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A simple discussion of the Berry phase
N. P. Ong, Department of Physics, Princeton University

What 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 2pRsinq, 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.  Berry's theorem states, that y actually acquires a geometric phase g (the Berry phase) which is the analog of a.  The Berry phase is the integral of (n|id/dt|n) with respect to time.

         

        For more details, see N. P. Ong and Wei-Li Lee, “Geometry and the Anomalous Hall Effect in Ferromagnets”, 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.