What We Talk About When We Talk About Symmetry: A Simple Hopping Model of Mono- and Bilayer Graphene

Synopsis

Symmetry constrains what physical phenomena can happen. From Noether’s theorem to group theory, physicists have long used different mathematical tools to describe the symmetries of various systems, and each new symmetry discovery reveals a deeper pattern in nature. In this talk, I will briefly introduce Lie algebras and representation theory, and explain how they mathematically describe the continuous symmetries in a system. Then, I will apply these ideas to a simple, nearest-neighbor hopping model of monolayer graphene and find the continuous symmetries in the first and second order expansions of the monolayer graphene hopping Hamiltonian. The continuous symmetries lead to the Hamiltonian conservation of higher-dimensional rotations and the discovery of the effective masses, which separate the energy band in monolayer graphene. I will also extend the same approach to bilayer graphene and show how a unitary transformation to a more explicit basis (Majorana basis) reveals certain subtle symmetries of the graphene system. The aim is to formalize the physical symmetries mathematically through group theory and demonstrate how this viewpoint helps us discover physical results in quantum mechanics, quantum field theory and condensed matter physics.