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Recent experiments on multilayer rhombohedral graphene have unearthed a number of interesting phenomena in the regime where integer and fractional quantum anomalous Hall phenomena were previously reported. Specifically, at low temperature (𝑇) and low applied currents, an “extended” integer quantum anomalous Hall (EIQAH) state is seen over a wide range of the phase diagram. As the current is increased, at low 𝑇, the EIQAH state undergoes a phase transition to a metallic state at generic fillings, and to the fractional quantum anomalous Hall (FQAH) state at the Jain fillings. Increasing temperature at the Jain fillings also leads to an evolution out of the EIQAH state to the Jain state. Here we provide an interpretation of many of these observations. We describe the EIQAH state as a crystalline state (either of holes doped into the 𝜈=1 state or an anomalous Hall crystal of electrons) that breaks moiré translation symmetry. At generic fillings, we show how an electric current-induced depinning transition of the crystalline order leads to peculiar nonlinear current-voltage curves consistent with the experiment. At Jain fillings, we propose that the depinning transition is preempted by an equilibrium transition between EIQAH and Jain FQAH states. This transition occurs due to the large polarizability of the Jain FQAH states, which enables them to effectively lower their energy in an applied electric field compared to the crystal states. We also discuss the finite-temperature evolution in terms of the relative entropies of the crystalline and FQAH states.
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Stability Theory of Stochastic Models in Opinion Dynamics
We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, that are inspired by the DeGroot-Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as nonlinear Markov chains. In this paper we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the l1-metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear, respectively–both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations
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- PAR ID:
- 10106348
- Date Published:
- Journal Name:
- IEEE Transactions on Automatic Control
- ISSN:
- 0018-9286
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TAC.2019.2912490
