Interacting Hofstadter Interface

Two-dimensional topological insulators possess conducting edge states at their boundary while being insulating in the bulk. We investigate the edge state emergent at a smooth topological phase boundary of interacting fermions within a full real-space analysis of the time-reversal invariant Hofstadter-Hubbard model. We characterize the localization of the edge state and the topological phase boundary by means of the local compressibility, the spectral density, a generalized local spin Chern marker as well as the Hall response and find good agreement between all these quantities. Computing the edge state spectra at the interface we observe robustness of the edge state against fermionic two-body interactions and conclude that interactions only shift its position. Hence the bulk-boundary correspondence for the interacting system is confirmed. Since experimental probing of edge states remains a challenge in ultracold atom setups, we propose the detection of the local compressibility by measuring correlations with a quantum gas microscope.

Figure 1

Schematic setup: Hamiltonian Eq. (1) in cylinder geometry. See text for details.

Figure 2

Subfigure (c): momentum-integrated spectral density ${\tilde{\rho }}_x^{\sigma }$ and local compressibility ${\tilde{\kappa }}_x^{\sigma }$ in the virtual spin basis Eq. (6) as functions of the lattice site index $x$ for interaction strengths $U=0$ and $U=2$ and staggering parameters ${\lambda }_L=0$ and ${\lambda }_R=2.5$. Edge state of the interface shown in (a) for $U=0$, computed from the noninteracting spectral function of the blue region marked in (c), and in (b) for $U=2$, computed from the analytically continued interacting spectral function of the orange region in (c).

Figure 3

Local compressibility computed from RDMFT results as a function of the site index $x$ and the interaction strength $U$ for three different sets of staggering parameters ${\lambda }_L$ and ${\lambda }_R$. The red dashed lines in (c) denote the interaction strengths $U=0$ and $U=2$, which are also used in Fig. 2 and Fig. 4.

Figure 4

Particle density $n_x^{\uparrow }(k_y)$ as function of site index $x$ and momentum $k_y$ for ${\lambda }_L=0$, and ${\lambda }_R=2.5$ for (a) $U=0$ as well as (b) $U=2$ corresponding to the two red dashed lines in Fig. 3.

Figure 5

Local Chern marker averaged over spatial dimension $y$ and smoothed over three lattice sites along the $x$ direction as a function of lattice site $x$ and interaction strength $U$ for ${\lambda }_L=0$ and ${\lambda }_R=2.5$. Red stars denote the maxima of the compressibility in Fig. 3.