Higher-order topological insulators (HOTIs), that are classified as obstructed atomic insulators (OAIs) in the topological quantum chemistry (TQC) theory, attract great interest owning to the lower-dimensional topological boundary states. However, the boundary states in HOTIs reported so far are often fragile, manifested as strongly depending on crystalline symmetries and cleavage terminations in the disk or cylinder geometry. Here, using the TQC theory, we propose an intuitive strategy to establish the connection between the obstructed Wannier charge centers of OAIs and the emergence of robust corner states in two-dimensional systems. Based on first-principles calculations and real space invariant theory, we extend the concept of OAIs to phonon systems and thereby predict that the robust corner states can be realized in the phonon spectra of $M{X}_3$ ($M$ = Bi, Sb, As, Sc, Y; $X$ = I, Br, Cl) monolayers. The phonon corner modes in different shapes of nanodisks are investigated, and their robustness facilitates the detection in experiments and further applications. This work suggests a promising avenue to explore more attractive features of higher-order band topology beyond the existing paradigm.

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• Edge states
• First-principles calculations
• Phonons
• Topological materials
• Topological phases of matter
• 2-dimensional systems
• Topological insulators
• High-throughput calculations
• Wannier function methods