Molecular-orbital representation with random U(1) variables

We propose random tight-binding models that host macroscopically degenerate zero-energy modes and belong to the unitary class. Specifically, we employ the molecular-orbital representation, where a Hamiltonian is constructed by a set of nonorthogonal orbitals composed of linear combinations of atomic orbitals. By setting the coefficients appearing in molecular orbitals to be random U(1) variables, we can make the models belong to the unitary class. We find two characteristic behaviors that are distinct from the random real-valued molecular-orbital model. Firstly, a finite-energy gap opens on top of the degenerate zero-energy modes. Secondly, besides the zero-energy modes, we also argue that the band center of the finite-energy modes is critical, which is inherited from the dual counterpart, namely, the random-phase model on a bipartite lattice. Furthermore, as a by-product of this model-construction scheme, we also construct the random tight-binding model on a composite lattice, where we also find a realization of critical states.