Topology-related phenomena are at the forefront of contemporary physics, and the topological charges of defects constitute one of the central topics in this field. Such charges are often represented by the winding numbers of associated vector-field distributions. Therefore, distinguishing vector-field patterns and their corresponding winding numbers provides an intuitive way to identify the topological charges of defects. It also suggests a route to designing specific vector-field distributions that can generate defect structures carrying prescribed topological charges. In this regard, it is essential for researchers to develop a clear understanding of vector fields with different winding numbers, namely different topological charges. In this paper, we first clarify the relationship between the topological charge of defects arising from a Schrödinger wave-function field in quantum mechanics and the winding number of the corresponding two-dimensional vector-field distribution. We then present in detail the vector-field patterns associated with arbitrary positive and negative integer winding numbers in two dimensions. Finally, we provide a quantitative calculation of the vector orientation at an arbitrary point in the two-dimensional plane. This paper may serve as a useful reference for junior graduate students and senior undergraduate students who are beginning to study topological physics.

Complex analysis acts as a key part of the undergraduate course, Methods of Mathematical Physics. One of its kernel concepts is the Cauchy integral defined in the complex plane $\mathbb{C}$. The aim of this paper is to introduce a new point of knowledge to undergraduate students and junior postgraduate students, that is, a third axis (the real axis $\mathbb{R}$, denoted as the time axis) can be introduced into the 2-dimensional complex plane $\mathbb{C}$ to form a (2+1)-dimensional space, $\mathbb{C}\times \mathbb{R}$, such that the Cauchy integral in the complex plane is transformed to be a description for an entangle with knot winding number Lk= ±1. Winding numbers are known as a weak topological invariant in knot theory, able to be regarded as the simplest case of a higher-order tool, the Kontsevich integral. Furthermore, if “multiple correlators” are considered simultaneously in a Cauchy integral, it means obtaining the general form of the Kontsevich integral (where the so-called group factor is ignored). This new tool plays a crucial role in both knot theory in mathematics and quantum field theory in physics. In this paper we will detail the above mentioned relevant concepts and theoretical methods, in the hope of providing for the interested reader a shortcut from undergraduate fundamental courses of theoretical physics to the frontiers of scientific research in modern mathematics and physics.

We aim to provide a way for interested undergraduate and postgraduate students to launch from college physics to access the cutting edge of the current international research on two-dimensional and three-dimensional new topological materials. This topic contains two parts, this paper is the second part. Based on the quantum Hall effects (integer, fractional, anomalous, spin) reviewed in the first paper of this topic, we give a brief introduction in this paper to several new topological materials: graphene, topological insulators and topological semimetals. As the examples of the two-band model describing the quantum anomalous Hall effects, we discuss the tight-binding model in graphene and square lattices, with emphasis placed on the relationship between energy bands and topological invariants, as well as the resulting bulk-boundary correspondence. The reader is suggested to strengthen the capability of theoretical calculation by paying attention to spin-orbit coupling, Fourier transform of Hamiltonian, solving energy bands, and the calculation of topological invariants and edge states. Finally a brief introduction to the energy bands and the corresponding surface states are also presented for different types of semimetals.

We aim to provide a way for interested undergraduate and postgraduate students to launch from college physics to access the cutting edge of the current international research on two-dimensional and three-dimensional new topological materials. This topic contains two parts, this paper is the first part. Undergraduates have learned electrodynamics, quantum mechanics, solid state physics, and other courses in the university stage. However, many textbooks often stay in the introduction of the traditional content of the course, fail to extend the scope to the latest progress of modern quantum physics. This paper serves as an attempt to fill this gap. Starting from the classical Hall effect in college physics, we first introduce the technique how to apply quantum mechanics to the study of a Hall system in 2-dimension or (2+1) dimensions. Then various quantum Hall effects (integral, fractional, spin and anomalous) and topological invariants existence in them are introduced.