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It is sometimes argued that spontaneous spin polarization cannot occur in quantum wires in zero magnetic field becuase of the Lieb-Mattis theorem (Ref. 27) for ideal 1D systems. In the present case, however, we generally focus on multisubbands wires in finite magnetic field, and with spin-dependent forces present. Therefore the Lieb-Mattis does not apply here. In by-passing, we also find polarization effects at zero field. Strictly speeking, such polarized solutions should not be accepted from a mathematical point of view if the wire is in the ideal 1D limit. However, a real device is never strictly 1D. Guided by experimental evidence we therefore take a pragmatic view by accepting also these solutions and give them physical significance. In fact, Fig. 1 shows a smooth, regular behavior of the observed data as the magnetic field is turned on. From a more formal point of view, we suggest that the polarized solutions at zero field indicate that local spin order may extend over a large (but not infinite) distance. In fact, the correlation length may exceed the dimensions of a real device, and for this reason the question about the Lieb-Mattis theorem appears less interesting. Similar aspects on the relation between symmetry and broken and symmetry-adapted solutions for many-electron quantum dots are discussed in Refs. 28 and 29.
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