## A simple discussion of the Berry phase

N. P. Ong, Department of Physics, Princeton UniversityWhat is the Berry phase?

1. Geometric angle and parallel transport

Suppose we travel on a closed
path *C* on a sphere (Earth) while holding a vector **V** parallel to the
surface, i.e. in the local tangent plane (Fig. 1a). At each point **R**,
we adopt the strict precaution of ensuring that **V** does not twist around
the local vertical axis (the local normal vector **n**) as we move along *C*.
This is known as parallel transport of **V** around *C*. When we
return to the starting point, we find that in general **V** makes an angle a(*C*) with its initial direction. This angle, which
depends only on the particular path *C*, is called the geometric
angle. It is the classical analog of the Berry phase in quantum mechanics.

[The angle a may be found
by inspection if *C* is along a latitude as shown in Fig. 1a.
Consider the cone with apex A that touches the sphere at the latitude (defined
by its polar angle q). At each point **R** the local patch of the
cone coincides with the local tangent plane. If we lay the cone flat on a
table, **n** is everywhere parallel to the normal vector of the table **z**
(Fig. 1b). The path *C* on the sphere corresponds to a line *C*'
on the cone. Since, under parallel transport, **V** does not twist
around **n** (i.e. **z**), it always points in the same direction as we
trace *C*' on the table. In the example drawn, **V** is initially
parallel to the latitude. When we return to the starting point, **V** is at
an angle a to the latitude (on the table top it is unchanged
from its initial direction, but because we live on the sphere, it is the angle
measured from the latitude that counts). Since the arc-length of *C*
equals 2p*R*sinq, the radius of the cone *a* = *R* tan q, and a also equals the angle of the missing wedge in (b), we
find

a = 2p(1-cosq)

The geometric angle equals
the solid angle subtended by the path *C* at the center of the
sphere. As a check, if *C* is the equator (q = p/2),a = 2p.]

2. Geometric phase under adiabatic change

In quantum mechanics, an effect that is
mathematically similar to the example in Fig. 1 leads to the appearance of a
phase (the Berry phase) in the wave function of a (sub)system in which a parameter is slowly
changed. Let us consider a system comprised of two subsystems.
Subsystem 1 is ponderous and described by a 'slow' variable **R **while
subsystem 2 is agile and described by a 'fast' variable **x**. For
example, 1 could be an atomic nucleus whose position is **R** while 2 is an
orbiting electron with coordinate **x **(Fig. 2a). In a second example
(Fig. 2b), 1 could be a heavy magnet coil that can be adjusted to orient its
magnetic field **B** in any direction (specified by **R**) while 2 is an
electron whose spin is aligned with **B **by the Zeeman energy.

In an adiabatic process, the
variable **R** undergoes a slow change and then returns to its initial
value, i.e. **R** executes a closed path in R-space. This path is the
analog of path *C* in Fig. 1. Because the change is gradual, we may assume
that subsystem 2 (the electron) remains in its original eigenstate throughout
(the analog of parallel transport). After **R** completes its loop, we
expect the electronic wave function y to recover its
initial value.

*Foundations of Quantum Mechanics in the Light of New Technology*(Proceedings of ISQM-Tokyo’05), ed. Sachio Ishioka and Kazuo Fujikawa (World Scientific 2006)p. 121, cond-mat/0508236.