On a new definition of Ricci curvature on Alexandrov spaces

Recently, in [49], a new definition for lower Ricci curvature bounds on Alexandrov spaces was introduced by the authors. In this article, we extend our research to summarize the geometric and analytic results under this Ricci condition. In particular, two new results, the rigidity result of Bishop-Gromov volume comparison and Lipschitz continuity of heat kernel, are obtained.Comment: Clarify a citation on page 20, add 2 reference

The effect of in-plane magnetic field and applied strain in quantum spin Hall systems: application to InAs/GaSb quantum wells

Motivated by the recent discovery of quantized spin Hall effect in InAs/GaSb quantum wells\cite{du2013}$^,$\cite{xu2014}, we theoretically study the effects of in-plane magnetic field and strain effect to the quantization of charge conductance by using Landauer-Butikker formalism. Our theory predicts a robustness of the conductance quantization against the magnetic field up to a very high field of 20 tesla. We use a disordered hopping term to model the strain and show that the strain may help the quantization of the conductance. Relevance to the experiments will be discussed.Comment: 8 pages, 10 figures. Comments are welcome

Ricci Curvature on Alexandrov spaces and Rigidity Theorems

In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.Comment: final versio

Lipschitz continuity of harmonic maps between Alexandrov spaces

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally H\"older continuous. In [39], F. H. Lin proposed a challenge problem: Can the H\"older continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.Comment: We remove the assumption in the previous version that the domain space has nonnegative generalized Ricci curvature. This solves Lin's conjecture completely. To appear in Invent. Mat

Yau's gradient estimates on Alexandrov spaces

In this paper, we establish a Bochner type formula on Alexandrov spaces with Ricci curvature bounded below. Yau's gradient estimate for harmonic functions is also obtained on Alexandrov spaces.Comment: Final version, to appear in J. Differential Geo

Helical damping and anomalous critical non-Hermitian skin effect

Non-Hermitian skin effect and critical skin effect are unique features of non-Hermitian systems. In this Letter, we study an open system with its dynamics of single-particle correlation function effectively dominated by a non-Hermitian damping matrix, which exhibits $\mathbb{Z}_2$ skin effect, and uncover the existence of a novel phenomenon of helical damping. When adding perturbations that break anomalous time reversal symmetry to the system, the critical skin effect occurs, which causes the disappearance of the helical damping in the thermodynamic limit although it can exist in small size systems. We also demonstrate the existence of anomalous critical skin effect when we couple two identical systems with $\mathbb{Z}_2$ skin effect. With the help of non-Bloch band theory, we unveil that the change of generalized Brillouin zone equation is the necessary condition of critical skin effect.Comment: 7+5 pages, 4+5 figure

Theory for Spin Selective Andreev Reflection in Vortex Core of Topological Superconductor: Majorana Zero Modes on Spherical Surface and Application to Spin Polarized Scanning Tunneling Microscope Probe

Majorana zero modes (MZMs) have been predicted to exist in the topological insulator (TI)/superconductor (SC) heterostructure. Recent spin polarized scanning tunneling microscope (STM) experiment$^{1}$ has observed spin-polarization dependence of the zero bias differential tunneling conductance at the center of vortex core, which may be attributed to the spin selective Andreev reflection, a novel property of the MZMs theoretically predicted in 1-dimensional nanowire$^{2}$. Here we consider a helical electron system described by a Rashba spin orbit coupling Hamiltonian on a spherical surface with a s-wave superconducting pairing due to proximity effect. We examine in-gap excitations of a pair of vortices with one at the north pole and the other at the south pole. While the MZM is not a spin eigenstate, the spin wavefunction of the MZM at the center of the vortex core, r = 0, is parallel to the magnetic field, and the local Andreev reflection of the MZM is spin selective, namely occurs only when the STM tip has the spin polarization parallel to the magnetic field, similar to the case in 1-dimensional nanowire2. The total local differential tunneling conductance consists of the normal term proportional to the local density of states and an additional term arising from the Andreev reflection. We also discuss the finite size effect, for which the MZM at the north pole is hybridized with the MZM at the south pole. We apply our theory to examine the recently reported spin-polarized STM experiments and show good agreement with the experiments.Comment: 14 pages, 14 figures, 1 table. Comments are welcome