Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

Lower olefins-generally referring to ethylene, propylene and butylene-are basic carbon-based building blocks that are widely used in the chemical industry, and are traditionally produced through thermal or catalytic cracking of a range of hydrocarbon feedstocks, such as naphtha, gas oil, condensates and light alkanes. With the rapid depletion of the limited petroleum reserves that serve as the source of these hydrocarbons, there is an urgent need for processes that can produce lower olefins from alternative feedstocks. The 'Fischer-Tropsch to olefins' (FTO) process has long offered a way of producing lower olefins directly from syngas-a mixture of hydrogen and carbon monoxide that is readily derived from coal, biomass and natural gas. But the hydrocarbons obtained with the FTO process typically follow the so-called Anderson-Schulz-Flory distribution, which is characterized by a maximum C-C hydrocarbon fraction of about 56.7 per cent and an undesired methane fraction of about 29.2 per cent (refs 1, 10, 11, 12). Here we show that, under mild reaction conditions, cobalt carbide quadrangular nanoprisms catalyse the FTO conversion of syngas with high selectivity for the production of lower olefins (constituting around 60.8 per cent of the carbon products), while generating little methane (about 5.0 per cent), with the ratio of desired unsaturated hydrocarbons to less valuable saturated hydrocarbons amongst the C-C products being as high as 30. Detailed catalyst characterization during the initial reaction stage and theoretical calculations indicate that preferentially exposed {101} and {020} facets play a pivotal role during syngas conversion, in that they favour olefin production and inhibit methane formation, and thereby render cobalt carbide nanoprisms a promising new catalyst system for directly converting syngas into lower olefins.

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The schematic diagram of different Sn-5s and I-5p antibonding strengths of FASnI3 and MASnI3 due to various bond lengths of Sn–I.