| Summary: | Short-range antiferromagnetic correlations are known to open a spin gap in
the repulsive Hubbard model on ladders with $M$ legs, when $M$ is even. We show
that the spin gap originates from the formation of correlated pairs of
electrons with opposite spin, captured by the hidden ordering of a spin-parity
operator. Since both spin gap and parity vanish in the two-dimensional limit,
we introduce the fractional generalization of spin parity and prove that it
remains finite in the thermodynamic limit. Our results are based upon
variational wave functions and Monte Carlo calculations: performing a finite
size-scaling analysis with growing $M$, we show that the doping region where
the parity is finite coincides with the range in which superconductivity is
observed in two spatial dimensions. Our observations support the idea that
superconductivity emerges out of spin gapped phases on ladders, driven by a
spin-pairing mechanism, in which the ordering is conveniently captured by the
finiteness of the fractional spin-parity operator.
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