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By Avron J.E.

The adiabatic quantum shipping in multiply hooked up platforms is tested. The structures thought of have numerous holes, often 3 or extra, threaded by means of self reliant flux tubes, the delivery houses of that are defined via 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 area may be pointed out with the multitorus, the adiabatic conductances have topological value, relating to the 1st Chern personality. particularly, they've got quantized averages. The authors describe a variety of sessions of quantum Hamiltonians that describe multiply hooked up structures and examine their simple homes. They pay attention to versions that decrease to the learn of finite-dimensional matrices. particularly, the relief 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 houses and their dependence at the collection of flux tubes. They introduce a style of topological type 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. a lot of those networks prove to have nontrivial topological shipping houses for either the free-electron and the tight-binding versions. The authors finish with a few open difficulties and questions.

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G. Drinfeld (1985) Sov. Math. Dokl. 32, 254–258. V. G. Drinfeld (1988) Sov. Math. Dokl. 36, 212–216. V. G. Drinfeld (1987) Quantum Groups, in Proceedings of the International Congress of Mathematicians, Berkeley, 1986, American Mathematical Society, 798–820. L. A. Takhtajan (1989) in Nankai Lectures on Mathematical Physics, Introduction to Quantum Group and Integrable Massive Models of Quantum Field Theory, eds. -L. -H. 69–197, Singapore: World Scientific, 1990. 2 follows the paper N. Yu. Reshetikhin, P.

69–197, Singapore: World Scientific, 1990. 2 follows the paper N. Yu. Reshetikhin, P. P. Kulish, E. K. Sklyanin (1981) Lett. Math. Phys. 5, 393–403. 3 is borrowed from B. Sutherland (1970) J. Math. Phys. 11, 3183–3186. 10), was proposed in R. J. Baxter (1972) Ann. Phys. 70, 323–337. M. L¨ uscher (1976) Nucl. Phys. B117, 475–492. The definition of t(k)(u) is taken from (Kulish, Sklyanin, 1982a), see above. For the master symmetries for quantum integrable chains see M. G. Tetelman (1982) Sov. Phys.

Baxter (1972) Ann. Phys. 70, 323–337. M. L¨ uscher (1976) Nucl. Phys. B117, 475–492. The definition of t(k)(u) is taken from (Kulish, Sklyanin, 1982a), see above. For the master symmetries for quantum integrable chains see M. G. Tetelman (1982) Sov. Phys. JETP 55(2), 306–310. E. Barouch, B. Fuchssteiner (1985) Stud. Appl. Math. 73, 221–237. H. Araki (1990) Commun. Math. Phys. 132, 155–176. The boost operator B was shown recently to have close relation to Baxter’s corner transfer matrices H. Itoyama, H.

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