Homotopy characterization of non-Hermitian Hamiltonians

We revisit the problem of classifying topological band structures in non-Hermitian systems. Recently, a solution has been proposed, which is based on redefining the notion of energy band gap in two different ways, leading to the so-called “point-gap” and “line-gap” schemes. However, simple Hamiltonians without band degeneracies can be constructed which correspond to neither of the two schemes. Here, we resolve this shortcoming of the existing classifications by developing the most general topological characterization of non-Hermitian bands for systems without a symmetry. Our approach, which is based on homotopy theory, makes no particular assumptions on the band gap, and predicts significant extensions to the previous classification frameworks. In particular, we show that the one-dimensional invariant generalizes from $\mathbb{Z}$ winding number to the non-Abelian braid group, and that depending on the braid group invariants, the two-dimensional invariants can be cyclic groups ${\mathbb{Z}}_n$ (rather than $\mathbb{Z}$ Chern number). We interpret these results in terms of a correspondence with gapless systems, and we illustrate them in terms of analogies with other problems in band topology, namely, the fragile topological invariants in Hermitian systems and the topological defects and textures of nematic liquids.

Figure 1

The space of eigenvalues in the Hermitian vs non-Hermitian case; ${\mathrm{Conf}}_2\left(\mathbb{R}\right)\sim {\mathbb{Z}}_2$ while ${\mathrm{Conf}}_2\left(\mathbb{C}\right)\sim S^1$.

Figure 2

The torus can be constructed out of a rectangle by gluing together the two red lines and the two blue lines. These lines become closed loops on the torus. Taken together, these two loops form the one-skeleton of the torus, while the cyan interior of the rectangle forms the two-cell of the torus. The one-skeleton $T_1^2=S^1\vee S^1$ consists of two circles joined at a common base point, i.e., the “bouquet” of two circles; we refer to these circles as $a$ and $b$ following the mathematical literature.

Figure 3

A texture on the sphere is merged with a texture on the torus to form a new texture on the torus. Two configurations are shown, a skyrmion and an antiskyrmion.

Figure 4

An $\mathbb{R}{P}^2$ texture on a sphere and a torus, drawn as a field of ellipsoids; the figure illustrates the mechanism by which the action of ${\pi }_1\left(X\right)$ on ${\pi }_2\left(X\right)$ leads to a reduction from a $\mathbb{Z}$ invariant to a ${\mathbb{Z}}_2$ invariant on the torus. The texture on the torus has nontrivial winding in the $x$ direction, while the texture on the sphere has nontrivial “Chern number.” The sphere is moved around a complete cycle in the $x$ direction, and meanwhile each ellipsoid undergoes a $\pi$ rotation around the $y$ axis. In the end, the texture on the sphere is equivalent to the texture on the $xz$ mirror of the original sphere. The color indicates the angle of rotation around the $y$ axis.

Figure 5

(a) Comparison of the three physical settings discussed in Secs. 4b, 4c, and 4d. Although the terminology differs, the description of their singularities (topological defects) is very similar. All three systems exhibit topologically protected line and point defects in 3D, mathematically described respectively by ${\pi }_1$ and ${\pi }_2$ of the order-parameter space. In each instance, the 1D and 2D invariants interact nontrivially, meaning that the ${\pi }_2$ charge describing a point node flips sign if it is carried along a closed path with nontrivial ${\pi }_1$ charge. This interaction leads to $\mathbb{Z}⟶{\mathbb{Z}}_2$ lowering of the topological invariant on the torus, if there is a nontrivial ${\pi }_1$ charge in some direction. (b) A 2D torus embedded in 3D, with a green curve indicating an exceptional line, and red and blue points indicating Weyl points of opposite chirality. The two angular coordinates on the torus are denoted $\theta$ and $\varphi$. Since the $\varphi$ direction links with the exceptional line, the torus has nontrivial 1D invariant in the $\varphi$ direction. By moving one of the Weyl points around the exceptional line, we can change its chirality and obtain a configuration with total charge 2 from this initial configuration of total charge 0; thus, total charge 2 represents a topologically trivial configuration, as we understand in terms of braiding Weyl points around an exceptional line.

Figure 6

The Wilson loop eigenvalue flow for the ${\mathbb{Z}}_2$ invariant. (a), (c) Show nontrivial ${\mathbb{Z}}_2$ invariant, while (b) and (d) show trivial ${\mathbb{Z}}_2$ invariant. We construct 2D models starting with a 3D model, and restricting to a 2D surface inside 3D $k$ space. The 3D model has four Weyl points (red) located at $k_z=0$ plane as shown in (a) and (b). We choose two different cylinders (topologically, toruses) whose projections are shown as orange loops in (a) and (b) for the Wilson loop calculation in (c) and (d). (c) Shows the flow of Wilson loop eigenvalues when the center of the cylinder lies at $(k_x,k_y)=(0,0)$. The cylinder encloses one Weyl point and the invariant is nontrivial. We see one crossing (odd) at $\pi$ and two crossings (even) at 0. (d) Shows the flow of Wilson loop eigenvalues when the center of the cylinder lies at $(k_x,k_y)=(1.2,1.2)$. The cylinder encloses no Weyl points and the invariant is trivial. We see zero crossing (even) at $\pi$ and three crossings (odd) at 0.