Topological Edge States with Zero Hall Conductivity in a Dimerized Hofstadter Model

The Hofstadter model is a simple yet powerful Hamiltonian to study quantum Hall physics in a lattice system, manifesting its essential topological states. Lattice dimerization in the Hofstadter model opens an energy gap at half filling. Here we show that even if the ensuing insulator has a Chern number equal to zero, concomitantly a doublet of edge states appear that are pinned at specific momenta. We demonstrate that these states are topologically protected by inversion symmetry in specific one-dimensional cuts in momentum space, define and calculate the corresponding invariants, and identify a platform for the experimental detection of these novel topological states.

Figure 1

Scheme of the hopping amplitudes in the dimerized Hofstadter model in the Landau gauge. All plaquettes of size $a\times b$, partially indicated by green rectangles, are penetrated by a magnetic flux of $\varphi =\alpha {\varphi }_0$, where ${\varphi }_0=hc/e$ is a magnetic flux quantum. In addition, we show a magnetic unit cell for the case $\alpha =1/4$ (red dashed rectangle). Note that the unit cell is penetrated by exactly one flux quantum ${\varphi }_0$.

Figure 2

Band structures for the dimerized Hofstadter model in a ribbon geometry of width $W=4N_{x}a$: $\alpha =1/4$, $N_x=100$. Parameters are (in units of $t_x$) (a) $t_y=0.5$, $\delta t=0$ (no dimerization), (b) $t_y=0.5$, $\delta t=0.4$ (trivial dimerization), (c) $t_y=0.5$, $\delta t=-0.4$ (nontrivial dimerization). States localized at the edges of the system are highlighted in red. Hall conductivities $\sigma$ for Fermi levels inside the bulk energy gaps are displayed in units of $e^2/h$. Relevant inversion-symmetric AAH cuts are indicated by dashed vertical lines. Note that in (c) there are nontrivial edge states at half filling although the corresponding Hall conductivity is zero. To explain this, we show in (d) a half-filling $t_y\text{−}\delta t$ phase diagram for the inversion-symmetric AAH model with $k_{y}b=-\pi /4$ and with respect to the 1D invariant $\mathcal{N}$ of Eq. (3). Points corresponding to (a)–(c) are indicated by red circles, including a possible path connecting them.

Figure 3

Bulk band structures for inversion-symmetric AAH models with and without dimerization: $\alpha =1/4$, $t_y=0.5t_x$; $\delta t=0.4t_x$ in (a) and (d), $\delta t=\delta t_c=-1/8t_x$ in (b) and (e), $\delta t=-0.4t_x$ in (c) and (f). The parities at the inversion-invariant momenta $k_x=0$ and $\pi /4a$ are indicated by green circles ($\zeta =+1$) and red squares ($\zeta =-1$). We also display the topological invariants $\mathcal{N}$ corresponding to Fermi levels inside the respective bulk energy gaps.