Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems

Weyl points in three spatial dimensions are characterized by a $\mathbb{Z}$-valued charge—the Chern number—which makes them stable against a wide range of perturbations. A set of Weyl points can mutually annihilate only if their net charge vanishes, a property we refer to as robustness. While nodal loops are usually not robust in this sense, it has recently been shown using homotopy arguments that in the centrosymmetric extension of the $\mathrm{AI}$ symmetry class they nevertheless develop a ${\mathbb{Z}}_2$ charge analogous to the Chern number. Nodal loops carrying a nontrivial value of this ${\mathbb{Z}}_2$ charge are robust, i.e., they can be gapped out only by a pairwise annihilation and not on their own. As this is an additional charge independent of the Berry $\pi$-phase flowing along the band degeneracy, such nodal loops are, in fact, doubly charged. In this manuscript, we generalize the homotopy discussion to the centrosymmetric extensions of all Atland-Zirnbauer classes. We develop a tailored mathematical framework dubbed the $\mathrm{AZ}+\mathcal{I}$ classification and show that in three spatial dimensions such robust and multiply charged nodes appear in four of such centrosymmetric extensions, namely, $\mathrm{AZ}+\mathcal{I}$ classes $\mathrm{CI}$ and $\mathrm{AI}$ lead to doubly charged nodal lines, while $\mathrm{D}$ and $\mathrm{BDI}$ support doubly charged nodal surfaces. We remark that no further crystalline symmetries apart from the spatial inversion are necessary for their stability. We provide a description of the corresponding topological charges, and develop simple tight-binding models of various semimetallic and superconducting phases that exhibit these nodes. We also indicate how the concept of robust and multiply charged nodes generalizes to other spatial dimensions.

Figure 1

Enclosing nodes (red) by $p$-spheres $S^p$ (blue) for $D=3$. (a) The only noncontractible $S^p$ wrapping around a nodal point is $S^2$. Any $S^1$ would be trivially shrinkable to a point as indicated, and similarly for a pair of points $S^0$. (b) A nodal line can be wrapped by both $S^1$ and $S^2$, but not by $S^0$. It follows that nodal lines may be doubly charged. (c) A spherical nodal surface can be clearly encased by $S^2$ (shown), as well as by a pair of points $S^0$ located on the opposite sides of the surface. (d) If the nodal surface takes the form of a cylinder winding around BZ (black frame) or of a torus, it can be wrapped by $S^1$ too. If the nodal cylinder is characterized by a nontrivial charge on $S^1$, it becomes robust and a Nielsen-Ninomiya type of argument [65] implies the presence of another such a cylinder. The same is not true for torical nodal surface which can be gapped out by itself. The pair of points indicated by arrows is an example of $S^0$ “enclosing” the cylinder. (e) Two kinds of nodal lines. The dashed ones wind around BZ, while the solid one does not and is hence called a nodal loop. Similar to the previous case, a nontrivial charge on $S^1$ makes only the winding loops robust.

Figure 2

Two bipartite lattices with a different representation of the inversion operator $\mathcal{I}$. The red (light) and blue (dark) discs represent atoms on the two sublattices—each unit cell contains one atom of every color. The black dots in example (b) indicate atoms that do not enter the effective TB model, but that nevertheless influence the hopping amplitudes by distorting the crystal field. The atoms of both colors are identical in all respects, such that the Hamiltonian is block-off-diagonal in the sublattice basis, leading to $\mathcal{C}={\tau }_z$. While in (a) the inversion symmetry ${\mathcal{I}}_{\left(\mathsf{a}\right)}={\tau }_x$ anticommutes with $\mathcal{C}$ ($\mathcal{I}$-odd SLS), in (b) the inversion operator ${\mathcal{I}}_{\left(\mathsf{b}\right)}={\mathbb{1}}_{\tau }$ commutes with it ($\mathcal{I}$-even SLS). In case (a), the Cartan labels of the relevant AZ and $\mathrm{AZ}+\mathcal{I}$ classes are different.

Figure 3

A schematic summary of Sec. 5. Here, $\mathcal{T}$, $\mathcal{P}$, $\mathcal{C}$ are the usual AZ symmetries, while the assigned Cartan labels correspond to the $\mathrm{AZ}+\mathcal{I}$ classes characterized by symmetries $\mathfrak{T}$, $\mathfrak{P}$, $\mathcal{C}$. The even/odd sublattice parity corresponds to $\mathcal{I}\mathcal{C}=\pm \mathcal{C}\mathcal{I}$ (where $\mathcal{I}$ is the inversion operator), and the even/odd gap function corresponds to ${\mathcal{I}}_0{\mathrm{\Delta }}_{\mathit{k}}=\pm {\mathrm{\Delta }}_{-\mathit{k}}{\mathcal{I}}_0$ (where ${\mathcal{I}}_0$ is the normal state inversion operator). With gray font we indicate the relevant $\mathrm{AZ}+\mathcal{I}$ class for noncentrosymmetric systems (assuming for simplicity that $\mathfrak{T}$ and $\mathfrak{P}$ are also absent), which are not discussed explicitly in the text. The cells with green background indicate the example models discussed explicitly in Secs. 6, 7, 8, 9.

Figure 4

Spectrum of Hamiltonian (35) consists of two Weyl points (black dots) of the same chirality with energy offset $\pm m$, crossing on a nodal sphere with radius $\left|m\right|$. The two species of lines indicate Berry curvatures of opposite signs, which integrate to opposite Chern numbers $\pm 1$. The total Chern number over the occupied states is zero inside and equals $\pm 2$ outside the nodal sphere, meaning that the nodal surface is a source of a pair of Berry phase quanta.

Figure 5

(a) Crystalline structure of $\mathrm{SrPtAs}$ consists of three elements: strontium (Sr: large, bright gray), platinum (Pt: large, light red), and arsenic (As: small, dark blue). The TB model of the present Sec. 6 considers $s$-like orbitals located at the $\mathrm{Pt}$ sites, while the TB models of the subsequent Secs. 7 to 9 capture $s$-like orbitals at both the $\mathrm{Pt}$ and the $\mathrm{As}$ sites. The thick green arrows ${\mathit{R}}_{1,2,3}$ indicate Bravais vectors (36a), while the thin arrows $a,c$ indicate unit cell dimensions. Six atoms ($2\times \mathrm{Sr},2\times \mathrm{Pt},2\times \mathrm{As}$) belonging to a single Bravais vector are highlighted with black thick circles in the bottom left part of the panel. (b) Top view of a single hexagonal layer of $\mathrm{SrPtAs}$. The triplets of vectors $\mathbf{t}={\left\{{\mathit{t}}_n\right\}}_{n=1}^3$ (solid purple) and $\mathbf{T}={\left\{{\mathit{T}}_n\right\}}_{n=1}^3$ (dashed orange) are used to achieve a compact formulation of the TB Hamiltonians.

Figure 6

Doubly charged nodal surfaces of $\mathrm{AZ}+\mathcal{I}$ class $\mathrm{D}$. (a) Fermi surface of TB model (37) with parameters (38) consists of a pair of pockets centered on the corners of BZ. (b) The $d+\mathrm{i}d$ order parameter (39a) creates double Weyl points where the vertical BZ edges cross the Fermi pockets. Admixing a $p$-wave order parameter (39b) inflates them into tiny nodal surfaces. (c) Plot of $signPf\left[\mathrm{i}\mathcal{H}\left(\mathit{k}\right)\right]$ for a horizontal slice through the nodal surface at the indicated value of $k_{z}c$. (d) Flow of the Berry curvature across an ellipsoid described in the text, which encloses the nodal surface. Integration reveals that the nodal surface is a source of two Berry phase quanta.

Figure 7

Doubly charged nodal surfaces of $\mathrm{AZ}+\mathcal{I}$ class $\mathrm{BDI}$. (a) Fermi surface of model (56) with parameters (57) consists of a pair of nodal cylinders centered on the vertical BZ edges. (b) and (c) Evolution of the $q\left(\mathit{k}\right)\in \mathsf{SO}\left(2\right)$ eigenvalues along a path encircling the nodal surface (green) and a path inside of it (red) indicate a nontrivial ${\pi }_1$ charge (53). (d) Plot of the sign of $det\left[q\left(\mathit{k}\right)\right]$ for horizontal planes at the indicated values of $k_{z}c$ demonstrate the nontrivial value of ${\pi }_0$ charge (47).

Figure 8

(a) Nontrivial eigenvalue winding of Wilson loop operators (65) for the nodal loop exhibited by model (67), and (b) the trivial winding for the nodal loop of model (69). Both spectra were calculated on a sphere with radius $2m$ centered at $\mathit{k}=\mathit{0}$. (c) Torical Fermi surface of a nodal loop metal. If the system develops a singlet SC order parameter with a gap function ${\mathrm{\Delta }}_{\mathit{k}}$ that changes sign along the torus (red and blue regions), a pair of SC nodal loops appear (cyan). By locally adjusting the energy of the metallic nodal loop to the chemical potential, it is possible to shrink the SC node to a single point. However, further energy variation of the metallic node leads to a regrowth of the SC nodal loop, thus manifesting a nontrivial ${\pi }_2$ charge. (d) The same consideration for a torical Fermi surface without an underlying nodal loop lead to removable SC nodal loops with a trivial ${\pi }_2$ charge. (e) Both torical Fermi surfaces admit a more complicated geometry of the SC nodal loops. Such linked nodal loops were argued to exhibit anomalous gravitomagnetoelectric response [114, 115].

Figure 9

Doubly charged nodal lines of $\mathrm{AZ}+\mathcal{I}$ class $\mathrm{CI}$. (a) Fermi surface of the developed TB model with parameters (71) consists of touching electron (blue) and hole (red) pockets. The SC order parameter (72) creates two pairs of SC nodal loops when $\left|\delta \right|<1$. (b) Winding of $\mathcal{W}\left(\theta \right)$ eigenvalues on an ellipsoid described in the text which contains a pair of nodal loops, and (c) the same calculation on an ellipsoid containing a single nodal loop reveal a nontrivial ${\pi }_2$ charge (65).

Figure 10

(a) Wilson loop spectrum over the occupied and (b) the unoccupied states of the (2+1)-band Hamiltonian (77). The pair of plots looks identical for the (2+2)-band Hamiltonian (79). (c,d) The corresponding Wilson loop spectra for the (2+2)-band Hamiltonian (80).

Figure 11

Doubly charged nodal loops of $\mathrm{AZ}+\mathcal{I}$ class $\mathrm{AI}$. (a) The transparent green sheets indicate the trajectory traced out by nodal loops of TB model (81) for parameters (83a) for $m\in [-2,2]$. Snapshots of these nodal loops for $m\in \{-2,-1.6,-0.8,0,0.8,1.6,2\}$ are shown in various shades of blue to red. The large circular gray loops inside the BZ correspond to nodes of the same model with $t_2=1.5$. (b) The Wilson loop spectrum for one of the nodal lines for parameters (83) exposes the nontrivial value of their ${\pi }_2$ charge (76). This is related to the underlying nodal lines formed within the (un)occupied bands along the vertical BZ edges, running through the interior of the plotted nodal loops. (c) The ${\pi }_2$ charge is trivial for nodal loops with $t_2=1.5$ on the other side of the topological phase transition (84).