# Berry Phase and Berry Curvature

The most horrible moment is not when you find something you have not learned and calculated; it is when you find something you have learned and calculated, but have no idea of this…

(In fact I have learned the intrinsic anomalous hall effect, and calculated the unquantized Hall conductivity in Cenke’s homework 4. But at that time I had no concept of AHE… I have also learned berry phases in undergrad, but at during work on Cenke’s homework I can make no connection to the old one I learned, nor could I relate berry phase to topology…)

Of course, Berry phase effects on electronic properties (Rev. Mod. Phys. 82, 1959 (2010)) is the reference for berry phase. This is an undergrad-level review paper, and whose content should be known to any undergrad who wants to do theoretical physics, by the end of their undergrad. I should have read it!

Condition under which berry phase cannot be neglected: “in systems with broken either time-reversal or inversion symmetries, their proper description requires the use of the full velocity formula 3.6.”

What is berry phase: note that a Hamiltonian is always a map from some parameter space to some target space; and the eigenstates are also map from this parameter space to a target space. The parameter space can be k space, or magnetic field space and also on. (The important thing is: a parameter space is a manifold, so we can consider topology on it!)

If we examine the map of an eigenstate on a closed path in the parameter space, we will find that this eigenstate, when coming back to its original place, accumulates a phase $\gamma = i\oint d\mathbf{R}\cdot \langle \psi(\mathbf{R})|\nabla_{\mathbf{R}}|\psi(\mathbf{R})\rangle$. (Note in this description, we have used adiabatic assumtion.) Of course, this phase is unique to quantum mechanics – it reflects that U(1) projective nature of physical states in the description of physical states (i.e. the Hilbert space). This $\gamma$ is the berry phase.

As noted above, the phase $\gamma$ is obtained only when we have a closed path in the parameter space. So how do we create a closed path in it? The review paper tells us this. In the case of the parameter space being a BZ, we can create a closed path by either applying a magnetic field (so that electrons go under cyclic motion) or apply high enough electric field (so that electrons drift on the dispersion, and due to the periodic nature of BZ they go back to their original spots). For a spin coupled to a magnetic field, we can let the magnetic field rotate in time (so that the magnetic field draws a circle in the parameter space of itself, which is simply $S^2$).

But the more intrinsic concept than berry phase is berry curvature — using Stoke’s theorem we can always transform the line integral of berry phase (it is really $A_\mu(\mathbf{R})=i \langle \psi(\mathbf{R})|\nabla_{\mathbf{R}_\mu}|\psi(\mathbf{R})\rangle$ that we are integrating over the closed path!) to a surface integral of  $F_{\mu\nu} = \partial_{[\mu}A_{\nu]}$ (in a more mathematically correct sense, a two-form integral on the parameter manifold). This object $F_{\mu\nu}$ is the berry curvature of the parameter space. The fundamental idea is that this berry curvature is intrinsic to the parameter space — it is the geometry of the parameter space.  (Note: strictly speaking, we cannot say a BZ is a parameter space — what we should say is that a dispersion band on BZ is a parameter space. This is because a BZ can have many bands on it, and each band has its own geometry and can define its own berry curvature!)

Now that we have defined berry curvature, what can we do with it? Note that this parameter space of $\mathbf{R}$, together with the berry curvature $F_{\mu\nu}$ on it, is a space with a gauge field. So any way we study a gauge theory — we will use it to study the parameter space and berry curvature!

For derivation of TKNN invariant (quantization of $\sigma_{xy}$ in QHE) and AHE (where $\sigma_{xy}$ is not quantized), and other examples, see the review paper.