Fig. 4: Entanglement spectrum and topology in 2D honeycomb phononic crystals.

a To obtain the k₂-dependent entanglement spectrum, we construct the correlation matrix of a rhombus subsystem A with the side length L = 5a, which has zigzag-type boundaries. b Schematic of the k-space in the rhombic first Brillouin zone. For each k₂, we construct the correlation matrix of the rhombus subsystem A for half-filling and obtain the entanglement spectrum. c Entanglement spectrum versus k₂ for the gapless phononic graphene. Both experimental data and the results from full-wave simulations without the dissipation effect are presented. d Measured phononic dispersion on the zigzag edge obtained from analyzing the pump-probe response around a zigzag edge boundary. The colormap gives the intensity of the detected acoustic signal after Fourier transformation for the pump-probe measurements along an edge boundary, i.e., it gives the spectral intensity along the edge boundary at various wavevectors and frequencies. e Entanglement entropy from both experiments and simulations versus k₂. Here the full-wave simulation also does not include the dissipation effect. f Entanglement spectrum versus k₂ for the gapped phase with broken inversion symmetry. Orange dashed lines in c and e denote the projection of the two Dirac points in the k₂ axis. Geometry parameters are the same as in Fig. 3.