Orbital Description of Landau Levels

Abstract

The pursuit of a lattice analogue for Landau levels has been a central theme in condensed matter physics. Although the correspondence between Chern bands and the lowest Landau level has been widely studied, a lattice realization of the first Landau level remains elusive. Here we construct a minimal lattice model that provides a concrete orbital description of both the lowest and first Landau levels. Using maximally localized Wannier functions with $s$, $p_-$, and $p_+$ orbital character, we develop a three-orbital model in which the two lowest Chern bands are flat and each carries a Chern number $\mathcal{C}=1$. The band topology arises from a sequence of ideal band inversions between Wannier states at the $Γ$ and $K$ points in momentum space, establishing an adiabatic connection between the atomic insulator limit and Landau level physics. Notably, many-body exact diagonalization reveals that the non-Abelian state can appear in the half-filled first Chern band. This construction can be further generalized to realize flat Chern bands analogous to higher Landau levels. Our results provide a new perspective on lattice analogues of Landau levels and may enable the exploration of fascinating topological phenomena at elevated temperatures.

Figures (4)

• Figure 1: Band structures and adiabatic connection from atomic insulator to LL. (a)-(e) The lowest few bands of $\mathcal{H}_0$ with different normalization parameter $\lambda$, where the overlap between Bloch state and Wannier function is labeled as different colors, blue for $s$ orbital, green for $p_-$ orbital, red for $p_+$ orbital and $\mathcal{C}$ denotes the Chern number of lowest three bands. The inset in (c) shows the Brillouin zone (BZ). (f) Band structure of the minimal three-orbital tight-binding model. (g) Real space distribution of $-N(\mathbf{r})$. The parameters are $\mathcal{J}/(\hbar^2/ma^2)=52\pi^2$, $\alpha=1$ and $N_0=0.28$.
• Figure 2: The MLWFs of the lowest few bands in Fig. \ref{['fig1']}(e) with $s$, $p_-$, $p_+$ orbital characteristics, and the white hexagon labels Wigner-Seitz cell.
• Figure 3: Distribution of Berry curvature $\mathcal{B}(\mathbf{k})$ and trace of Fubini–Study metric $\text{Tr}[g(\mathbf{k})]$ for the two lowest bands of both continuum model and tight-binding model across the Brillouin zone (BZ). The BZ is outlined by dashed hexagon. The bands derived from the continuum model are designated as LLL and 1LL, reflecting their highly ideal character and close resemblance to flat LLs. In contrast, the bands from the tight-binding model are labeled LCB and 1CB. The Berry curvature fluctuation $\delta\mathcal{B}$ is noticeably larger in LCB and 1CB compared to their LLL and 1LL counterparts, resulting in a stronger deviation of the trace condition measure $\mathbb{T}$ from the ideal value.
• Figure 4: Exact diagonalization and PES for fractional filled CB. (a) Low energy many-body energy spectrum for $1/3$ filled LCB with $N_{\text{uc}}=24$. (b) PES with $N_A=3$ for the three degenerate ground states in (a), where $\rho=(1/d)\sum_{i}^{d}\left|\Psi_i\right>\left<\Psi_i\right|$. $d$ is the degeneracy of ground states. (c), (d) Energy spectrum for $1/2$ filled 1CB with $N_{\text{uc}}=26$ and $N_{\text{uc}}=28$. (e), (f) PES with $N_A=3$ for the six degenerate ground states in $N_{\text{uc}}=20$ and $N_{\text{uc}}=28$. Here we only show the lowest energy per momentum sectors in addition to the degenerate ground state.