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

TL;DR

This work identifies a new two-dimensional topological semimetal state, the hidden-essential nodal line (HENL), which is essentially degenerate yet resides on high-symmetry planes and is enforced by horizontal glide-mirror symmetry. Through symmetry analysis, lattice-model construction, and extensive screening of 528 magnetic layer groups, the authors show that HENLs occur in hundreds of groups for both spinless ($4N+2$ filling) and spinful ($2N+1$ filling) electrons, with additional enforcement possible from ordinary mirrors and time-reversal in certain type-II/IV groups. They provide minimal lattice models displaying HENLs, and identify concrete material candidates—MoPO$_5$ (spinless) and Sb$_{5}$O$_7$ (spinful)—where HENLs form and can be observed, the latter exhibiting a persistent spin texture (PST) that acts as a spin-domain boundary across HENLs. The discovery of HENLs expands the landscape of topological semimetals, offering new routes for observing and manipulating topological band crossings in 2D systems and motivating extensions to three-dimensional and engineered platforms.

Abstract

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.

Figures (5)

• Figure 1: Schematic showing the electronic band of nodal line. The red solid line denotes the nodal lines. (a) Accidental nodal line on a high-symmetry plane. (b) The essential nodal line resides on a high-symmetry line. (c) The hidden-essential nodal lines are distributed on a high-symmetry plane.
• Figure 2: Mechanism of emergence of the HENL (a)-(c) Schematic diagrams of the distribution of HENLs enforced by the symmetry of three glide mirrors, where $G_{1(2)}$ denotes the reciprocal lattice vector. (d)-(f) Schematic figure of the energy bands corresponding to (a)-(c) along the curve $C$. The signs $\pm$ of glide eigenvalues are represented by solid lines of different colors (yellow and green). Glide eigenvalues are exchanged from $\boldsymbol{k}$ to $\boldsymbol{k^{'}}$ point along the path $C$, and a band crossing appears at $\boldsymbol{k}_{c}$. (g) Schematic figure depicting the spinless HENL enforced by $\left\{M_{z}|\frac{1}{2}00\right\}$ with time-reversal symmetry. (h) The spinful HENL guaranteed by ordinary mirror and time-reversal symmetry $\mathcal{T}$ [or $\mathcal{T}\tau_{1/2}$], and its typical energy band along curve $C$ is illustrated in (i).
• Figure 3: (a), (d) and (g) show the band structures along high-symmetry lines for lattice models (\ref{['tb1']}), (\ref{['tb2']}), and (\ref{['tb3']}). (b), (e) and (h) display the BZs as well as HENL distributions for the same models. (c), (f) and (i) present the HENL distributions in the unit cells of the reciprocal lattices.
• Figure 4: (a) Stucture and (c) BZ of monolayer MoPO$_5$. (b) The band structure of MoPO$_5$ without SOC. (d) 3D band plotting of HENL.
• Figure 5: (a) Top and side view of monolayer Sb$_{5}$O$_7$ and (c) corresponding to BZ with the distribution of HENLs. (b) Band structure of Sb$_{5}$O$_7$ with SOC. (d) Spin-resolved band structures along high-symmetry lines. The spin configurations of $S_{z}$ for (e) lower-band and (f) upper-band within the BZ.