Essentially degenerate hidden nodal lines in two-dimensional magnetic layer groups

According to the theory of group representations, the types of band degeneracy can be divided into accidental degeneracy and essential degeneracy. The essentially degenerate nodal lines (NLs) are typically resided on the high-symmetry lines of the Brillouin zone. Here, we propose a type of NL in two dimension that is essentially degenerate but is hidden within the high-symmetry planes, making it less observable, dubbed a hidden-essential nodal line (HENL). The existence of HENL is guaranteed as long as the system hosts a horizontal glide-mirror symmetry, hence such NLs can be widely found in both non-magnetic and magnetic systems. We perform an exhaustive search over all 528 magnetic layer groups (MLGs) for HENL that can be enforced by glide-mirror symmetry with both spinless and spinfull systems. We find that 122 candidate MLGs host spinless HENL, while 63 candidate MLGs demonstrate spinful HENL. In addition, we reveal that horizontal mirror and time-reversal symmetry in type-II and type-IV MLGs with spin-orbital coupling can enforce HENL formed. The 15 corresponding candidate MLGs have also been presented. Furthemore, we derive a few typical lattice models to characterize the existence for the HENL. For specific electronic fillings in real materials, namely 4$N$+2 in spinless systems (and 2$N$+1 in spinful systems), the presence of the HENLs in candidate MLGs is required regardless of the details of the systems. Using \emph{ab-initio} calculations, we further identify possible material candidates that realize spinless and spinful HENL. Moreover, spinful HENLs exhibit a novel persistent spin texture wih the characteristic of momentum-independent spin configuration. Our findings uncover a new type of topological semimetal state and offer an ideal platform to study the related physics of HENLs.