Topological nodal superconducting phases and topological phase transition in the hyperhoneycomb lattice

We establish the topology of the spin-singlet superconducting states in the bare hyperhoneycomb lattice, and we derive analytically the full phase diagram using only symmetry and topology in combination with simple energy arguments. The phase diagram is dominated by two states preserving time-reversal symmetry. We find a line-nodal state dominating at low doping levels that is topologically nontrivial and exhibits surface Majorana flatbands, which we show perfectly match the bulk-boundary correspondence using the Berry phase approach. At higher doping levels, we find a fully gapped state with trivial topology. By analytically calculating the topological invariant of the nodal lines, we derive the critical point between the line-nodal and fully gapped states as a function of both pairing parameters and doping. We find that the line-nodal state is favored not only at lower doping levels but also if symmetry-allowed deformations of the lattice are present. Adding simple energy arguments, we establish that a fully gapped state with broken time-reversal symmetry likely appears covering the actual phase transition. We find this fully gapped state to be topologically trivial, while we find an additional point-nodal state at very low doing levels that also break time-reversal symmetry and has nontrivial topology with associated Fermi surface arcs. We eventually address the robustness of the phase diagram to generalized models also including adiabatic spin-orbit coupling, and we show how all but the point-nodal state are reasonably stable.

Figure 1

(a) Hyperhoneycomb lattice belonging to SG70 for the Wyckoff's position $16e$ and spanned by the primitive lattice vectors ${\mathit{a}}_i$. (b) BZ for SG70 with primitive reciprocal-lattice vectors ${\mathit{b}}_i$ [18]. $\mathrm{\Gamma }, Y, T$, and $Z$ are high-symmetry points of the first BZ. ${\mathrm{\Gamma }}_2, Y_2, T_2$, and $Z_2$ are the equivalent high-symmetry points of the next BZ.

Figure 2

Toroidal Fermi surface of the normal state band structure for a finite doping ($\mu =0.06$) away from half-filling. The plotted domain spans two BZs such that two copies of the Fermi surface are visible (the second copy is split into eight eighths shifted by reciprocal-lattice vectors, e.g., $\mathit{k}\left({\mathrm{\Gamma }}_2\right)-\mathit{k}\left(\mathrm{\Gamma }\right)={\mathit{b}}_1+{\mathit{b}}_2+{\mathit{b}}_3$.

Figure 3

(a) BdG energy gap for the state ${\mathrm{\Gamma }}_{1,a}^{+}$ represented through the energy isosurfaces at $E\approx -{\mathrm{\Delta }}_{\mathrm{bulk}}=-0.5t$. (b) Nodal lines (zero-energy isolines) of state ${\mathrm{\Gamma }}_{1,b}^{+}$. Two BZs are shown, and thus two copies of the spectrum are visible in (a) and (b).

Figure 4

Surface BdG spectral function at zero energy for the line-nodal state ${\mathrm{\Gamma }}_{1,b}^{+}$ at a $\left(100\right)$ surface (a) and a $\left(001\right)$ surface (b). Yellow indicates high spectral weight (not seen), black indicates exponentially suppressed spectral weight.

Figure 5

(a) Schematic bulk nodal lines of the state ${\mathrm{\Gamma }}_{1,b}^{+}$ and surface normal vector $\mathit{n}\propto {\mathit{b}}_1$ (brown) for a surface spanned by ${\tilde{\mathit{a}}}_1={\mathit{a}}_3$ and ${\tilde{\mathit{a}}}_2=-{\mathit{a}}_2+{\mathit{a}}_3$. (b) Projection in parallel to $\mathit{n}\propto {\mathit{b}}_1$ of the bulk nodal lines onto the surface BZ with $\{{\tilde{\mathit{b}}}_1,{\tilde{\mathit{b}}}_2\}$ reciprocal primitive vectors (orange). (c) Surface BdG spectral function at zero energy over the surface BZ. (d) Bulk Berry phase over a noncontractible loop ${\gamma }_B\left[{\mathcal{L}}_{{\mathit{k}}_{\parallel }}\right]$, with ${\gamma }_B=0$ (dark) and ${\gamma }_B=\pi$ (light). (e) BdG spectrum for the full slab geometry as a function of ${\tilde{b}}_2$ and at fixed ${\tilde{b}}_1=0$, i.e., along a slice of the surface BZ marked by a light green line in (d).

Figure 6

(a) BdG energy gap for the state ${\mathrm{\Gamma }}_{1,c}^{+}$ represented through the maximum occupied iso-energy surface, here for ${\mathrm{\Delta }}_h={\mathrm{\Delta }}_z=0.1t$ and $\varphi =0.9\pi$. (b) Positions in the BZ of the nodal points of the state ${\mathrm{\Gamma }}_d^{+}$ (numerical accuracy makes them look like cigars). (c) BdG dispersion in the vicinity of the nodal points over the plane $k_z=0$ containing four nodal points, here for $|{\mathrm{\Delta }}_h|=|{\mathrm{\Delta }}_z|=0.5t$ and $\varphi =0.2\pi$.

Figure 7

(a) Schematic nodal points (blue, labeled from 1 to 4) for the ${\mathrm{\Gamma }}_d^{+}$ state with Chern numbers ($\pm 1$) and Berry flux lines joining them (pink). The orange sphere surrounding one nodal point illustrates the base loop ${\mathcal{L}}_{\theta }$ (red) at a fixed polar angle $\theta \in [0,\pi ]$. (b) Flow of the Berry phase as a base loop covers the full sphere leading to a Chern number $C_1=-1$.

Figure 8

Surface properties for the point-nodal state ${\mathrm{\Gamma }}_d^{+}$. (a) Surface BdG spectral function at zero energy for a $\left(100\right)$ surface. (b) Surface BdG spectral function at zero energy for a $\left(001\right)$ surface. (c) Surface BdG spectrum as a function of $k_z$ for fixed $k_y=-0.2[2\pi /b]$ for the $\left(100\right)$ surface, i.e., along the light green line of (a). (d) Surface BdG spectrum as a function of $k_x$ for fixed $k_y=0.1[2\pi /b]$ for the $\left(001\right)$ surface, i.e., along the light green line of (b).

Figure 9

(a) Winding number (blue) and Berry phase (red) computed over the base loop ${\mathcal{L}}_S=\mathrm{\Gamma }Y{Z}_2{T}_2\mathrm{\Gamma }$ [see Fig. 3] as a function of $\alpha ={\mathrm{\Delta }}_z/\mathrm{\Delta }$ for $(\mu ,t,\mathrm{\Delta },{\mathrm{\Delta }}_0)=(0.06,0.2,0.1,0)$. Both capture the topological phase transition between the fully gapped state ${\mathrm{\Gamma }}_{1,a}^{+}$ ($\alpha =+1$) and the line-nodal state ${\mathrm{\Gamma }}_{1,b}^{+}$ ($\alpha =-1$) at ${\alpha }_c\approx 0.77$. (b) Berry phase computed over ${\mathcal{L}}_S$ as a function of $\beta ={\mathrm{\Delta }}_h/\mathrm{\Delta }$ capturing the topological phase transition at ${\beta }_c=0.7$ for the same choice of remaining parameters as in (a).

Figure 10

Non-self-consistent phase diagram for the fully gapped (G, blue) and line-nodal (LN, red) phases with TRS as a function of $\alpha ={\mathrm{\Delta }}_z/\mathrm{\Delta }$ and $\beta ={\mathrm{\Delta }}_h/\mathrm{\Delta }$ with $\alpha ,\beta \in [-1,1]$ and $\mu /t$. (a) Phase diagram when $\alpha$ and $\beta$ are changed separately for fixed $\mu /t=0.3, {\mathrm{\Delta }}_0=0$, and $\mathrm{\Delta }=0.5t$. (b) Complete phase diagram for $\alpha , \beta$, and $\mu /t$. The imbedded horizontal plane (darker colors) gives the phase diagram at the fixed doping $\mu /t=0.3$. Remaining parameters as in (a).

Figure 11

Norm $\text{Re}\mathrm{Log}\left[z_{1,a}\left(k_y\right)\right]$ (dashed lines) and argument $\text{Im}\mathrm{Log}\left[z_{1,a}\left(k_y\right)\right]$ (solid lines) of the complex function $z_{1,a}\left(k_y\right)$ as a function of $k_y$ for two different values of the parameter $\alpha$ (red and blue). Remaining parameters are $(\mu ,t,\mathrm{\Delta },{\mathrm{\Delta }}_0)=(0.06,0.2,0.1,0)$ from which we find ${\alpha }_c\approx 0.769$.