Topological phase

Similarides Between Potential Ruid Dynamics and Quantum Mechanics Electrons in the Dirac Theory The Nearly Nonrelativistic Limit The Lagrangean-Density Correction Term Topological Phase for Dirac Electrons What Have We Learned About Spinor Phases ... 

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

One can trace the continuous evolution of 0 (or of 0/2) as <() describes the circle q = constant. This will yield the topological phase (as well as intermediate, open-path phase during the circling). We illustrate this in the next two figures for the case q > 1 (encircling the ci s). 

Experimental observation of topological phases is difficult, for one reason (among others) that the dynamic-phase part (which we have subtracted off in our formalism, but is present in any real situation) in general oscillates much faster than the topological phase and tends to dominate the amplitude behavior [306-312]. Several researches have addressed this difficulty, in particular, by neutron-interferornehic methods, which also can yield the open-path phase [123], though only under restricted conditions [313]. 

Dirac electrons, 266-268 topological phase, 270-272 Lagrangean-density correction term, 269-... 

In Figure 2a several important stages in the circling are labeled with Arabic numerals. In the adjacent Figure 2b the values of 0(4>, q) are plotted as < ) increases continuously. The labeled points in the two Figures correspond to each other. (The notation is that points that represent zeros of tan 0 are marked with numbers surrounded by small circles, those that represent poles are marked by numbers placed inside squares, other points of interest that are neither zero nor poles are labeled by free numbers.) The zero value of the topological phase (0/2) arises from the fact that at the point 3 (at which <(> = it/2), 0 retraces its values, rather than goes on to decrease. 

---