quasinormal modes (QNMs) by employing the pseudo-spectral method. The outburst of overtones is manifestly observed in the QNMs of these regular black holes, which can be attributed to the deviation of the Schwarzschild black hole by quantum effects of gravity. Furthermore, the QNMs under the perturbation of electromagnetic field exhibit smaller real and imaginary parts than those under scalar field perturbation. By comparing the QNMs of the regular black hole featured by Minkowskian core with those of Bardeen black hole featured by de Sitter core, we find they may be an effective tool to distinguish these BHs.

Chen Tang

Qing-Quan Jiang

Quasinormal modes of regular black holes with sub-Planckian curvature and Minkowskian core

We study a holographic model in which the striped structure of charge density is spontaneously formed over an ionic lattice which breaks the translational symmetry explicitly. The effect of commensurate lock-in between the spontaneous stripes and the ionic lattice is observed when the lattice amplitude is large enough. We investigate the optical conductivity as a function of frequency in commensurate state and compare its characteristics during the phase transition from metallic phase to insulating phase. Notably, we find that the DC resistivity in lock-in state increases algebraically with lowering temperature, which is in line with the phenomenon observed in the holographic model for simulating the experimental behavior of Mott insulator in [1]. In addition, at lower temperature the pinning effect is observed for both unlock-in and lock-in states. This holographic model successfully demonstrates the commensurate lock-in signatures, and provides more information for understanding the interplay between ionic lattices and electronic lattices by holography.

The commensurate state and lock-in in a holographic model

Qian Chen

Cheng Peng

Yu Tian

Addendum to: Entanglement of defect subregions in double holography

In the framework of double holography, we investigate the entanglement behavior of a subregion of the defect on the boundary of a CFT3. The entanglement entropy of this defect subregion is determined by the quantum extremal surface (QES) anchored at the two endpoints of the subregion from the brane perspective. We further analyze the entanglement entropy of the quantum matter within this QES, which can be extracted from the total entanglement entropy. We find there are two phases of the QES. To numerically distinguish these phases, we design a strategy for approaching the QES by progressively reducing the width of a semi-ellipse-like region within the CFT3, which is bounded by the defect. During this process, we discover an entanglement phase transition driven by the degree of freedom on the brane. In the shrinking phase, the entanglement wedge of the defect subregion sharply decreases to zero as the removal of the CFT3. In contrast, in the stable phase, the wedge almost remains constant. In this phase, the formulas of entanglement measures can be derived based on defect and CFT3 central charges in the semi-classical limit. For entanglement entropy, the classical geometry only contributes a subleading term with logarithmic divergence, but the matter entanglement exhibits a dominant linear divergence, even in the semi-classical limit. For the reflected entropy within the defect subregion, classical geometry contributes a leading term with logarithmic divergence, while the quantum matter within the entanglement wedge only contributes a finite term.

Entanglement of defect subregions in double holography

We investigate the evolution of entanglement within an open, strongly coupled system interacting with a heat bath as its environment, in the frameworks of both the doubly holographic model and the Sachdev-Ye-Kitaev (SYK) model. Generally, the entanglement within the system initially increases as due to internal interactions; however, it eventually dissipates into the environment. In the doubly holographic setup, we consider an end-of-the-world brane in the bulk to represent an eternal black hole coupled with its radiation and the evolution of the global thermofield double (TFD) state. For small black holes, the reflected entropy between the bipartition exhibits a ramp-plateau-slump behavior, where the plateau arises due to the phase transition of the entanglement wedge cross-section before the Page time. Similarly, the mutual information between the bipartition displays a ramp-slop-stabilizing behavior. In quantum mechanics, we consider a double copy of the SYK-plus-bath system in a global TFD state, resembling an eternal black hole interacting with an environment. The Rényi mutual information within the double-copied SYK clusters exhibits a ramp-plateau-slope-stabilizing behavior. The dynamic behaviors of the entanglement quantities observed in these two models are attributable to the competition between the internal interaction of the system and the external interaction with the baths. Our study provides a fine-grained picture of the entanglement dynamics inside black holes before their Page time.