Variable-order topological insulators

Abstract Topological insulators, either the first-order or the higher-order, experience generally a transition to a trivial phase or a topological one of the same order through the gap closing and reopening procedure. Here, we report a topological insulator, which switches directly between the first and higher orders, with only varying the hoppings and without breaking the symmetry. The phase transition of the first and higher orders is originated from a competition mechanism between the nearest and second-nearest neighbor interactions. This variable-order topological insulator is implemented in a two-dimensional phononic crystal, and the one-dimensional helical edge states, which signal the first-order phase, and the zero-dimensional corner states, which signal the second-order one, are demonstrated in the simulations and experiments. Our study gives insight to the topological states of different orders.