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The adiabatic quantum transport in multiply connected systems is examined. The systems considered have several holes, usually three or more, threaded by independent flux tubes, the transport properties of which are described by matrix-valued functions of the fluxes. The main theme is the differential-geometric interpretation of Kubo's formulas as curvatures. Because of this interpretation, and because flux space can be identified with the multitorus, the adiabatic conductances have topological significance, related to the first Chern character. In particular, they have quantized averages. The authors describe various classes of quantum Hamiltonians that describe multiply connected systems and investigate their basic properties. They concentrate on models that reduce to the study of finite-dimensional matrices. In particular, the reduction of the \"free-electron\" Schrödinger operator, on a network of thin wires, to a matrix problem is described in detail. The authors define \"loop currents\" and investigate their properties and their dependence on the choice of flux tubes. They introduce a method of topological classification of networks according to their transport. This leads to

Author

Avron J.E.

Published

1995

Language

EN