Localized-basis formulation of interacting Hamiltonians in flat topological bands: coherent states and coherent-like states for fractional physics

Abstract

In topological bands, it is impossible to construct exponentially localized Wannier functions while preserving the symmetries. Instead, in quantum Hall systems, one can define an overcomplete basis of spatially localized coherent states. In this work, we propose a unified framework for understanding the quantum Hall effect and Chern insulators from the perspective of localized bases, by extending the overcomplete basis of coherent states to Chern bands in terms of coherent-like states. Specifically, by representing both coherent states and coherent-like states as wave packets defined on a band, the difference between them can be encoded solely in the functional form of the wave packet in momentum space. Furthermore, for filling factor $ν=1/3$, we define a local repulsive interaction Hamiltonian based on these bases and discuss properties of its ground states. In particular, by relating this Hamiltonian to previously studied models, we show that in quantum Hall systems it possesses exactly zero-energy ground states with topological degeneracy, thereby confirming that it serves as a model for fractional quantum Hall systems. In addition, we numerically verify that the Hamiltonian possesses topological degeneracy for representative Chern insulator models. An advantage of this formulation is that it allows fractional quantum Hall systems and various fractional Chern insulator systems to be discussed within a unified framework using the same Hamiltonian form. In addition, we discuss that coherent-like states can also be defined in $\mathbb{Z}_2$ topological insulators. Corresponding to the fermionic time-reversal symmetry of the system, Kramers-degenerate coherent-like states can be naturally defined. The localized basis constructed from coherent-like states is expected to be useful for describing strongly correlated topological phases in flat-band systems.

Figures (5)

• Figure 1: (a) Zero-modes of $PZPZ^*P$ in checkerboard lattice model. (b) The ten smallest eigenvalues of $\hat{Z}\hat{Z}^{\dagger}$, except for one quasi-zero mode. The blue line corresponds to $\lambda_n=n/\pi$.
• Figure 2: Distributions $|a_0(\bm{k})|^2$ for (a) $\tilde{\bm{r}}_1=(0.5,0),~\tilde{\bm{r}}_2=(0,0.5)$, (b) $\tilde{\bm{r}}_1=(0,-0.5),~\tilde{\bm{r}}_2=(-0.5,0)$. The counterpart in the lowest Landau level is shown in (c). The comparison between (b) and (c) is shown in (d).
• Figure 3: $\min_{\bm{k}}\log |a_{\bm{k}}(\bm{\delta},0)|$ as a function of $\bm{\delta}$ for the checkerboard-lattice model with (a) $N_x\times N_y=6\times4$ and $12\times12$, and for the QWZ model with (c)$N_x\times N_y=6\times4$ and $12\times12$. $\bm{\delta}=\bm{0}$ corresponds to $\tilde{\bm{r}}_1=(0,-0.5),~\tilde{\bm{r}}_2=(-0.5,0)$ for the checkerboard-lattice model and $\tilde{\bm{r}}_1=(0,0),~\tilde{\bm{r}}_2=(0,0)$ for the QWZ model, respectively.
• Figure 4: Exact Diagonalizations of the coherent-state model with $U(\bm{\delta})=1$ and $U(\bm{\delta})=\delta_{\bm{\delta},\bm{0}}$. Total momentum sectors are labeled by $(n_1,n_2)$.
• Figure 5: Exact Diagonalizations of the coherent-like-state Hamiltonians for (a) the checkerboard lattice model and (b) the QWZ model. $\bm{\delta}=\bm{0}$ corresponds to $\tilde{\bm{r}}_1=(0,-0.5),~\tilde{\bm{r}}_2=(-0.5,0)$ for the checkerboard-lattice model and $\tilde{\bm{r}}_1=(0,0),~\tilde{\bm{r}}_2=(0,0)$ for the QWZ model, respectively.