One-dimensional $2^n$-root topological insulators and superconductors

Abstract

Square-root topology is a recently emerged sub-field describing a class of insulators and superconductors whose topological nature is only revealed upon squaring their Hamiltonians, i.e., the finite energy edge states of the starting square-root model inherit their topological features from the zero-energy edge states of a known topological insulator/superconductor present in the squared model. Focusing on one-dimensional models, we show how this concept can be generalized to 2 n -root topological insulators and superconductors, with n any positive integer, whose rules of construction are systematized here. Borrowing from graph theory, we introduce the concept of arborescence of 2 n -root topological insulators/superconductors which connects the Hamiltonian of the starting model for any n, through a series of squaring operations followed by constant energy shifts, to the Hamiltonian of the known topological insulator/superconductor, identified as the source of its topological features. Our work paves the way for an extension of 2 n -root topology to higher-dimensional systems.

Square-root topological phase with time-reversal and particle-hole symmetry

“…They have analyzed a onedimensional tight-binding model and have demonstrated that the system hosts edge modes induced by the topology of the squared Hamiltonian rather than the original Hamiltonian. After this proposal, analysis of square-root topological insulators in higher dimensions has been addressed [25][26][27][28][29][30][31][32] , which has elucidated ubiquity of the squareroot topological phases. For instance, a square-root counterpart of higher-order topological phases are reported by both theoretical 25 and experimental works 33,34 .…”