Bloch classification surface for three-band systems

Abstract

Topologically protected states can be found in physical systems, that show singularities in some energy contour diagram. These singularities can be characterized by winding numbers, defined on a classification surface, which maps physical state parameters. We have found a classification surface, which applies for three-band hamiltonian systems in the same way than standard Bloch surface does for two-band ones. This generalized Bloch surface is universal in the sense that it classifies a very large class of three-band systems, which we have exhaustively studied, finding specific classification surfaces, applying for each one.

Figures (19)

• Figure 1: Representation of $\mathcal{S}$, as explained in Ref. Abramovici.
• Figure 2: Contours of the highest eigenenergy of $H_{\rm gH}$ with $k_x$ and $k_y$ varying in $[-2\pi,2\pi]$. Discontinuities $K_0$, $K_1$, $K_2$ and $K_3$ are indicated. Both axis are necessarily $2\pi$-periodic but, according to the values of $({\delta_1},{\delta_3})$, the diagram can be $\pi$-periodic in $k_x$-direction or in $k_y$-direction or both or none. Here, $({\delta_1},{\delta_3})=(-\frac{1}{2},\frac{1}{2})$.
• Figure 3: Plots of $\alpha_i(\cos t,\sin t)$ for $\alpha=a,b$ and $i=4..7$ (withdrawing $b_7$) for arbitrary values of $({\delta_1},{\delta_3})$, here $({\delta_1},{\delta_3})=({-}0.5,1.2)$.
• Figure 4: Plot of $|\widehat{7_a}(Q_t)|$, the distance from $Q_t$ to $s_6$ for $t\in[0,2\pi]$ and arbitrary values of $({\delta_1},{\delta_3})$, here $({\delta_1},{\delta_3})=(0.5,0.3)$.
• Figure 5: Representation of the mapping by $\tau_1$ of several circles of $(k_x,k_y)$-plane. $\tau_1(s_6)$ is pinched along a line $\mathcal{L}$, which is indicated in the middle of the figure. Large cyan points represent $\tau_1(D_i)$ for $i=1..4$, mappings are in red when the circle is around a singularity, in blue when it avoids it. Red, yellow, green and purple points follow this order in non-trivial paths but may overlap. Plain lines correspond to singularity $K_0$ ($\tau_1(D_0)$ is a singularity of the surface, as shown in Fig. \ref{['det00']}). Dotted lines correspond to singularity $K_1$; $\tau_1(D_1)$ is at the middle of $\mathcal{L}$ and the non-trivial path crosses $\mathcal{L}$ twice. Dashed lines correspond to singularity $K_2$ (the 8-shape is artificially created by the two-dimensional projection of the drawing). Dot-dashed lines correspond to singularity $K_3$, one observes that the non-trivial one makes a loop with 8-shape, which collapses at one extremity of $\mathcal{L}$.