The solar minimum 23/24 is considered to be unusual because it exhibits features that differ notably from those commonly seen in pervious minima. In this letter, we analyze the solar polar magnetic field, the potential-field solution of the solar cor
ona, and the in-situ solar wind measurements to see whether the recent solar minimum 24/25 is another unusual one. While the dipolar configuration that are commonly seen during minimum 22/23 and earlier minima persist for about half a year after the absolute minimum of solar cycle 24, the corona has a morphology more complex than a simple dipole before the absolute minimum. The fast solar wind streams are less dominant than minimum 23/24. The IMF strength, density and mass flux that are historically low in the minimum 23/24 are regained during minimum 24/25, but still do not reach the minimum 22/23 level. From the analysis of this Letter, it seems that the minimum 24/25 is only partially unusual, and the recovery of the commonly minimum features may result from the enhancement of the polar field.
The quantum spin Hall (QSH) effect is known to be unstable to perturbations violating time-reversal symmetry. We show that creating a narrow ferromagnetic (FM) region near the edge of a QSH sample can push one of the counterpropagating edge states to
the inner boundary of the FM region, and leave the other at the outer boundary, without changing their spin polarizations and propagation directions. Since the two edge states are spatially separated into different lanes, the QSH effect becomes robust against symmetry-breaking perturbations.
Topological phase transitions in a three-dimensional (3D) topological insulator (TI) with an exchange field of strength $g$ are studied by calculating spin Chern numbers $C^pm(k_z)$ with momentum $k_z$ as a parameter. When $|g|$ exceeds a critical va
lue $g_c$, a transition of the 3D TI into a Weyl semimetal occurs, where two Weyl points appear as critical points separating $k_z$ regions with different first Chern numbers. For $|g|<g_c$, $C^pm(k_z)$ undergo a transition from $pm 1$ to 0 with increasing $|k_z|$ to a critical value $k_z^{tiny C}$. Correspondingly, surface states exist for $|k_z| < k_z^{tiny C}$, and vanish for $|k_z| ge k_z^{tiny C}$. The transition at $|k_z| = k_z^{tiny C}$ is acompanied by closing of spin spectrum gap rather than energy gap.
We propose a topological understanding of the quantum spin Hall state without considering any symmetries, and it follows from the gauge invariance that either the energy gap or the spin spectrum gap needs to close on the system edges, the former scen
ario generally resulting in counterpropagating gapless edge states. Based upon the Kane-Mele model with a uniform exchange field and a sublattice staggered confining potential near the sample boundaries, we demonstrate the existence of such gapless edge states and their robust properties in the presence of impurities. These gapless edge states are protected by the band topology alone, rather than any symmetries.
We show that a thin film of a three-dimensional topological insulator (3DTI) with an exchange field is a realization of the famous Haldane model for quantum Hall effect (QHE) without Landau levels. The exchange field plays the role of staggered fluxe
s on the honeycomb lattice, and the hybridization gap of the surface states is equivalent to alternating on-site energies on the AB sublattices. A peculiar phase diagram for the QHE is predicted in 3DTI thin films under an applied magnetic field, which is quite different from that either in traditional QHE systems or in graphene.
