Vacancy-Induced Tunable Kondo Effect in Twisted Bilayer Graphene.

In single sheets of graphene, vacancy-induced states have been shown to host an effective spin-1/2 hole that can be Kondo screened at low temperatures. Here, we show how these vacancy-induced impurity states survive in twisted bilayer graphene (TBG), which thus provides a tunable system to probe the critical destruction of the Kondo effect in pseudogap hosts. Ab initio calculations and atomic-scale modeling are used to determine the nature of the vacancy states in the vicinity of the magic angle in TBG, demonstrating that the vacancy can be treated as a quantum impurity. Utilizing this insight, we construct an Anderson impurity model with a TBG host that we solve using the numerical renormalization group combined with the kernel polynomial method. We determine the phase diagram of the model and show how there is a strict dichotomy between vacancies in the AA/BB versus AB/BA tunneling regions. In AB/BA vacancies, the Kondo temperature at the magic angle develops a broad distribution with a tail to vanishing temperatures due to multifractal wave functions at the magic angle. We argue that scanning tunneling microscopy in the vicinity of the vacancy can act as a probe of both the critical single-particle states and the underlying many-body ground state in magic-angle TBG.

Covering Problems and Core Percolations on Hypergraphs.

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

Deciphering functional redundancy in the human microbiome.

Although the taxonomic composition of the human microbiome varies tremendously across individuals, its gene composition or functional capacity is highly conserved - implying an ecological property known as functional redundancy. Such functional redundancy has been hypothesized to underlie the stability and resilience of the human microbiome, but this hypothesis has never been quantitatively tested. The origin of functional redundancy is still elusive. Here, we investigate the basis for functional redundancy in the human microbiome by analyzing its genomic content network - a bipartite graph that links microbes to the genes in their genomes. We find that this network exhibits several topological features that favor high functional redundancy. Furthermore, we develop a simple genome evolution model to generate genomic content network, finding that moderate selection pressure and high horizontal gene transfer rate are necessary to generate genomic content networks with key topological features that favor high functional redundancy. Finally, we analyze data from two published studies of fecal microbiota transplantation (FMT), finding that high functional redundancy of the recipient's pre-FMT microbiota raises barriers to donor microbiota engraftment. This work elucidates the potential ecological and evolutionary processes that create and maintain functional redundancy in the human microbiome and contribute to its resilience.

Bridges in complex networks.

A bridge in a graph is an edge whose removal disconnects the graph and increases the number of connected components. We calculate the fraction of bridges in a wide range of real-world networks and their randomized counterparts. We find that real networks typically have more bridges than their completely randomized counterparts, but they have a fraction of bridges that is very similar to their degree-preserving randomizations. We define an edge centrality measure, called bridgeness, to quantify the importance of a bridge in damaging a network. We find that certain real networks have a very large average and variance of bridgeness compared to their degree-preserving randomizations and other real networks. Finally, we offer an analytical framework to calculate the bridge fraction and the average and variance of bridgeness for uncorrelated random networks with arbitrary degree distributions.