  Entanglement entropy is a fundamental concept with rising importance in
different fields ranging from quantum information science, black holes to
materials science. In complex materials and systems, entanglement entropy
provides insight into the collective degrees of freedom that underlie the
systems' complex behaviours. As well-known predictions, the entanglement
entropy exhibits area laws for systems with gapped excitations, whereas it
follows the Gioev-Klich-Widom scaling law in gapless fermion systems.
Furthermore, the entanglement spectrum provides salient characterizations of
topological phases and phase transitions beyond the conventional paradigms.
However, many of these fundamental predictions have not yet been confirmed in
experiments due to the difficulties in measuring entanglement entropy in
physical systems. Here, we report the experimental verification of the above
predictions by probing the nonlocal correlations in phononic systems. From the
pump-probe responses in phononic crystals, we obtain the entanglement entropy
and entanglement spectrum for phononic systems with the fermion filling analog.
With these measurements, we verify the Gioev-Klich-Widom scaling law of
entanglement entropy for various quasiparticle dispersions in one- and
two-dimensions. Moreover, we observe the salient signatures of topological
phases in the entanglement spectrum and entanglement entropy which unveil an
unprecedented probe of topological phases without relying on the bulk-boundary
correspondence. The progress here opens a frontier where entanglement entropy
serves as an important experimental tool in the study of emergent phases and
phase transitions which can be generalized to non-Hermitian and other
unconventional regimes.
