We develop a chiral anomalous fermion hamiltonian proposal to study the
higher order topological (HOT) phase with chiral symmetry $\mathcal{C}$
fractionalized like $\mathcal{C}_{x}\mathcal{C}_{y}\mathcal{C}_{z}$. First, we
solve the $\mathcal{C}$-chiral symmetry constraint for eight band models and
describe those induced by the partial $\mathcal{C}_{i}$’s. Then, we determine
the explicit expression of fractional states characterising HOT matter and
comment on the relationships amongst them and with the standard
Altland-Zirnbauer gapless modes. We also give characteristic properties of the
gapless fractional states and compute their contribution to the topological
index of the chiral model. The findings of this work are shown to be crucial
for investigating and handling high order topological phase.
