Emergent Z₂ gauge symmetry and spin-charge separation in one dimensional physics

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We demonstrate that in the scaling limit the phenomenon of spin-charge separation as encountered in Luttinger liquids defined on the lattice can be associated with the emergence of local Ising symmetry. This Z₂ gauge field is of geometrical nature (sublattice parity) and the Luttinger liquid corresponds with the critical point where the symmetry is on the verge of becoming local. We argue that in a true one dimensional superconductor the local symmetry will be fully realized as long as the geometrical structure associated with spin-charge separation is maintained.