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note
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It is often argued that magnetization in QPC's cannot occur because it would violate the Lieb-Mattis theorem for strictly one-dimensional systems [E. Lieb and D. Mattis, Phys. Rev. 125, 164 (1962)]. As in our model, however, the QPC is in close contact with two-dimensional reservoirs and can hardly be viewed as a one-dimensional systems unless the connecting channel is very long.
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