A similar-in-spirit angle shows up in the problem of "parallel transport" (jargony wiki) along curved surfaces. Imagine taking a walk while holding an arrow, initially pointed east, say. You walk, always keeping the arrow pointed in the

*local*direction of east, in the closed path shown at right. On a flat surface, when you get back to your starting point, the arrow is pointing in the same direction it did initially. On a curved (say spherical) surface, though, something different has happened. As shown, when you get back to your starting point, the arrow has rotated from its initial position, despite the fact that you always kept it pointed in the

*local*east direction. The angle of rotation is a geometric phase analogous to Berry phase. The issue is that the local definition of "east" varies over the surface of the sphere. In more mathematical language, the basis vectors (that point in the local cardinal directions) vary in space. If you want to keep track of how the arrow vector changes along the path, you have to account for both the changing of the numerical components of the vector along each basis direction,

*and*the change in the basis vectors themselves. This kind of thing crops up in general relativity, where it is calculated using Christoffel symbols.

So what about the actual Berry phase? To deal with this with a minimum of math, it's best to use some of the language that Feynman employed in his popular book QED. The actual math is laid out here. In Feynman's language, we can picture the quantum mechanical phase associated with some quantum state as the hand of a stopwatch, winding around. For a state \(| \psi\rangle \) (an energy eigenstate, one of the "energy levels" of our system) with energy \(E\), we learn in quantum mechanics that the phase accumulates at a rate of \(E/\hbar\), so that the phase angle after some time \(t\) is given by \(\Delta \phi = Et/\hbar\). Now suppose we were able to mess about with our system, so that energy levels varied as a function of some tuning parameter \(\lambda\). For example, maybe we can dial around an externally applied electric field by applying a voltage to some capacitor plates. If we do this slowly (adiabatically), then the system always stays in its instantaneous version of that state with instantaneous energy \(E(\lambda)\). So, in the Feynman watch picture, sometimes the stopwatch is winding fast, sometimes it's winding slow, depending on the instantaneous value of \(E(\lambda)\). You might think that the phase that would be racked up would just be found by adding up the little contributions, \(\Delta \phi = \int (E(\lambda(t))/\hbar) dt\).

However, this misses something! In the parallel transport problem above, to get the right total answer about how the vector rotates globally we have to keep track of how the basis vectors vary along the path. Here, it turns out that we have to keep track of how

*the state itself*, \(| \psi \rangle\), varies locally with \(\lambda\). To stretch the stopwatch analogy, imagine that the hand of the stopwatch can

*also*gain or lose time along the way because the positioning of the numbers on the watch face (determined by \(| \psi \rangle \) ) is actually also varying along the path.

[Mathematically, that

*second*contribution to the phase adds up to be \( \int \langle \psi(\lambda)| \partial_{\lambda}| \psi(\lambda) \rangle d \lambda\). Generally \(\lambda\) could be a vectorial thing with multiple components, so that \(\partial_{\lambda}\) would be a gradient operator with respect to \(\lambda\), and the integral would be a line integral along some trajectory of \(\lambda\). It turns out that if you want to, you can define the integrand to be an effective vector potential called the Berry connection. The curl of that vector potential is some effective magnetic field, called the Berry curvature. Then the line integral above, if it's around some closed path in \(\lambda\), is equal to the

*flux*of that effective magnetic field through the closed path, and the accumulated Berry phase around that closed path is then analogous to the Aharonov-Bohm phase.]

Why is any of this of interest in condensed matter?

Well, one approach to worrying about the electronic properties of conducting (crystalline) materials is to think about starting off some electronic wavepacket, initially centered around some particular Bloch state at an initial (crystal) momentum \(\mathbf{p} = \hbar \mathbf{k}\). Then we let that wavepacket propagate around, following the rules of "semiclassical dynamics" - the idea that there is some Fermi velocity \(\partial E(\mathbf{k})/\partial \mathbf{k}\) (related to how the wavepacket racks up phase as it propagates in space), and we basically write down \(\mathbf{F} = m\mathbf{a}\) using electric and magnetic fields. Here, there is the usual phase that adds up from the wavepacket propagating in space (the Fermi velocity piece), but there can be an additional Berry phase which here comes from how the Bloch states actually vary throughout \(\mathbf{k}\)-space. That can be written in terms of an "anomalous velocity" (anomalous because it's not from the usual Fermi velocity picture), and can lead to things like the anomalous Hall effect and a bunch of other measurable consequences, including topological fun.