  The existence of $1/2$ modes in the entanglement spectrum (ES) has been shown
to be a powerful quantum-informative signature of boundary states of gapped
topological phases of matter, e.g., topological insulators and topological
superconductors, where the finite bulk gap allows us to establish a
crystal-clear correspondence between $1/2$ modes and boundary states. Here we
investigate the recently proposed higher-order Weyl semimetals (HOWSM), where
bulk supports gapless higher-order Weyl nodes and boundary supports hinge arcs
and Fermi arcs. We find that the aim of unambiguously identifying higher-order
boundary states ultimately drives us to make full use of eigen quantities of
the entanglement Hamiltonian: ES as well as Schmidt vectors (entanglement
wavefunctions, abbr. EWF). We demonstrate that, while both hinge arcs and Fermi
arcs contribute to $1/2$ modes, the EWFs corresponding to hinge arcs and Fermi
arcs are respectively localized on the virtual hinges and surfaces of the
partition. Besides, by means of various symmetry-breaking partitions, we can
identify the minimal crystalline symmetries that protect boundary states.
Therefore, for gapless topological phases such as HOWSMs, we can combine ES and
EWF to universally identify boundary states and potential symmetry requirement.
While HOWSMs are prototypical examples of gapless phases, our work sheds light
on general theory of entanglement signature in gapless topological phases of
matter.
