  The celebrated Dyson singularity signals the relative delocalization of
single-particle wave functions at the zero-energy symmetry point of disordered
systems with a chiral symmetry. Here we show that analogous zero modes in
interacting quantum systems can fully localize at sufficiently large disorder,
but do so less strongly than nonzero modes, as signifed by their real-space and
Fock-space localization characteristics. We demonstrate this effect in a spin-1
Ising chain, which naturally provides a chiral symmetry in an odd-dimensional
Hilbert space, thereby guaranteeing the existence of a many-body zero mode at
all disorder strengths. In the localized phase, the bipartite entanglement
entropy of the zero mode follows an area law, but is enhanced by a
system-size-independent factor of order unity when compared to the nonzero
modes. Analytically, this feature can be attributed to a specific zero-mode
hybridization pattern on neighboring spins. The zero mode also displays a
symmetry-induced even-odd and spin-orientation fragmentation of excitations,
characterized by real-space spin correlation functions, which generalizes the
sublattice polarization of topological zero modes in noninteracting systems,
and holds at any disorder strength.
