  Topological semimetals (TSMs) in which conduction and valence bands cross at
zero-dimensional (0D) Dirac nodal points (DNPs) or 1D Dirac nodal lines (DNLs),
in 3D momentum space, have recently drawn much attention due to their exotic
electronic properties. Here we generalize the TSM state further to a
higher-symmetry and higher-dimensional pseudo Dirac nodal sphere (PDNS) state,
with the band crossings forming a 2D closed sphere at the Fermi level. The PDNS
state is characterized with a spherical backbone consisting of multiple
crossing DNLs while band degeneracy in between the DNLs is approximately
maintained by weak interactions. It exhibits some unique electronic properties
and low-energy excitations, such as collective plasmons different from DNPs and
DNLs. Based on crystalline symmetries, we theoretically demonstrate two
possible types of PDNS states, and identify all the possible band crossings
with pairs of 1D irreducible representations to form the PDNS states in 32
point groups. Importantly, we discover that strained MH3 (M= Y, Ho, Tb, Nd) and
Si3N2 are materials candidates to realize these two types of PDNS states,
respectively. As a high-symmetry-required state, the PDNS semimetal can be
regarded as the "parent phase" for other topological gapped and gapless states.
