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I mean to have q-fold degenerate ground states on a ring which could not be lifted by local perturbation.

If the answer is no, then what is the physical (or mathematical) reason against having such a state in 1D ?

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Perhaps if you have $N$ segments of the wire which are topological superconductors ($p$-wave), all of these segments have non-topological segments between them, and the boundaries between the non-topological and topological sections are well separated then you would have a topologically protected $2^N$-fold degenerate ground state. However, it won't have topological order (i.e. long-range entanglement); I'm not 100% sure why. In a recent talk Kitaev mentioned that the 1-D $p$-wave chain does not have topological order. –  NanoPhys May 2 '13 at 7:16

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