In this talk, I will first discuss several progresses made in our group about fundamental properties of skyrmions in chiral magnetic films. These include 1) skyrmion sizes in isolated, in crystal, or in stripy forms; 2) skyrmion nucleation, formation, and potential barrier energies; 3) the roles of magnetic field in skyrmion crystal formation; 4) the stability and existing conditions of composite skyrmions such as target skyrmions and skyrmion bags/cluster; 5) topological equivalence of stripy phases and skyrmion crystals. Then I will discuss a new theory about widely observed unusual anisotropic magnetoresistance (UAMR) in bilayers which leads to the notion of the spin-Hall MR (SMR) in the famous SMR theory. The theory is based on the universal features in all bilayer heterostructure: resistivity tensor depends on magnetization and interfacial field. I will show that the angular dependencies of UAMR do not depend on the microscopic details, thus are universal. Experiments that can test this theory against the SMR theory are also proposed. ** This work is supported by the National Key Research and Development Program (No. 2020YFA0309600), the NSFC Grant (No. 11974296), and HK RGC Grants (No. 16300523, 16300522, and 16302321).

Topology, a well-established concept in mathematics, has nowadays become essential to describe condensed matter. At its core are chiral electron states on the bulk, surfaces and edges of the condensed matter systems, in which spin and momentum of the electrons are locked parallel or anti-parallel to each other. Magnetic and non-magnetic Weyl semimetals, for example, exhibit chiral bulk states that have enabled the realization of predictions from high energy and astrophysics involving the chiral quantum number, such as the chiral anomaly, the mixed axial-gravitational anomaly and axions. The potential for connecting chirality as a quantum number to other chiral phenomena across different areas of science, including the asymmetry of matter and antimatter and the homochirality of life, brings topological materials to the fore.
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The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. The seminar introduces a definition of universal scaling on a non-homogeneous graph. The corresponding scaling theory is expected to depend only on a single parameter, the spectral dimension ds, which plays the role of the relevant parameter on complex geometries. We will then focus on a concrete example, the long-range diluted graph (LRDG), which allows to tune the value of the spectral dimension continuously. By means of extensive numerical simulations, we probe the scaling exponents of a simple instance of O(N) symmetric models on the LRDG showing quantitative agreement with the theoretical prediction of universal scaling in fractional dimensions.