Correspondence between dynamic higher-order topological insulator and synthetic higher-order Dirac semimetal

With the recent discovery of the correspondence between low-dimensional dynamical systems and high-dimensional topological insulators through the pumping process, many efforts have been made to generalize these correspondences to other topological phases. Yet, there is ongoing ambiguity about the correspondence of higher-order topological semimetals. Here, we propose a correspondence between a two-dimensional (2D) dynamic higher-order system and three-dimensional (3D) higher-order Dirac semimetals (HODSMs) through synthetic space consisting of a 2D lattice and a one-dimensional parameter dimension. We explore the evolution of the 2D higher-order topological insulator in a hexagonal acoustic crystal and find all the hallmarks of 3D HODSM in it, including Dirac points, surface states, and higher-order hinge states. By measuring the local density of states and acoustic pressure fields evolved in the 2D acoustic samples, we show that the system undergoes band gap-closing points, mapping to Dirac points in 3D synthetic space. The corner states evolving in nontrivial topological phase constitute hinge states connecting synthetic Dirac points. Our research deepens the understanding of the connections between different topological phases and may inspire further exploration of other topological effects in high-dimensional systems.