TOPOLOGICAL QUANTUM NUMBERS AND PHASE TRANSITIONS IN MATTER

In the early years of the last century the discreteness of matter, of electric charge, and of mechanical action became firmly established, and slowly some of the more subtle implications of the interplay of these three were worked out. Dirac showed that magnetic monopoles also had to be quantized, the importance of dislocations in solids was shown, and the quantization of circulation in neutral superfluids and of magnetic flux in superconductors was predicted and demonstrated. Such topological defects can be a sign of a symmetry broken by a phase transition, or, as Onsager suggested in his first exposition of quantized circulation, can themselves drive a phase transition. I discuss circulation in superfluids, flux in superconductors and Hall conductance in inversion layers as examples of such quantum numbers. I show why there is a topological quantum number, and ask how the mathematical quantum number is related to measurable quantities. Recently there has been interest in whether the robustness of such topological defects make them suitable for quantum manipulation.

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