Exact Parent Hamiltonians for All Landau Level States in a Half-flux Lattice

Abstract

Realizing topological flat bands with tailored single-particle Hilbert spaces is a critical step toward exploring many-body phases, such as those featuring anyonic excitations. One prominent example is the Kapit-Mueller model, a variant of the Harper-Hofstadter model that stabilizes lattice analogs of the lowest Landau level states. The Kapit-Mueller model is constructed based on the Poisson summation rule, an exact lattice sum rule for coherent states. In this work, we consider higher Landau-level generalizations of the Poisson summation rule, from which we derive families of parent Hamiltonians on a half-flux lattice which have exact flat bands whose flatband wavefunctions are lattice version of higher Landau level states. Focusing on generic Bravais lattices with only translation and inversion symmetries, we discuss how these symmetries enforced gaplessness and singular points for odd Landau level series, and how to achieve fully gapped parent Hamiltonians by mixing even and odd series. Our model points to a large class of tight-binding models with suitable energetic and quantum geometries that are potentially useful for realizing non-Abelian fractionalized states when interactions are included. The model exhibits fast decay hopping amplitudes, making it potentially realizable with neutral atoms in optical lattices.

Figures (2)

• Figure 1: Energy spectrum plotted under standard Landau gauge, and hopping amplitudes of $H_1$ and $H_2$ on rectangular lattice with aspect ratio $|\bm a_2| = 1.2 |\bm a_1|$. The $\tilde{\bm b}_{1}=\bm b_1/2$ and $\tilde{\bm b}_2 = \bm b_2$ are reciprocal vectors spanning the reduced Brillouin zone. (a), (b) are energy bands for $H_{1}$ and $H_{2}$, respectively. Their ground states form exact flat bands. The odd series $\hat{H}_{2n+1}$ have symmetry enforced quadratic band touching points. (c), (d) are the corresponding energy bands along high symmetric points. For $H_1$, it is gapless at high symmetry points $\bm{k}_{*}{=}(0,0)$ and $\tilde{\bm{b}}_2/2$ due to symmetries. Figure (e), (f) are the hopping amplitudes $\vert J(\bm d) \vert$ for $H_{1}$, $H_2$, respectively. The hoppings $J(\bm d)$ are defined in $H = \sum_{\bm r,\bm d\in\Lambda} J(\bm d) |\bm r-\bm d\rangle\langle\bm r|$. They have a concentrated distribution around the origin and decay fast away from the origin. The hopping amplitudes beyond NNN order are nearly negligible.
• Figure 2: The evolution of band properties with respect to the mixing $\lambda$ and the hopping truncation $\xi$. The truncation sets the upper bound of the hopping range such that we only retain hopping terms $J(m\bm a_1+n\bm a_2)$ with $\vert m \vert {+} \vert n \vert {\le} \xi$ in model Eqn. (\ref{['def_lambda_model']}). (a) Band gap, which is defined as the minimal energy of the higher band minus the maximal energy of the lower band. (b) Bandwidth of the lower band. (c), (d) Chern number and the integration of the trace of metric tensor. While the exact model with $\xi = \infty$ is fully gapped with a $C = 1$ lower band whenever $\lambda \neq 0$, the truncated model with finite $\xi$ has topological transition at finite $\lambda_c$ (marked with dash lines) where gap closes and reopens. The critical $\lambda_c$ moves towards $0$ when $\xi$ increases.