This lecture focuses on basic understanding of the topological properties of materials and discusses the applications of information devices and topological quantum computers based on topological materials. To achieve this, we first study basic quantum mechanics and methods for calculating the band structure of materials, and discuss the topological properties of wave functions. Next, we study about the topological properties of two-dimensional semiconductor materials and graphene, and understand the theoretical descriptions of topological systems such as topological insulators, topological superconductors, and the quantum Hall effect. Based on this knowledge, we further explore the recent device applications through reviewing information devices based on the topological quantum materials, and the implementation of topological quantum computers.

Basic knowledge of quantum mechanics, Basic knowledge of Condensed matter physics or physical electronics
: Quantum Mechanics, Physics of Materials, Physical Electronics, Quantum Chemistry, or any equivalents.

Week 1 : Basic Quantum Mechanics
-Quantum state, Hilbert space, Observable as operator, Measurement, …
Week 2 : Band theory of Quantum Matter
-Band theory, Bloch Hamiltonian, Tight binding model, Many-body systems
Week 3 : Adiabatic Evolution and Berry Phase
-Adiabatic theorem, Berry connection, Berry Phase, Berry Curvature
Week 4: Topological Phase
-Topological Invariants, Continuous Deformation, Topological Phase transition
Week 5 : Symmetry and Topologies
- TRS/ PHS/chiral Symmetries, Symmetry-protected/Intrinsic Topology

Week 6 : Integer Quantum Hall System
-Landau level, Hall Conductance, TKNN invariants
Week 7 : Topological Insulator I
- Haldane model, Bulk-edge Correspondence, Chern Insulator
Week 8 : Topological insulator II
-Z2 topological insulator, Kramer’s degeneracy, Spin-momentum Locking
Week 9-10: Interaction-driven Topology
-Fractional Quantum Hall state, Charge Fractionalization, Moire system

Week 11 : Anyons
-Abelian Anyon, Non-abelian Anyon
Week 12: Topological Superconductors
-Brief description of Superconductor, BdG Hamiltonian, SC+TI
Week 13: Majorana Zero Modes
-Kitaev Chain Model, ZBCP in quantum matters, Experimental Studies
Week 14 : Braiding
-Non-abelian Statistics, Majorana Braiding
Week 15 : Towards Topological Quantum Computing
-Fusion rules, Unitary gates, fault tolerance
Week 16 : Final Exam

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