By Avron J.E.

The adiabatic quantum shipping in multiply hooked up platforms is tested. The platforms thought of have numerous holes, frequently 3 or extra, threaded by means of self sufficient flux tubes, the delivery houses of that are defined by way of matrix-valued capabilities of the fluxes. the most subject is the differential-geometric interpretation of Kubo's formulation as curvatures. due to this interpretation, and since flux house could be pointed out with the multitorus, the adiabatic conductances have topological importance, concerning the 1st Chern personality. specifically, they've got quantized averages. The authors describe quite a few periods of quantum Hamiltonians that describe multiply attached platforms and examine their uncomplicated houses. They be aware of types that lessen to the examine of finite-dimensional matrices. particularly, the aid of the "free-electron" Schrödinger operator, on a community of skinny wires, to a matrix challenge is defined intimately. The authors outline "loop currents" and examine their homes and their dependence at the selection of flux tubes. They introduce a style of topological category of networks in response to their delivery. This results in the research of point crossings and to the organization of "charges" with crossing issues. Networks made with 3 equilateral triangles are investigated and labeled, either numerically and analytically. lots of those networks prove to have nontrivial topological shipping homes for either the free-electron and the tight-binding types. The authors finish with a few open difficulties and questions.

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Let TT be a family of mappings of E onto itself defined by mapping every a G E to another point Tra according to the following coordinate expression: x(a) H-> x(TTa) — x(a) + r, r € (—oo, oo). 125) One can see that TT form a one-parameter transformation group of E, translating every point in E by an amount r. This group generates a curve a starting from each point a(0) by x(a(r)) =x(a(0))+r, Matsushima (1972) p. 79. r e (-00,00). 3. DIFFERENTIAL OPERATORS, VECTORS AND FIELDS 37 This curve in turn gives rise to a complete vector field X = X± where X = ^ dx » dr = 1 =* X = #.

With this introduction we can now define further operations on the functions in C°°(a) and in C°°(En). 48) where a,b £ M, is called a linear operator at a point a £ En on the set of smooth functions C°°(a). 1(1) Properties Cl A linear operator Aa gives rise to a real number Aa(f) for any / £ C'x{a). If / is a zero function then Aa(f) = 0. In contrast a linear operator A on C°°(En) gives rise to a new smooth function A(f) on En. C2 A linear operator Aa at a point a depends only on the behaviour of local functions in the neighbourhood of a.

It follows that we can express Aa(f) (1-56) ™ in coordinates x'i as where We conclude that Aa is a coordinate independent quantity in the sense that: 1. Aa maps every function / e C°°(a) to a real number Aa(f), value Aa(f) is the same in all coordinate systems. and this 2. , we have where A'Qfc to A>a are related by Eq. 59). 24 CHAPTER 1. MANIFOLDS AND DYNAMICAL SYSTEMS What has been said above applies to A. We have */>-X>'(£)-f>*(£). <""> 3= 1 x ' x R=l ' where In fact, differential operators can be introduced in a more abstract manner without explicit reference to any coordinate system.