Peculiar symmetry-protected electronic dispersions in two-dimensional materials

Groups hosting new dispersions are listed in table 1. Besides intrinsic layer group notation (the first part), the space group of the system obtained by periodic repetition of the layer along axis perpendicular to it (column plane) according to Bilbao Crystallographic Server is also given (second part), where the directions x, y and z are along axes of orthorombic/tetragonal 3D primitive unit cell. On the other hand, in POLSym approach we used convention that layers are in xy-plane. Orthogonal lattice vectors a₁ and a₂ span primitive rectangular/square 2D unit cell, while reciprocal lattice vectors k₁ and k₂ satisfy a_j⋅k_l = 2πδ_j,l and q₁, q₂ are projections of q along k₁ and k₂. Relevant BZs are in figure 2.

Table 1. Groups providing dispersions (3.1). Notations for layer (columns 1 and 2) and space groups (columns 4, 5 and 6) are according to [37, 38] respectively. IR notation in the eighth column is as in Bilbao Crystallographic Server [33]. Effective Hamiltonian is indicated in the last column by the nonzero parameters (and their interrelations) of (2.5). For the last four groups a = c while ${\overline{M}}_{6}$ and ${\overline{M}}_{7}$ are conjugated pair of IRs.

Layer double group			Corresponding space double group
Group		IR	Group			Plane	IR	Dispersion	Nonzero ${v}_{pq}^{i}$
21	p21212	${\overline{S}}_{5}$	18	P21212	${D}_{2}^{3}$	z = 0	${\overline{S}}_{5}$	(3.1a)	${v}_{13}^{1}$, ${v}_{23}^{1}$, ${v}_{33}^{1}$, ${v}_{11}^{2}$, ${v}_{21}^{2}$, ${v}_{31}^{2}$
25	pba2	${\overline{S}}_{5}$	32	Pba2	${C}_{2v}^{8}$	z = 0	${\overline{S}}_{5}$	(3.1a)	${v}_{13}^{1}$, ${v}_{23}^{1}$, ${v}_{33}^{1}$, ${v}_{11}^{2}$, ${v}_{21}^{2}$, ${v}_{31}^{2}$
28	pm21b	${\bar{Y} }_{5},{\overline{S}}_{5}$	26	Pmc21	${C}_{2v}^{2}$	y = 0	${\overline{Z}}_{5}$, Ū5	(3.1a)	${v}_{11}^{1}$, ${v}_{21}^{1}$, ${v}_{31}^{1}$, ${v}_{10}^{2}$, ${v}_{20}^{2}$, ${v}_{30}^{2}$
29	pb21m	${\bar{Y} }_{5},{\overline{S}}_{5}$	26	Pmc21	${C}_{2v}^{2}$	x = 0	${\overline{Z}}_{5}$, ${\overline{T}}_{5}$	(3.1b)	${v}_{02}^{1}$, ${v}_{10}^{2}$, ${v}_{20}^{2}$, ${v}_{30}^{2}$
30	pb2b	${\bar{Y} }_{5},{\overline{S}}_{5}$	27	Pcc2	${C}_{2v}^{3}$	x = 0	${\overline{Z}}_{5}$, ${\overline{T}}_{5}$	(3.1a)	${v}_{11}^{1}$, ${v}_{21}^{1}$, ${v}_{31}^{1}$, ${v}_{10}^{2}$, ${v}_{20}^{2}$, ${v}_{30}^{2}$
32	pm21n	5	31	Pmn21	${C}_{2v}^{7}$	y = 0	${\overline{Z}}_{5}$	(3.1a)	${v}_{13}^{1}$, ${v}_{23}^{1}$, ${v}_{33}^{1}$, ${v}_{10}^{2}$, ${v}_{20}^{2}$, ${v}_{30}^{2}$
33	pb21a	5	29	Pca21	${C}_{2v}^{5}$	y = 0	${\overline{Z}}_{5}$	(3.1b)	${v}_{33}^{1}$, ${v}_{11}^{2}$, ${v}_{21}^{2}$, ${v}_{30}^{2}$
34	pb2n	5	30	Pnc2	${C}_{2v}^{6}$	x = 0	${\overline{Z}}_{5}$	(3.1a)	${v}_{13}^{1}$, ${v}_{23}^{1}$, ${v}_{33}^{1}$, ${v}_{10}^{2}$, ${v}_{20}^{2}$, ${v}_{30}^{2}$
54	p4212	(${\overline{M}}_{6},{\overline{M}}_{7}$)	90	P4212	${D}_{4}^{2}$	z = 0	(${\overline{M}}_{6},{\overline{M}}_{7}$)	(3.1a)	$\left\{\begin{aligned}\hfill {v}_{02}^{1}& ={v}_{02}^{2}={v}_{31}^{1}=-{v}_{31}^{2}\hfill \\ \hfill {v}_{10}^{1}& ={v}_{10}^{2}={v}_{23}^{1}=-{v}_{23}^{2}\hfill \\ \hfill {v}_{13}^{1}& =-{v}_{13}^{2}=-{v}_{20}^{1}=-{v}_{20}^{2}\hfill \end{aligned}\right\}$
56	p4bm	(${\overline{M}}_{6},{\overline{M}}_{7}$)	100	P4bm	${C}_{4v}^{2}$	z = 0	(${\overline{M}}_{6},{\overline{M}}_{7}$)	(3.1a)
58	$p\overline{4}{2}_{1}m$	(${\overline{M}}_{6},{\overline{M}}_{7}$)	113	$P\overline{4}{2}_{1}m$	${D}_{2d}^{3}$	z = 0	(${\overline{M}}_{6},{\overline{M}}_{7}$)	(3.1a)
60	$p\overline{4}b2$	(${\overline{M}}_{6},{\overline{M}}_{7}$)	117	$P\overline{4}b2$	${D}_{2d}^{7}$	z = 0	(${\overline{M}}_{6},{\overline{M}}_{7}$)	(3.1a)

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Effective Hamiltonians allowed by symmetry group in the special points of Brillouin's zone are presented in the last column of the table 1: the nonzero real coefficients ${v}_{pq}^{i}$ in the expansion (2.5) are specified, together with the constraints among them. The listed forms correspond to the group settings (lattice vectors and coordinate origin) and double valued irreducible co-representations obtained by POLSym code. In fact, this enabled flexibility in the choice of generators (coordinate system and translational periods), which finally results in the form of irreducible co-representations. These are chosen such to get the same form of the effective Hamiltonian whenever it is possible (for different groups). Equivalent (but different) settings (and co-representations) produce different (still equivalent with respect to dispersions) Hamiltonian forms. Clearly, the exact values of the nonzero coefficients ${v}_{pq}^{i}$ (listed in the last column of the table 1) are material dependent. The groups' generators and their representative matrices in the allowed co-representations associated to the specified high-symmetry points are in the table 2. It should be remarked that in all the considered cases this point is fixed by the whole gray group, i.e. the little group is the gray (double) group, and the allowed co-representations of the little group are simultaneously the irreducible co-representations of the gray group. The matrices of the relevant co-representations are four-dimensional. In all the cases time-reversal corresponds to the matrix $\hat{T}$; all other generators are represented by the block-diagonal matrices $\hat{D}=\mathrm{diag}\left(\hat{R},{\hat{R}}^{{\ast}}\right)$, with mutually conjugated 2 × 2 blocks. Therefore, only this block, $\hat{R}$, is given in the table 2.

Table 2. Allowed irreducible co-representations: for each group and corresponding HSP, the generators are listed, and the block-diagonal part $\hat{R}$ of double valued co-representation $\hat{D}$ representing these generators (in the same order). Here, ${C}_{n\hat{n}}$ is rotation for 2π/n around axis $\hat{n}$ (which is $\hat{x}$, , , or $\hat{c}=\frac{1}{\sqrt{2}}\left(\hat{x}+\hat{y}\right)$), ${m}_{\hat{n}}$ is vertical mirror plane which contains $\hat{n}$ axis, m_h is horizontal mirror plane, and S_n = C_nm_h.

Group	HSP	Generators		$\hat{R}$
21	S	$\left({C}_{2\hat{x}}\vert \frac{1}{2}0\right)$	$\left({C}_{2\hat{y}}\vert 0\frac{1}{2}\right)$		${\hat{\sigma }}_{3}$	${\hat{\sigma }}_{1}$
25	S	$\left({m}_{\hat{x}}\vert \frac{1}{2}0\right)$	$\left({m}_{\hat{y}}\vert 0\frac{1}{2}\right)$		${\hat{\sigma }}_{3}$	${\hat{\sigma }}_{1}$
28	Y	(I|10)	$\left({C}_{2\hat{y}}\vert 0\frac{1}{2}\right)$	m	${\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
28	S	(I|10)	$\left({C}_{2\hat{y}}\vert 0\frac{1}{2}\right)$	m	$-{\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
29	Y	(I|10)	$\left({C}_{2\hat{y}}\vert 0\frac{1}{2}\right)$	mh	${\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
29	S	(I|10)	$\left({C}_{2\hat{y}}\vert 0\frac{1}{2}\right)$	mh	$-{\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
30	Y	(I|10)	$\left({m}_{\hat{y}}\vert 0\frac{1}{2}\right)$	C2	${\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
30	S	(I|10)	$\left({m}_{\hat{y}}\vert 0\frac{1}{2}\right)$	C2	$-{\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
32	Y	(I|10)	$\left({m}_{\text{h}}\vert \frac{1}{2}\frac{1}{2}\right)$	m	${\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
33	Y	$\left({m}_{\text{h}}\vert \frac{1}{2}0\right)$	$\left({C}_{2\hat{y}}\vert 0\frac{1}{2}\right)$		$\mathrm{i}{\hat{\sigma }}_{3}$	${\hat{\sigma }}_{1}$
34	Y	(I|10)	$\left({m}_{\text{h}}\vert \frac{1}{2}\frac{1}{2}\right)$	C2	${\hat{\sigma }}_{0}$	$-{\hat{\sigma }}_{3}$	$-\mathrm{i}{\hat{\sigma }}_{2}$
54	M	(I|10)	$\left({C}_{2\hat{c}}\vert \frac{1}{2}\frac{1}{2}\right)$	C4	$-{\hat{\sigma }}_{0}$	$-\mathrm{i}{\hat{\sigma }}_{2}$	${\mathrm{e}}^{-\mathrm{i}\frac{3\pi }{4}}\enspace \mathrm{diag}\left(1,\mathrm{i}\right)$
56	M	(I|10)	$\left({m}_{\hat{c}}\vert \frac{1}{2}\frac{1}{2}\right)$	C4	$-{\hat{\sigma }}_{0}$	$-\mathrm{i}{\hat{\sigma }}_{2}$	${\mathrm{e}}^{-\mathrm{i}\frac{3\pi }{4}}\enspace \mathrm{diag}\left(1,\mathrm{i}\right)$
58	M	(I|10)	$\left({m}_{\hat{c}}\vert \frac{1}{2}\frac{1}{2}\right)$	S4	$-{\hat{\sigma }}_{0}$	$-\mathrm{i}{\hat{\sigma }}_{2}$	${\mathrm{e}}^{-\mathrm{i}\frac{3\pi }{4}}\enspace \mathrm{diag}\left(1,\mathrm{i}\right)$
60	M	(I|10)	$\left({C}_{2\hat{c}}\vert \frac{1}{2}\frac{1}{2}\right)$	S4	$-{\hat{\sigma }}_{0}$	$-\mathrm{i}{\hat{\sigma }}_{2}$	${\mathrm{e}}^{-\mathrm{i}\frac{3\pi }{4}}\enspace \mathrm{diag}\left(1,\mathrm{i}\right)$

The described technique leads to two new types of dispersions (figure 1; crossings are taken at E = 0). The first one is PF, with four bands (obtained for u, v = ±1):

$$\begin{equation}{E}_{v,u}\left(\boldsymbol{q}\right)=v\sqrt{a{q}_{1}^{2}+c{q}_{2}^{2}+u\enspace b\left\vert {q}_{1}{q}_{2}\right\vert }.\end{equation} \tag{ 3.1a }$$

The expression under the square root is non-negative since a, b and c are positive quantities (functions of ${v}_{pq}^{i}$) such that b² − 4ac < 0. For quadratic layer groups (54, 56, 58, 60) c = a, and above dispersion degenerates to the isotropic one ${E}_{v,u}=v\sqrt{a{\boldsymbol{q}}^{2}+u\enspace b\left\vert {q}_{1}{q}_{2}\right\vert }$. Two groups, 29 and 33, enforce b² − 4ac = 0, hosting thus FT dispersions (with bands counted by u, v = ±1):

$$\begin{equation}{E}_{v,u}\left(\boldsymbol{q}\right)=v\left\vert \right.f\left\vert {q}_{1}\right\vert +u\enspace g\left\vert {q}_{2}\right\vert \left\vert \right.,\end{equation} \tag{ 3.1b }$$

with f, g positive quantities, also functions of ${v}_{pq}^{i}$. Note that on the other side, the limit b → 0⁺ gives Dirac dispersion.

Dispersions (3.1), differing from the well-known Dirac, semi-Dirac or quadratic, impose specific physical properties. In this context, one must take into account the range of validity of these forms, describing the realistic band structures only in the vicinity of high-symmetry point. In particular, corresponding density of states (DOS) near E = 0 are:

$$\begin{equation}{\rho }_{\text{PF}}^{\text{SOC}}=\frac{2\vert E\vert }{\pi \sqrt{4ac-{b}^{2}}},\end{equation} \tag{ 3.2a }$$

$$\begin{equation}{\rho }_{\text{FT}}^{\text{SOC}}\approx \frac{L}{4{\pi }^{2}\sqrt{{f}^{2}+{g}^{2}}}.\end{equation} \tag{ 3.2b }$$

Unlike to PF, but similarly to 3D nodal semimetals [39], exact calculation of DOS of FT is prevented due to the non-circular iso-energetic lines (figure 1). Thus, the last expression corresponds to realistic situations where the horizontal parts of band crossing lines are of the length L (this is an effective range of approximation). In non-SOC case calculation of DOS gives doubled results (3.2), since each energy is spin degenerate, which is then decoupled from the orbital one. Non zero DOS of FT near E = 0 is in contrast to DOS of Dirac or PF dispersions being proportional to $\left\vert E\right\vert$, as well as to semi-Dirac which is proportional to $\sqrt{\left\vert E\right\vert }$. This affects many properties, to mention only charge and spin transport. Further, it can be shown that the electron effective mass, obtained from band curvatures, for all dispersions (3.1) vanishes. Let us emphasize that the higher order terms, neglected in derivation cannot change the obtained band topology (figure 1), though may distort bands slightly.

Despite the obtained dispersions are essential, i.e. resistant to symmetry preserving perturbation, an interesting additional insight is gained by considering symmetry breaking. Herein, taking into account group–subgroup relations, we discuss the possibilities of robustness or switching between various dispersions at the same BZ-point by lowering the symmetry, e.g. due to strain. It is expected that decreasing the number of symmetry elements leads to relaxing the constraints imposed on Hamiltonians, and consequently increasing (or preserving) the number of independent parameters. In this context, taking into account the number of non-zero parameters ${v}_{pq}^{i}$ of (2.5) given in table 1, it is meaningful to consider the transitions from FT to anisotropic PF, as well as from isotropic PF to FT, when the symmetry is lowered. Precisely, the allowed four-band model Hamiltonian diagonalizing in PF dispersion have six real independent parameters, which are reduced to three for quadratic groups; similarly, there are 4 real independent Hamiltonian parameters for FT. Before proceeding, let us take a brief look into the robustness of FT and PF.

Regarding groups 29 and 33 supporting FT dispersion, symmetry reduction in which either nonsymmorphic glide plane or screw axis (but not both) is retained causes that FT at the Y point splits into two non-degenerate conical dispersions. Opposite out-of-plane shifts of the adjacent nuclei positioned in the mirror plane, transforms mirror into a glide plane, while doubling the lattice constant; this in turn halves primitive vector k₁ of the reciprocal lattice. Group 29 reduces to 33 and the S point in 29 becomes Y point in 33. Consequently, FT in Y and S points in 29 are robust against lowering the symmetry to group 33. Similarly, concerning the PF, any homogenous stretching along a₁ or a₂ axis deforms square primitive cell to rectangular, reducing the symmetries of layer groups 54 and 58 (56 and 60) to the group 21 (25) and causes PF to change from isotropic to anisotropic form, which implies direction-dependent electronic and related properties.

Since PF is a generalized form of FT, one could expect that the parameters of these dispersions can be interrelated by tuning. However, continuous transformation from FT to PF at the same point of the BZ is not possible, since neither of groups supporting FT is a subgroup of any of groups allowing PF, nor vice-versa. The expression (3.2a) for DOS of PF shows that the changing parameters such that PF approaches to FT results in a singularity at zero energy. In the other words, if opposite would hold, arbitrarily small displacements of nuclei, being sufficient to lower the symmetry, would cause a jump of (graphene-like) negligible DOS of PF to a finite and constant DOS of FT, which we found unlikely. At the same time, such obstruction from DOS does not forbid the transition between Dirac (double degenerate cones with four-fold degenerate point) and PF, nor it forbids splitting of FT and PF into two non-degenerate conical dispersions (with double degenerate point).

Following the above arguments, it is expected that transition from Dirac cone to PF may be realized by lowering the symmetry, since Dirac dispersions has less independent parameters than PF. According to [5] Dirac semimetals in time-reversal invariant two-dimensional systems with strong SOC are possible in nonsymmorphic groups with inversion symmetry. E.g. let us consider the layer group 46 (pmmn), hosting Dirac cones at X, Y and S HSPs (the BZ is the same as this one given on the left panel in figure 2). It is expected that the violation of the inversion symmetry leads to Weyl points or node [5]. However, listing all subgroups, it turns out that the two of the subgroups, 32 and 21, actually host PF in the points Y and S, respectively. Indeed, in [46], using spinfull tight-binding model with four sites (with s-orbitals) per unit cell, authors show that at fillings 2, 6, system invariant under double layer group 21 is semimetal, which hosts one fourfold degenerate and four Weyl points. A plethora of such cases, where groups allowing PF from the table 1 are subgroups of symmetry groups of Dirac semimetals, indicates candidates for transitions between centrosymmetric and noncentrosymmetric crystals with protected four-fold band crossing point. Moreover, the existence of such essential fourfold degenerate point simultaneously with double degenerate Weyl points in the same system, makes that the layers from our list represent possible two-dimensional materials suitable for the study of their interplay.

Despite the fabrication of freestanding layers is not always feasible, the above theoretical predictions required material realizations, or at least numerical simulations. To find realistic material with layer groups from table 1 we searched the list [41] of 3D layered materials, synthesis of which has been reported in the literature. In the table 3 we listed potential material candidates with symmetry groups allowing the predicted peculiar dispersions. These are laboratory fabricated 3D crystals with layered structures, which could be easily or potentially exfoliated into layers.

Table 3. Material candidates: layered systems with symmetry groups hosting the dispersions (3.1). Layer and corresponding space groups are listed for materials given by a formula and materials project ID. Abbreviations EE and PE stand for easily and potentially exfoliable, respectively, according to [41].

Layer group		Space group		Formula	ID	EE/PE
21	p21212	18	P21212	As2SO6	mp-27230	EE
				MgMoTeO6	mp-1210722	EE
25	pba2	32	Pba2	Au2Se2O7	mp-28095	EE
				Re2S2O13–I	mp-974650	EE
28	pm21b	26	Pmc21	TlP5	mp-27411	EE
				KO2H4F	mp-983327	PE
				NaGe3P3	mp-1104707	PE
29	pb21m	26	Pmc21	WO2Cl2	mp-32539	EE
32	pm21n	31	Pmn21	CuCOCl	mp-562090	EE
33	pb21a	29	Pca21	BiIO4	mp-1191266	PE
				KPSe6	mp-18625	EE
58	$p\overline{4}{2}_{1}m$	113	$P\overline{4}{2}_{1}m$	LiReO2F4	mp-554108	EE

It is interesting to single out our group-theoretical findings indicated that dispersions (3.1) are not preserved when SOC is neglected, except for the LDG 33, which supports FT dispersion also in that case [29]. Inclusion of SOC moves FT from BZ corners to the Y-point. The material BiIO₄ belongs to corresponding space group 29 and has layers parallel to the y = 0 plane. Consequently, it should exfoliate to layer group 33 so we choose it for further DFT investigations, as an example of achievements of our theory. Since IRs from table 1 are the only extra IRs in these BZ points, the dispersions (3.1) are unavoidable for crystals with symmetry of these groups. On the other hand, the position of Fermi level cannot be determined solely by symmetry arguments, nor it can be guaranteed that no other bands cross or touch the Fermi level.

We determined crystal (figure 3) and band structure (figure 4) of BiIO₄ mono-layer configuration using DFT calculations: full relaxation and bands calculations were performed by QUANTUM ESPRESSO software package [42], full relativistic PAW pseudopotentials [43, 44], with the Perdew–Burke–Ernzerhof exchange–correlation functional [45]. The energy cutoff for electron wavefunction and charge density of 47 Ry and 476 Ry were chosen, respectively. The band structures were found in 500 k-points on selected path, and 2500 k-points for 2D band structure plots in the vicinity of HSPs.

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Crystal structure of mono-layer is shown in figure 3. It belongs to rectangular lattice of the group 33, with nearly equal a₁ = 0.566 nm and a₂ = 0.575 nm. Band structure of BiIO₄ mono-layer with and without SOC is shown in figure 4. It turns out that the system is insulating in undoped and ungated regime. The closest to Fermi level FT state is at −0.9 eV. When SOC is neglected energy at the point S is eightfold degenerate (including spin), which gives electron filling of 8n that is necessary for insulating systems [40]. With inclusion of SOC the eightfold spinfull degeneracy at S is lifted, but sets of eight non-degenerate bands each, form cat's cradle structure along ΓX line, as predicted in reference [46]. This gives again electron filling of 8n [46, 47]. Our electron filling of 184, derived from DFT calculations, is indeed divisible by 8. Electron filling for DLG 33 prevents FT to be the only dispersion at the Fermi level, while for remaining groups in table 1 the filling condition necessary for Fermi surface consisting of isolated points is ν = 4n + 2.

Behavior of FT states with inclusion of SOC is shown in figure 5. In non-SOC case, two pairs of Dirac lines meet at the point S and form the FT states. SOC splits eightfold degenerate band at S into four double degenerate ones. Near point Y, SOC splits fourfold spinfull degenerate Dirac line into one FT state. Since SOC strength is proportional to the fourth power of the atomic number [48], heavy elements in the material induced observable splitting.

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