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Register a new userA geometry-induced topological phase transition in random graphs
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Added by
Jasper Van Der Kolk
Publication date
2021
fields
Physics
and research's language is
English
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Clustering $unicode{x2013}$ the tendency for neighbors of nodes to be connected $unicode{x2013}$ quantifies the coupling of a complex network to its underlying latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric-to-nongeometric phase transition to be topological in nature, with atypical features such as diverging free energy and entropy as well as anomalous finite size scaling behavior. Moreover, a slow decay of clustering in the nongeometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.
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As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinatorics of the graph formed by the bars. We show that if this graph is an Erdos-Renyi random graph G(n,c/n), then there exists a sharp threshold for a giant rigid component to emerge. For c < c_2, w.h.p. all rigid components span one, two, or three vertices, and when c > c_2, w.h.p. there is a giant rigid component. The constant c_2 approx 3.588 is the threshold for 2-orientability, discovered independently by Fernholz and Ramachandran and Cain, Sanders, and Wormald in SODA07. We also give quantitative bounds on the size of the giant rigid component when it emerges, proving that it spans a (1-o(1))-fraction of the vertices in the (3+2)-core. Informally, the (3+2)-core is maximal induced subgraph obtained by starting from the 3-core and then inductively adding vertices with 2 neighbors in the graph obtained so far.
The transition from topologically nontrivial to a trivial state is studied by first-principles calculations on bulk zinc-blende type (Hg$_{1-x}$Zn$_x$)(Te$_{1-x}$S$_x$) disordered alloy series. The random chemical disorder was treated by means of the Coherent Potential Approximation. We found that although the phase transition occurs at the strongest disorder regime (${xapprox 0.5}$), it is still manifested by well-defined Bloch states forming a clear Dirac cone at the Fermi energy of the bulk disordered material. The computed residual resistivity tensor confirm the topologically-nontrivial state of the HgTe-rich (${x<0.5}$), and the trivial state of the ZnS-rich alloy series (${x>0.5}$) by exhibiting the quantized behavior of the off-diagonal spin-projected component, independently on the concentration $x$.
The recently discovered three dimensional or bulk topological insulators are expected to exhibit exotic quantum phenomena. It is believed that a trivial insulator can be twisted into a topological state by modulating the spin-orbit interaction or the crystal lattice via odd number of band
Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are all intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale-invariance, self-similarity, and other forms of fundamental symmetries in networks. Network geometry is also of great utility in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments dealing with these approaches to network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.
We investigate the possibility of using structural disorder to induce a topological phase in a solid state system. Using first-principles calculations, we introduce structural disorder in the trivial insulator BiTeI and observe the emergence of a topological insulating phase. By modifying the bonding environments, the crystal-field splitting is enhanced and the spin-orbit interaction produces a band inversion in the bulk electronic structure. Analysis of the Wannier charge centers and the surface electronic structure reveals a strong topological insulator with Dirac surface states. Finally, we propose a prescription for inducing topological states from disorder in crystalline materials. Understanding how local environments produce topological phases is a key step for predicting disordered and amorphous topological materials.
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