$2^n$-root weak, Chern, and higher-order topological insulators, and $2^n$-root topological semimetals

Recently, we have introduced [A. M. Marques et al., Phys. Rev. B 103, 235425 (2021)] the concept of $2^n$-root topology and applied it to one-dimensional systems. These models require $n$ squaring operations to their Hamiltonians, intercalated with different constant energy downshifts at each level, in order to arrive at a decoupled block corresponding to a known topological insulator (TI) that acts as the source of the topological features of the starting $2^n\text{−}\mathrm{root}$ TI $\left(\sqrt[2^n]{\text{TI}}\right)$. In the process, $n$ nontopological residual models with degenerate spectra and in-gap impurity states appear, which dilute the topologically protected component of the starting edge states. Here, we generalize this method to several two-dimensional models, by finding the 4-root version of lattices hosting weak and higher-order boundary modes (both topological and nontopological) of a Chern insulator and of a topological semimetal. We further show that a starting model with a non-Hermitian region in parameter space and a complex energy spectrum can nevertheless display a purely real spectrum for all its successive squared versions, allowing for an exact mapping between certain non-Hermitian models and their Hermitian lower root-degree counterparts. A comment is made on the possible realization of these models in artificial lattices.

Figure 1

Schematic depiction of the main ingredients for the construction of a higher root-degree version ($\sqrt[2^n]{\text{TI}}$) from its lower root-degree chain ($\sqrt[2^{n-1}]{\text{TI}}$). (a) Subdivision of the $\sqrt[2^{n-1}]{\text{TI}}$ by introducing a site in the middle of each hopping term. (b) Renormalization of the new doubled hopping terms, both in amplitude and in phase. (C) Uniformization of the squared onsite energy at the relevant blue sublattice the $\sqrt[2^n]{\text{TI}}$ which, if not achieved by default, requires the introduction of extra sites connected to the blue sites with lower squared onsite energies with properly tuned hopping strengths. Additional subtleties of this method can be found in [14] and here at the end of Sec. 4.

Figure 2

(a) Asymmetric 2D SSH lattice. Red (green) shaded region indicates the localization of the horizontal (vertical) edge states, while the gray shaded region indicates the localization of the corner states. The purple shaded square in the middle delimits the unit cell. (b) Energy spectrum for the model in (a) for an open square lattice of $8\times 8$ unit cells, with $t_{2x}=1$ (the energy unit) and $t_{2y}=0.25$, as a function of a linear variation of $\sqrt[4]{t_{1x}}=\sqrt[4]{t_{1y}}$. Solid blue curves correspond to bulk states, solid red (green) curves to horizontal (vertical) edge states, and the fourfold-degenerate dashed black zero-energy curve to the corner states. (c) Polarization profile, in units of $e$, along the $x$ (red squares) and $y$ (green dots) direction for the bulk band $j=1,2,3,4$ (it is the same for all bands).

Figure 3

Triptych of energy spectra for the asymmetric 2D SSH model open along $x$ and periodic along $y$ with 15 supercells, as a function of $k_y$. Parameters: $t_{1x}=1$, $t_{2y}=0.25$, and $t_{1x}=t_{1y}=0.2$ for the left panel, $t_{1x}=t_{1y}=0.25$ for the middle panel, and $t_{1x}=t_{1y}=0.3$ for the right panel. From one side to the other there is a gap closing and reopening between the two red bands of horizontal edge states.

Figure 4

(a) Square-root version of the asymmetric 2D SSH lattice. The meaning of the shaded regions is explained in Fig. 2. (b) Energy spectrum for the model in (a) for an open square lattice of $8\times 8$ unit cells for the same parameters and color scheme for the curves as in Fig. 2. (c) Polarization profile, in units of $e$, along the $\mu =x,y$ direction for the lowest bulk band. The dashed line at $P_{\mu }^1=0.5$ is a guide to the eye.

Figure 5

Schematic illustration of the result of squaring the $\sqrt{\text{HOTI}}$ lattice (and the respective Hamiltonian below) with an integer number of unit cells in $x$ and $y$. Up to a constant overall energy shift $c_1$, the squared model is composed of two decoupled lattices, one yielding the HOTI in the blue sublattice and the other yielding the residual model in the gray sublattice. The meaning of the shaded areas is given in Fig. 2. Orange impurity sites with an onsite energy offset in relation to the other sites of the same lattice appear at different boundaries with opposite effects: they gap out the boundary modes in the HOTI lattice but drive the boundary modes in the residual lattice to the same gaps of the corresponding ones in the HOTI.

Figure 6

Unit cell of the $\sqrt[4]{\text{HOTI}}$. Blue sites form one sublattice and the green and gray sites form another. Open sites at the edges belong to adjacent unit cells.

Figure 7

Real part of the energy spectrum, with the same parameters and color scheme for the curves as in Fig. 2, obtained from diagonalization of (a) $H_{\sqrt[4]{\text{HOTI}}}$, the Hamiltonian of the open $\sqrt[4]{\text{HOTI}}$ lattice with $8\times 8$ unit cells plus 36 extra sites from incomplete unit cells (16 extra gray sites with index 35 and 36, from Fig. 6, placed on top and 16 extra gray sites with index 16 and 28 placed at the left); (b) $H_{\sqrt[4]{\text{HOTI}}}^{2^{\prime }}=H_{\sqrt[4]{\text{HOTI}}}^2-c_{2}I$, with $c_2$ given in (45); (c) $H_{\sqrt[4]{\text{HOTI}}}^{4^{\prime }}=H_{\sqrt[4]{\text{HOTI}}}^{2^{\prime }}{H}_{\sqrt[4]{\text{HOTI}}}^{2^{\prime }}-c_{1}I$, with $c_1$ given below (34). (d), (e) Imaginary part of the spectrum for the corresponding cases above. The dashed orange vertical line crossing (a) and (d) at $\sqrt[4]{t_{1x}}=0.5$ separates the non-Hermitian region to the left from the Hermitian region to the right.

Figure 8

Plaquette of the (a) quartic-root, (b) square-root, and (c) original version of the CI, corresponding to the Haldane model without a mass term. Numbered sites indicate the respective unit cells. Energy spectrum as a function of $k_2$, with $(t,\lambda ,\varphi )=(1,\frac{0.2}{3\sqrt{3}},\frac{\pi }{2})$ of (d) $H_{\sqrt[4]{\text{CI}}}$, (e) $H_{\sqrt[4]{\text{CI}}}^{2^{\prime }}=H_{\sqrt[4]{\text{CI}}}^2-c_{2}I$, with $c_2$ given below (57), and (f) $H_{\sqrt[4]{\text{CI}}}^{4^{\prime }}=H_{\sqrt[4]{\text{CI}}}^{2^{\prime }}{H}_{\sqrt[4]{\text{CI}}}^{2^{\prime }}-c_{1}I$, with $c_1$ given below (55) and $H_{\sqrt[4]{\text{CI}}}$ the Hamiltonian of the $\sqrt[4]{\text{HOTI}}$ lattice open along ${\mathbf{a}}_{\mathbf{1}}$ and periodic along ${\mathbf{a}}_{\mathbf{2}}$, with 24 extra sites in incomplete unit cells above the top edge in ${\mathbf{a}}_{\mathbf{1}}$, given by the 6 blue sites $\{4,7,8,9,10,11\}$ plus the 18 gray and green sites connected to them in (a). Blue (red) bands correspond to bulk (edge) bands.

Figure 9

(a) Schematic illustration of the result of squaring the $\sqrt{\text{BK}}$ triangular lattice (and the respective Hamiltonian above), with the unit cell shown at the lower left. Up to a constant overall energy shift $c_1$, the squared model is composed of two decoupled lattices, one yielding the BK in the blue sublattice and the other yielding the residual model in the gray sublattice with edge (corner) impurity sites with an onsite energy offset of $-t_2$ ($-2t_2$), in relation to the red bulk sites. Shaded red (gray) regions indicate the localization of the edge (corner) states. (b), (c) Energy spectrum of the bulk $\sqrt{\text{BK}}$ model, in units of $\sqrt{t_1}=1$, along the high-symmetry lines delimiting the orange region of the Brillouin zone depicted at the right for (b) $\sqrt{t_2}=\sqrt{t_1}$ and (c) $\sqrt{t_2}=1.2$. High-symmetry points of the Brillouin zone: $\mathrm{\Gamma }=(0,0)$, $K=(\frac{4\pi }{3},0)$, $K^{\prime }=(\frac{2\pi }{3},\frac{2\pi }{\sqrt{3}})$, and $M=(\pi ,\frac{\pi }{\sqrt{3}})$.

Figure 10

Energy spectrum, in units of $t_2=1$, as a function of a linear variation of $\sqrt[4]{t_1}$, obtained from diagonalization of (a) $H_{\sqrt[4]{\text{BK}}}$, the Hamiltonian of the $\sqrt[4]{\text{BK}}$ triangular lattice with the unit cell of Fig. 12; (b) $H_{\sqrt[4]{\text{BK}}}^{2^{\prime }}=H_{\sqrt[4]{\text{BK}}}^2-c_{2}I$, with $c_2=3(\sqrt{t_1}+\sqrt{t_2})$; (c) $H_{\sqrt[4]{\text{BK}}}^{4^{\prime }}=H_{\sqrt[4]{\text{BK}}}^{2^{\prime }}{H}_{\sqrt[4]{\text{BK}}}^{2^{\prime }}-c_{1}I$, with $c_1=t_1+t_2$. The different types of states are indicated in (c). Green labeled points along the dashed orange vertical line in (a) correspond to the respective states depicted in Fig. 12.

Figure 11

Polarization profile, in units of $e$, along both the ${\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{a}}_{\mathbf{2}}$ directions for the lowest bulk band of the $\sqrt{\text{BK}}$ model (green dots) and of the $\sqrt[4]{\text{BK}}$ model (red squares).

Figure 12

(a) Unit cell of the $\sqrt[4]{\text{BK}}$. Blue sites form one sublattice and the green and gray sites form another. (b), (c) Energy spectrum of the bulk $\sqrt[4]{\text{BK}}$ model, in units of $\sqrt{t_1}=1$, along the high-symmetry lines delimiting the orange region of the Brillouin zone depicted in Fig. 9, for the same parameters as the corresponding cases there. The four gap closings at the Dirac $K$ points in (b) are opened in (c). (d) Profile of the lower-left corner states highlighted in Fig. 10 of the $\sqrt[4]{\text{BK}}$ triangular lattice, where the radius of the circle represents the amplitude of the wave function at the respective site and the color represents its phase, coded by the color bar below. Each state has two others degenerate in energy lying at the top and lower-right corners. The arrows indicate the site with the same component in all four states and corresponds, after two squaring and energy downshifting operations, to the lower-left corner site of the BK lattice depicted in the middle of Fig. 9.

Figure 13

Energy spectrum, in units of $t$, as a function of $k_2$, obtained from diagonalization of (d) $H_{\sqrt[4]{\text{TS}}}$, the Hamiltonian of the $\sqrt[4]{\text{TS}}$ lattice with the unit cell of Fig. 12 with $t_1=t_2=t$, open along ${\mathbf{a}}_{\mathbf{1}}$ and periodic along ${\mathbf{a}}_{\mathbf{2}}$; (b) $H_{\sqrt[4]{\text{TS}}}^{2^{\prime }}=H_{\sqrt[4]{\text{TS}}}^2-c_{2}I$, with $c_2=6\sqrt{t}$; (f) $H_{\sqrt[4]{\text{TS}}}^{4^{\prime }}=H_{\sqrt[4]{\text{TS}}}^{2^{\prime }}{H}_{\sqrt[4]{\text{TS}}}^{2^{\prime }}-c_{1}I$, with $c_1=3t$. Blue (red) bands correspond to bulk (edge) bands.

Figure 14

Zak's phase along ${\mathbf{a}}_{\mathbf{1}}={\widehat{e}}_x$ as a function of $k_2$ for the lowest bulk band of the TS model (solid blue dots) and of the $\sqrt{\text{TS}}$ model (open red dots).