Derivative Formula and Applications for Degenerate Diffusion Semigroups

By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack inequalities are derived.Comment: 18 page

Nucleation of membrane adhesions

Recent experimental and theoretical studies of biomimetic membrane adhesions [Bruinsma et al., Phys. Rev. E 61, 4253 (2000); Boulbitch et al., Biophys. J. 81, 2743 (2001)] suggested that adhesion mediated by receptor interactions is due to the interplay between membrane undulations and a double-well adhesion potential, and should be a first-order transition. We study the nucleation of membrane adhesion by finding the minimum-energy path on the free energy surface constructed from the bending free energy of the membrane and the double-well adhesion potential. We find a nucleation free energy barrier around 20kBT for adhesion of flexible membranes, which corresponds to fast nucleation kinetics with a time scale of the order of seconds. For cell membranes with a larger bending rigidity due to the actin network, the nucleation barrier is higher and may require active processes such as the reorganization of the cortex network to overcome this barrier. Our scaling analysis suggests that the geometry of the membrane shapes of the adhesion contact is controlled by the adhesion length that is determined by the membrane rigidity, the barrier height, and the length scale of the double-well potential, while the energetics of adhesion is determined by the depths of the adhesion potential. These results are verified by numerical calculations

Quantum anomalous Hall effect in magnetic topological insulators

The search for topologically non-trivial states of matter has become an important goal for condensed matter physics. Here, we give a theoretical introduction to the quantum anomalous Hall (QAH) effect based on magnetic topological insulators in two-dimension (2D) and three-dimension (3D). In 2D topological insulators, magnetic order breaks the symmetry between the counter-propagating helical edge states, and as a result, the quantum spin Hall effect can evolve into the QAH effect. In 3D, magnetic order opens up a gap for the topological surface states, and chiral edge state has been predicted to exist on the magnetic domain walls. We present the phase diagram in thin films of a magnetic topological insulator and review the basic mechanism of ferromagnetic order in magnetically doped topological insulators. We also review the recent experimental observation of the QAH effect. We discuss more recent theoretical work on the coexistence of the helical and chiral edge states, multi-channel chiral edge states, the theory of the plateau transition, and the thickness dependence in the QAH effect.Comment: 13 pages, 11 figures. Invited Review to Physica Scripta, Nobel Physica Symposium on New Forms of Matter: Topological Insulators and Superconductor

Simplified topological invariants for interacting insulators

We propose general topological order parameters for interacting insulators in terms of the Green's function at zero frequency. They provide an unified description of various interacting topological insulators including the quantum anomalous Hall insulators and the time reversal invariant insulators in four, three and two dimensions. Since only Green's function at zero frequency is used, these topological order parameters can be evaluated efficiently by most numerical and analytical algorithms for strongly interacting systems.Comment: Published versio