Response function of four-dimensional topological insulator

• Main Author: Theofilus, Dustin Erhard
• Other Authors: Chong Yidong
• Format: Final Year Project
• Language: English
• Published: Nanyang Technological University 2024
• Subjects: Physics; Topological insulator; Haldane lattice; Four-dimensional lattice; Linear response theory; First chern number; Hall conductivity; Edge currents
• Online Access: https://hdl.handle.net/10356/175654
• Institution: Nanyang Technological University

In this paper, we apply linear response theory to investigate the current-current response in four-dimensional topological insulator (TI). For lattices with periodic boundary conditions, we show that the bulk DC response function is proportional to the first Chern Number (1CN) and equal to the 2D Hall conductivity. We confirmed that the Haldane lattice exhibits non-zero DC response function, characterised by 1CN equal to ±1 since time-reversal symmetry is broken; whereas the four-dimensional lattice shows zero DC response function, characterised by vanishing 1CN since time-reversal symmetry remains unbroken. For lattices with open boundary conditions, we demonstrate that the energy spectrum of the trivial phase is gapped like that of ordinary insulators. The currentcurrent response function is therefore zero as expected. The energy spectrum of the topological phase however, hosts discrete, zero-energy, edge states that are responsible for carrying edge currents. The current-current response function is zero in the bulk, but non-zero at the edges. We also observe that the finite size of the lattice does affect the values of current-current response function.