  We show that mutual statistics between quantum particles can be tuned to
generate emergent novel few particle quantum mechanics for the boundary modes
of symmetry-protected topological phases of matter. As a concrete setting, we
study a system of pseudofermions, defined as quantum particles with tunable
algebra, which lie on two distinct Su-Schrieffer-Heeger (SSH) chains. We find
that as the mutual statistics of the particles are tuned -- the boundary modes
present in the two chains gets non-trivially entangled showing a sudden jump in
their mutual entanglement entropy. We further show that, such tuning of
statistics engenders a first-order transition between two topologically
non-trivial phases which differ in the behavior of inter-chain entanglement.
Using a combination of analytical and numerical techniques and effective
modeling, we uncover the rich physics that this system hosts. The results are
of particular relevance in context of the study of the effective low energy
quantum mechanics of topological edge modes in one hand and their recent
realization in ultracold atoms on the other. This then provides for controlled
manipulation of such low energy modes.
