Berry Phase & Topological Phases of Matter
Geometric phase from adiabatic transport, and how it gives rise to topological invariants.
Consider a Hamiltonian $\hat H(\mathbf R)$ depending on parameters $\mathbf R$. Slowly varying $\mathbf R$ along a closed loop, an instantaneous eigenstate $|n(\mathbf R)\rangle$ acquires not only a dynamical phase but also a Berry phase
$$\gamma_n = i \oint \langle n(\mathbf R) | \nabla_{\mathbf R} | n(\mathbf R)\rangle \cdot d\mathbf R = \oint \mathbf A_n \cdot d\mathbf R,$$where $\mathbf A_n = i \langle n | \nabla_{\mathbf R} | n\rangle$ is the Berry connection. By Stokes,
$$\gamma_n = \int_S \mathbf F_n \cdot d\mathbf S, \quad \mathbf F_n = \nabla \times \mathbf A_n \quad (\text{Berry curvature}).$$For a spin-$\tfrac{1}{2}$ in a slowly rotating magnetic field, the Berry phase is half the solid angle swept: $\gamma = \Omega/2$. The Berry curvature acts as a magnetic field in parameter space.
Topological band insulators have nontrivial integrated Berry curvature (Chern number $C \in \mathbb Z$) over the Brillouin zone. The integer quantum Hall conductivity is $\sigma_{xy} = (e^2/h)\, C$. Edge states protected by topology cannot be removed by smooth deformations that preserve the gap.
Interactive: Berry phase for a spin in a rotating field
The path traces a cone of half-angle $\theta$ on the parameter sphere; the Berry phase equals half the enclosed solid angle.