A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in ...gauge-gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Rényi entropies. Here we show that all Rényi entropies satisfy a similar area law in holographic theories and are given by the areas of dual cosmic branes. This geometric prescription is a one-parameter generalization of the minimal surface prescription for entanglement entropy. Applying this we provide the first holographic calculation of mutual Rényi information between two disks of arbitrary dimension. Our results provide a framework for efficiently studying Rényi entropies and understanding entanglement structures in strongly coupled systems and quantum gravity.
A
bstract
Recent work showed holographic error correcting codes to have simple universal features at
O
(1
/G
). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated ...with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (
O
(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in
G
, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
A
bstract
We study the Euclidean gravitational path integral computing the Rényi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition ...can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton’s constant
G
N
, providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in
G
N
. We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.
Metal–organic frameworks (MOFs) have recently emerged as a type of uniformly and periodically atom‐distributed precursor and efficient self‐sacrificial template to fabricate hierarchical ...porous‐carbon‐related nanostructured functional materials. For the first time, a Cu‐based MOF, i.e., Cu‐NPMOF is used, whose linkers contain nitrogen and phosphorus heteroatoms, as a single precursor and template to prepare novel Cu3P nanoparticles (NPs) coated by a N,P‐codoped carbon shell that is extended to a hierarchical porous carbon matrix with identical uniform N and P doping (termed Cu3P@NPPC) as an electrocatalyst. Cu3P@NPPC demonstrates outstanding activity for both the hydrogen evolution and oxygen reduction reaction, representing the first example of a Cu3P‐based bifunctional catalyst for energy‐conversion reactions. The high performances are ascribed to the high specific surface area, the synergistic effects of the Cu3P NPs with intrinsic activity, the protection of the carbon shell, and the hierarchical porous carbon matrix doped by multiheteroatoms. This strategy of using a diverse MOF as a structural and compositional material to create a new multifunctional composite/hybrid may expand the opportunities to explore highly efficient and robust non‐noble‐metal catalysts for energy‐conversion reactions.
Cu3P nanoparticles coated by a N,P‐codoped carbon shell (hierarchical porous carbon matrix) are prepared using a novel Cu‐based metal–organic framework (MOF) containing dual linkers as a template and single precursor. The Cu3P@NPPC catalyst demonstrates a high specific surface area and affords remarkable bifunctional electrocatalytic performance for hydrogen evolution reaction and oxygen reduction reaction with long‐term durability for both reactions.
A
bstract
Recent work found an enhanced correction to the entanglement entropy of a subsystem in a chaotic energy eigenstate. The enhanced correction appears near a phase transition in the ...entanglement entropy that happens when the subsystem size is half of the entire system size. Here we study the appearance of such enhanced corrections holo-graphically. We show explicitly how to find these corrections in the example of chaotic eigenstates by summing over contributions of all bulk saddle point solutions, including those that break the replica symmetry. With the help of an emergent rotational symmetry, the sum over all saddle points is written in terms of an effective action for cosmic branes. The resulting Renyi and entanglement entropies are then naturally organized in a basis of fixed-area states and can be evaluated directly, showing an enhanced correction near holographic entanglement transitions. We comment on several intriguing features of our tractable example and discuss the implications for finding a convincing derivation of the enhanced corrections in other, more general holographic examples.
Continuing research on the preparation and structural determination of monolayer-protected silver clusters has been performed. The compounds include mixed-valence Ag
0/1+
clusters and single-valence ...Ag
1+
clusters, which contain a few to tens or hundreds of Ag atoms that are protected by organic ligands. Sometimes, counter ions and extraneous species appear in their crystalline state. These non-metal parts define the shell layers of silver clusters. Strong coordination bonds and weak supramolecular interactions have been employed not only to modify the shell configurations and components of discrete silver clusters but also to hierarchically assemble silver clusters, producing novel cluster-based functional materials with unexpected physical and chemical properties. Atomically-precise structures help to map out definite electronic structures and structure-property correlations, enabling precise control of shell layers to achieve desired stability and specific functionalities. In this Tutorial Review, based on classic silver cluster paradigms, we first summarize the strategies and recent advances in precise modification and hierarchical assembly of well-defined silver clusters through shell engineering. Second, the correlations of structure-property and structure-functionality are summarized. Of these, the most important is structure-luminescence relationship, which is discussed in detail. In this topic, the uniqueness and prospect of silver clusters as potential lighting materials are scrutinized. Finally, the existing challenges and perspectives of functional silver clusters are presented. The general strategic design presented in this Review will motivate researchers to exploit the development of functionality-oriented materials based on nanosized building blocks in the enrichment of nanotechnology and material science.
This tutorial review focuses on the modification and assembly of atomically-precise silver clusters by changing shell layers for more stability and functionalities, especially for brighter luminescence.
A
bstract
We point out a connection between the emergence of bulk locality in AdS/CFT and the theory of quantum error correction. Bulk notions such as Bogoliubov transformations, location in the ...radial direction, and the holographic entropy bound all have natural CFT interpretations in the language of quantum error correction. We also show that the question of whether bulk operator reconstruction works only in the causal wedge or all the way to the extremal surface is related to the question of whether or not the quantum error correcting code realized by AdS/CFT is also a “quantum secret sharing scheme”, and suggest a tensor network calculation that may settle the issue. Interestingly, the version of quantum error correction which is best suited to our analysis is the somewhat nonstandard “operator algebra quantum error correction” of Beny, Kempf, and Kribs. Our proposal gives a precise formulation of the idea of “subregion-subregion” duality in AdS/CFT, and clarifies the limits of its validity.
Silver chalcogenolate cluster assembled materials (SCAMs) are a category of promising light‐emitting materials the luminescence of which can be modulated by variation of their building blocks ...(cluster nodes and organic linkers). The transformation of a singly emissive Ag12(SBut)8(CF3COO)4(bpy)4n (Ag12bpy, bpy=4,4′‐bipyridine) into a dual‐emissive (Ag12(SBut)6(CF3COO)6(bpy)3)n (Ag12bpy‐2) via cluster‐node isomerization, the critical importance of which was highlighted in dictating the photoluminescence properties of SCAMs. Moreover, the newly obtained Ag12bpy‐2 served to construct visual thermochromic Ag12bpy‐2/NH2 by a mixed‐linker synthesis, together with dichromatic core–shell Ag12bpy‐2@Ag12bpy‐NH2‐2 via solvent‐assisted linker exchange. This work provides insight into the significance of metal arrangement on physical properties of nanoclusters.
Tandem varying of Ag cluster structure and tuning mixed linkers in a stepwise fashion are used to achieve dual‐emitting homogeneous and core–shell silver chalcogenolate cluster assembled materials. The arrangements of silver in the cluster are demonstrated to play a fundamentally important role in luminescence.
Atomically precise enantiomeric metal clusters are scarce, and copper(I) alkynyl clusters with intense circularly polarized luminescence (CPL) responses have not been reported. A pair of chiral ...alkynyl ligands, (R/S)‐2‐diphenyl‐2‐hydroxylmethylpyrrolidine‐1‐propyne (abbreviated as R/S‐DPM) we successfully prepared and single crystals were characterized of optically pure enantiomeric pair of atomically‐precise copper(I) clusters, Cu14(R/S‐DPM)8(PF6)6 (denoted as R/S‐Cu14), which feature bright red luminescence and CPL with a high luminescence anisotropy factor (glum). A dilute solution containing R/S‐Cu14 was nonluminescent and CPL inactive at room temperature. Crystallization‐ and aggregation‐induced emission (CIE and AIE, respectively) contribute to the triggering of the CPL of R/S‐Cu14 in the crystalline and aggregated states. Their AIE behavior and good biocompatibility indicated applications of these copper(I) clusters in cell imaging in HeLa and NG108‐15 cells.
Atomically precise chiral CuI alkynyl nanoclusters R/S‐Cu14 with inherent chirality were synthesized for the first time. Crystallization‐ and aggregation‐induced emission (CIE and AIE, respectively) trigger circularly polarized luminescence (CPL) with an unprecedented luminescence anisotropy factor (glum).
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