Alice strings in non-Hermitian systems

An Alice string is a topological defect with a very peculiar feature. When a defect with a monopole charge encircles an Alice string, the monopole charge changes sign. In this paper, we generalize this notion to the momentum space of periodic media with loss and gain. In particular, we find that the generic band-structure node for a three-dimensional non-Hermitian crystalline system acts as an Alice string, which can flip the Chern number charge carried by Weyl points and by exceptional-line rings. We discuss signatures of this topological structure for a lattice model with one tuning parameter, including nontrivial braiding of bulk band nodes, and the spectroscopic features of both the bulk and the surface states. We also explore how an Alice string affects the validity of the Nielsen-Ninomiya theorem, and present a mathematical description of the braiding phenomenon.

Figure 1

(a) Two Weyl points A and B (green dots) near an exceptional line (vertical green). The short blue vs the long red “sausages” represent two topologically distinct surfaces on which one can compute the total charge of the Weyl points. The total charge indicates whether the Weyl points annihilate if brought together along the blue vs the red trajectory contained inside the two surfaces. (b) Circumnavigating the exceptional line leads to reordering of the energy bands, which effectively flips the Chern number of the transported nodes. Therefore, if the topological charges of the Weyl points add up on the blue surface, they cancel out on the red surface.

Figure 2

Band nodes of the model in Eq. (1) for the displayed values of $\alpha$. The orange (pink) sheet indicates the plane $k_z=0$ ($k_y=0)$. The band structure exhibits an exceptional line (green line) inside the pink plane. Furthermore, a pair of Weyl points (green dots) is ejected from the exceptional line at $\alpha =\pi /4$. We show in Fig. 3 that the ejected Weyl points locally carry the same chirality. Nevertheless, they annihilate at $\alpha =3\pi /4$ after encircling the exceptional line.

Figure 3

(a) The band structure of the model in Eq. (1) for $\alpha =\pi /4+0.3$ exhibits an exceptional line (green line) and a pair of Weyl points (green dots). The total charge of the two Weyl points on the blue and red surface, respectively, is determined by plotting the Wilson loop eigenvalues for paths (vertical circles) with a fixed angle $\varphi =\arg (k_x+i{k}_y)$, as $\varphi$ sweeps along the surface. We find that the Chern number is (b) $+2$ on the blue surface and (c) zero on the red surface.

Figure 4

(a)–(c) Band nodes (green) of the Hamiltonian in Eq. (3) for the indicated values of parameters $\lambda$ and $m$. For $m=0$, the model is Hermitian, and one can assign a unique chirality to all the Weyl points (the $+$ vs $-$ signs displayed with blue font). For $m\ne 0$, the Chern number of a node becomes ambiguous and depends on the choice of the enclosing surface. The non-Hermitian perturbation is set up such that it inflates into an exceptional ring only the Weyl point at $k_z=0$. (d) The band nodes of panel (c) inside the pink sheet $k_y=0$. To study the monopole charge of the exceptional ring, we consider surfaces of revolution obtained by rotating around the $k_z$ axis the blue loop [$k_x=1.3cos\varphi$ and $k_z=\frac{1}{3}sin\varphi$ for $\varphi \in [0,2\pi )$] and the red loop [$k_z$ further lowered by $\frac{3}{2}{({cos}^4\varphi -1)}^8$], respectively. We describe the obtained surfaces as an “ellipsoid” and a “bowl,” respectively. The Weyl point at $k_z<0$ makes it impossible to deform the ellipsoid into the the bowl without closing the energy gap, therefore the two surfaces may exhibit different Chern number. (e), (f) Wilson loop flow for the ellipsoid and the bowl surface, respectively. In both cases, the Wilson operators are computed along paths of constant $k_x$, and the parameter $\theta$ grows from zero at point $\mathrm{N}$ to $\pi$ at point $\mathrm{S}$, indicated for both surfaces in panel (d). The branch of the Riemann sheet is chosen such that both surfaces consider the same Bloch state at points $\mathrm{N}$ and $\mathrm{S}$, where both eigenvalues of the Hamiltonian remain real for all considered values of the model parameters.

Figure 5

(a)–(e) Position of Weyl points (green dots) of the model in Eq. (4b) inside the $k_x=0$ plane for the indicated values of parameter $m$. At $m=-3$, a pair of Weyl points of opposite chirality is created at $(k_y,k_z)=(0,0)$, and departs along the $k_z$ axis. The Weyl points collide and bounce for $m=-1$ at $(k_y,k_z)=(0,\pi )$ after circumnavigating the $k_z$ circle of the BZ torus, and then again for $m=+1$ at $(k_y,k_z)=(\pi ,\pi )$. The Weyl points do not annihilate at these two collisions because of the Alice string enclosed by the $k_z$ circle of the BZ torus. The Weyl points finally annihilate for $m=+3$ at $(k_y,k_z)=(\pi ,\pi )$ after the effect of the Alice string is undone by moving along the $k_z$ direction for the second time. (f)–(j) Zero-energy surface Fermi arcs of the same model for the corresponding values of $m$ assuming a system termination in the $x$ direction. The details of the computation and plotting are explained in the second paragraph of Sec. 3b.

Figure 6

(a) Abe homotopy considers maps from a cylinder, $S^1\times I$, with the boundary (red) mapped to the base point $\mathfrak{m}$. (b) By further requiring a line segment $\left\{x_0\right\}\times I$ to be also mapped to the base point, one recovers the second homotopy group, ${\pi }_2\left(M\right)$. (c) By narrowing attention to maps that only depend on the position along $I$, one recovers the first homotopy group, ${\pi }_1\left(M\right)$. (d) Abe homotopy allows us to compose elements of the second homotopy group (map on $S^2$) with elements of the first homotopy group (map on $S^1$). The green line indicates a band node. (e) We attach to the sphere $S^2$ based at $\mathfrak{m}$ a string from $\mathfrak{m}$ to ${\mathfrak{m}}^{\prime }$. This corresponds to conjugating the cylinder representing the map on $S^2$ with a cylinder representing the map on the string. Attaching a closed string (loop) based at $\mathfrak{m}$ corresponds to conjugating the element from ${\pi }_2\left(M\right)$ with an element from ${\pi }_1\left(M\right)$.