Higher order topological matter and fractional chiral states

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.