What Becomes of the Antiferromagnetic String When Long Range Neel Order Is Lost? Exact Results

Early string descriptions of doped antiferromagnets associate hole motion in a locally Néel-ordered background with a path-dependent antiferromagnetic string. We ask what becomes of this mechanism when long range Néel order is lost while short range antiferromagnetic correlations persist. In a singlet bond representation, the underlying basis states do not assign an oriented $A\mathrm{/}B$ spin registration to each bond. Therefore, any surviving long wavelength antiferromagnetic axis is naturally described as a director, $\mathbf{n}\sim -\mathbf{n}$. Long range Néel order can nevertheless provide a global $S^2$ lift of this director field; when that long range registration is lost, non liftable configurations of the $R{P}^2$ manifold become physically accessible, opening a topologically nontrivial ${\mathbb{Z}}_2$ sector. Its elementary defects are half vortices that globally minimize the $R{P}^2$ $\sigma$-model energy within the nontrivial ${\mathbb{Z}}_2$ sector. An $\mathit{\text{a}}\mathit{\text{posteriori}}$ energetic comparison further shows that the projective half vortex texture is softer than the corresponding vector reference texture. A pair of half vortices with opposite windings is logarithmically confined, $V(R)=\kappa \ln \left(R\mathrm{/}{r}_{\text{core}}\right)$, $\kappa =\pi {\rho }_s\mathrm{/}2$. The resulting confinement is carried not by a one dimensional string but by an extended two-dimensional spin texture. Thus, the loss of long-range Néel order does not eliminate antiferromagnetic confinement but removes the global Néel lift. This activates a distinct projective mechanism in which a local path dependent string is replaced, at long wavelengths, by a global topological texture with logarithmic confinement.