Three-dimensional $\mathcal{PT}$-symmetric topological phases with a Pontryagin index

We report on a certain class of three-dimensional topological insulators and semimetals protected by spinless $\mathcal{P}\mathcal{T}$ symmetry, hosting an integer-valued bulk invariant. We show using homotopy arguments that these phases host multigap topology, providing a realization of a single $\mathbb{Z}$ invariant in three spatial dimensions that is distinct from the Hopf index. We identify this invariant with the Pontryagin index, which describes Belavin-Polyakov-Schwartz-Tyupkin (BPST) instantons in particle physics contexts and corresponds to a three-sphere winding number. We study naturally arising multigap linked nodal rings, topologically characterized by split-biquaternion charges, which can be removed by non-Abelian braiding of nodal rings, even without closing a gap. We additionally recast the describing winding number in terms of gauge-invariant combinations of non-Abelian Berry connection elements, indicating relations to Pontryagin characteristic class in four dimensions. These topological configurations are furthermore related to fully nondegenerate multigap phases that are characterized by a pair of winding numbers relating to two isoclinic rotations in the case of four bands and can be generalized to an arbitrary number of bands. From a physical perspective, we also analyze the edge states corresponding to this Pontryagin index as well as their dissolution subject to the gap-closing disorder. Finally, we elaborate on the realization of these novel non-Abelian phases, their edge states, and linked nodal structures in acoustic metamaterials and trapped-ion experiments.

Figure 1

(a) Phase diagram of the minimal model with $p_x=p_y=p_z=1$ supporting nontrivial Pontryagin indices $Q$. The mass term controls the $\mathbb{Z}$-valued invariant, similarly to the two-dimensional models with Euler topology [41]. (b) Projection of bulk bands and edge states along a one-dimensional section of a three-dimensional insulator with nontrivial Pontryagin index $Q=1$ set in the minimal model, as compared with $Q=0$.

Figure 2

Nodal structures over the three-dimensional Brillouin zone for the minimal model of the Hamiltonian provided in Eq. (22). Nodes between bands $|u_1\rangle$ and $|u_2\rangle$ were denoted in blue, while band crossings between bands $|u_2\rangle$ and $|u_3\rangle$ were denoted in orange. (a) $Q=0$ with trivial nodal structure and vanishing linking numbers. (b) $Q=1$, with two nodal rings present. (c) $Q=2$, with four linked nodal rings. We show that the structures are not protected by the Pontryagin index $Q$, but are easily realized in the context of the presented models.

Figure 3

Explicit unlinking of nodal structure through the creation of adjacent nodal rings on closing a gap, as parametrized in Appendix pp3. The blue and orange rings show nodal lines between the bands ($|u_1\rangle$–$|u_2\rangle$, and $|u_2\rangle$–$|u_3\rangle$, correspondingly) in the occupied three-band subspace and the green rings show nodal lines between the highest occupied and unoccupied bands. (a) initial nodal links, (b)–(c) creation of additional rings, (d) connecting rings, (e)–(f) splitting and disentangling green and blue rings, (g) removal of green rings, (h) contraction and annihilation of blue rings to enter the gapped flag limit. Numerically, the unbraiding on closing the principal gap can be achieved by adding a diagonal mass term to the Hamiltonian, similarly to the two-dimensional Euler insulators and semimetals, where an effective mass to unbraid the nodes can also be realized by adding on-site disorder [56].

Figure 4

Visualization of the process of unbraiding of nodal structures without closing the principal gap in a model with $Q=1$. The blue and orange rings show nodal lines in the occupied three-band subspace, as in Fig. 3. (a) Initial nodal links, (b) creation of additional rings, (c) connecting rings, (d)–(e) splitting and disentangling rings, (f)–(g) closure and removal of orange rings, (h) leftover blue ring, which can be contracted and removed, obtaining the fully gapped flag limit. The unbraiding was obtained on adding a parametrized dispersive band-splitting term to the Hamiltonian, contrary to the previously introduced unbraiding with a diagonal term closing the principal gap; see also Appendix pp3.

Figure 5

Non-Abelian Wilson loop winding for Pontryagin index (a) $Q=0$, (b) $Q=1$, (c) $Q=2$, and (d) $Q=3$. The loops were evaluated at $k_y=0$, and high-symmetry planes $k_y=\pi$. For $Q=0$ we find no winding of the Wilson loop eigenvalues $\varphi$, along with the winding of $Q=1, Q=2, Q=3$ being even, contrary to the Stiefel-Whitney insulators with odd Wilson loop winding. We observe that the combined total winding of the eigenvalues within two planes sums to $2Q$.

Figure 6

Comparison of edge states in the bulk gaps of insulators with Pontryagin indices $Q=2, Q=1$, and $Q=0$, realized in (a)–(c) the model Hamiltonians (22), (d) flag limit phase $(w_L,w_R)=(0,1)$, (e) $(w_L,w_R)=(1,1)$, (f) $(w_L,w_R)=(2,1)$. The edge states are evaluated under open boundary conditions in the $x$ direction with $(k_y,k_z)=(0.5,0.5)$. We find that the bulk-boundary correspondence of these three-dimensional $\mathcal{PT}$-symmetric phases yields the multiplicities of edge states $g=4Q$ in the principal gap of $3\oplus 1$ model, and $g_1=4(w_L+w_R), g_3=4(w_L-w_R)$ in bottom and top gaps in the flag limit. While the topologically trivial $\mathcal{PT}$-symmetric phases with $Q=0$ can host edge states, these are not topologically protected due to the triviality of the bulk invariant, as we verify against disorder (Sec. 6). On closing two lower gaps with a band inversion, $g=g_3$, which can be mapped to the correspondence of invariants. $Q=(w_L-w_R)$. The fourfold multiplicity of edge states can be understood in terms of the interplay of the symmetries: the essential $\mathcal{PT}$, as well as $\mathcal{T}$ (${\mathcal{T}}^2=1$), and $\mathcal{P}$ admitted by the models.

Figure 7

Edge and bulk states of (a)–(b) the Pontryagin $Q=1$ and (c)–(d) associated flag limit $(w_L,w_R)=(0,1)$ phases subject to weak disorder ($W=0.3$). The one-dimensional wave function sections of clean phases (black) were plotted against the same states perturbed with weak Anderson disorder (red). In both phases, the edge states remain exponentially localized (a), (c) as long as the disorder strength is not sufficient to close the gap. The bulk states (b), (d) do not show similar robustness, becoming distorted on adding disorder.

Figure 8

Wannier ladders for Pontryagin index (a) $Q=0$, (b) $Q=1$, (c) $Q=2$ and (d) $Q=3$. The flows of HWF centers (${\overline{w}}_z$) were evaluated along $k_y$ on setting $k_x=0$. For $Q=0$ we find no net flow of HWF centers in the ladder. For nontrivial $Q\in \mathbb{Z}$ (b)–(d), we observe two HWF centers flowing by $\pm 2Q$ real-space unit cells, along with a third HWF center without any flow, all of which correspond to the occupied three-band subspace. On the contrary, the unoccupied band subspace supports no flow in the associated HWF centers.

Figure 9

Surface Wilson loops for (a) $Q=0$, (b) $Q=1$, (c) $Q=2$, and (d) $Q=3$ along $k_x$. Here, the plotted Wilson loop eigenvalues $\varphi$ were found for two surface bands associated with two HWF center (${\overline{w}}_z$) sheets closest to the surface, on isolating these from the rest of the Wannier ladder. For original HWF sheets, see the Wannier ladder in Fig. 8. For $Q=0$ we find no Wilson loop winding. For non-trivial $Q\in \mathbb{Z}$ (b)–(d), we find two opposite eigenvalues $\pm 2Q$. While such pair does not carry a surface Chern number, as explained in the main text; under the ${\mathcal{C}}_2\mathcal{T}$ symmetry satisfied on the surface, such winding is topologically protected against the symmetry-preserving perturbations. Based on the deduced Wilson loop winding, protected under the symmetry enforcing a reality condition [32], we conclude that the surface bands host a surface Euler invariant ${\chi }_s$, which we find on two opposite surfaces under the open boundary conditions in $z$ direction.

Figure 10

Assigning the non-Abelian charges to different node configurations. The generators of the group are assigned to single nodes in each of the gaps and more complex configurations are achi- eved by multiplying the charges of the constituent nodes. Note that $e_1{e}_2{e}_3$ and $-e_1{e}_2{e}_3$ are in different conjugacy classes, as there is no braiding procedure that can flip the sign of only one of the nodes.