Unconventional Fractional Phases in Multi-Band Vortexable Systems

Abstract

In this Letter, we study topological flat bands with distinct features that deviate from conventional Landau level behavior. We show that even in the ideal quantum geometry limit, moire flat band systems can exhibit physical phenomena fundamentally different from Landau levels without lattices. In particular, we find new fractional quantum Hall states emerging from multi-band vortexable systems, where multiple exactly flat bands appear at the Fermi energy. While the set of bands as a whole exhibits ideal quantum geometry, individual bands separately lose vortexability, and thus making them very different from a stack of Landau levels. At certain filling fractions, we find fractional states whose Hall conductivity deviates from the filling factor. Through careful numerical and analytical studies, we rule out all known mechanisms--such as fractional quantum Hall crystals or separate filling of trivial and topological bands--as possible explanations. Leveraging the exact solvability of vortexable systems, we use analytic Bloch wavefunctions to uncover the origin of these new fractional states, which arises from the commensurability between the moire unit cell and the magnetic unit cell of an emergent effective magnetic field.

Figures (15)

• Figure 1: Schematic showing how an FCI state with unequal filling fraction and many-body Chern number arises in multiband vortexable systems due to inhomogeneous effective magnetic field distribution $B(\mathbf{r})$ and commensurability between moiré unit cell and magnetic length. The white dashed lines show the smallest unit cells.
• Figure 2: Four exact flat bands in single layer system with QBCP under periodic strain field. (a) Band structure for the Hamiltonian in Eq. \ref{['eq:hamiltonian']} with $\tilde{A}(\mathbf{r})= -\frac{\alpha}{2}\sum_{n=1}^{3}e^{i(4-n)\phi} \cos\left(\mathbf{b}_n \cdot \mathbf{r} \right)$ showing 4 exact flat bands at the "magic" value $\tilde{\alpha} =\frac{\alpha}{|\mathbf{b}| ^{2}} = -2.88$ where $\mathbf{b}_i (i=1, 2, 3)$ are the moiré reciprocal lattice vectors shown in (b). (b) also shows the moiré unit cell, lattice vectors $\mathbf{a}_1$ and $\mathbf{a}_2$, moiré Brillouin zone (mBZ). Density plot of $|\psi_{\Gamma}(\mathbf{r})|$ (normalized by its maximum) is shown at the bottom right corner of (a). The dark spots represent zeros of $|\psi_{\Gamma}(\mathbf{r})|$. There are two zeros at the corners of the unit cell. (c) Decomposition of the four flat bands in sublattice space. Two bands which are polarized on A sublattice together have $C=1$, while the other two bands polarized on B sublattice together have $C=-1$. The two sublattices are related by time reversal symmetry $\mathcal{T}$. The two bands polarized on A sublattice can be decomposed into one $C=1$ band and one $C=0$ band. The $C=0$ band corresponds to a very localized Wannier function $w(\mathbf{r})$. (d) Density plot of $|w(\mathbf{r})|^2$, where brighter color represent larger value of the norm of the wavefunction. (e) Wilson loop spectra $\tilde{\theta}({\mathbf{k}})=\frac{\theta({\mathbf{k}})}{2\pi}$ of the $C=1$ and $C=0$ band plotted in blue and red, respectively. (f) Effective magnetic field $B(\mathbf{r})/B_0$ in the moiré unit cell for the flatbands obtained by comparing the flatband wavefunctions to lowest Landau Level (LLL) wavefunctions.
• Figure 3: Many-body energy spectra (a,c,e,g) and occupation number in the Chern ($n_C(\mathbf{k}) = \langle c^\dag_C(\mathbf{k})c_C(\mathbf{k})\rangle$) and Wannier ($n_W(\mathbf{k}) = \langle c^\dag_W(\mathbf{k})c_W(\mathbf{k})\rangle$) bands of the ground-states (averaged over the three ground-states) (b,d,f,h) at filling $\nu=1$, $4/3$, $1/3$ and $2/3$, respectively. The momentum numbers are shown in the inset in (b). The ground-states in (a,c,e,g) are marked by red circles. At $\nu=1$, there is a single insulating ground-state, whereas for each of the other filling fractions, there are three quasi-degnerate ground states. The many-body Chern numbers of the ground states are $C_\text{mb}=0$ in (a) and (e), and are $C_\text{mb}=1/3$ for (c) and (g). For $\nu=1$ and $\nu=1/3$ ground states, (b) and (f) show that the Chern band is fully empty, all electrons are in the Wanier band. For $\nu=4/3$ ground state, (d) shows the Wannier band is fully filled, and the Chern band is $1/3$ filled. For $\nu=2/3$ ground state, (h) shows that both bands are partially occupied.
• Figure 4: Phase transition between FCIs with $C_\text{mb}=2/3$ and $C_\text{mb}=1/3$ at filling fraction $\nu=4/3$ in a $p4mm$ symmetric system. The single particle Hamiltonian is given in Eqs. \ref{['eq:hamiltonian']} and \ref{['eq:p4mmStrain']}. The unit cell, mBZ, (reciprocal) lattice vectors are shown in red in (b) and (c). However, when $\beta = 0$, the primitive unit cell and corresponding mBZ are shown in blue in (b) and (c). (a) Line (in red) of "magic" parameters $(\alpha,\beta)$, at which four exact flatbands (two per sublattice) appear. The two flatbands per sublattice have total Chern number $|C|=1$. At the point where the red line crosses $\alpha$ axis ($\alpha=0.5279$ and $\beta=0$), the four flatbands (two per sublattice) corresponding to the red mBZ in (c) can be unfolded into two flatbands (one per sublattice) corresponding to the blue mBZ in (c). (d) Band structure for $\alpha=0.6651,\beta=0.7183$ containing 4 exact flatbands at $\tilde{E}=0$. At the bottom right corner, the density plot of wavefunction $|\psi_\Gamma(\mathbf{r})|$ is shown. The dark spots in the wavefunctions are the zeros. (f, g) Many-body energy spectra at filling $\nu=4/3$ for $\alpha=0.5279,\beta=0$ and $\alpha=0.6651,\beta=0.7183$ respectively. In both cases, we find three quasi degenerate ground states (encircled with red circles). We numerically find the many-body Chern numbers of these quasi-degenerate ground states to be $C_\text{mb}=2/3$ and $C_\text{mb}=1/3$ for (f) and (g), respectively. (e) Many-body gap $\Delta E$ between the lowest energy excited state and highest energy ground state (among the three quasi-degnerate ground state) as the ratio $\beta/\alpha$ is varied along red line in (a).
• Figure A1: (a) shows the diagonal terms $n_C(\mathbf{k})=\langle c^\dag_C(\mathbf{k}) c_C(\mathbf{k})\rangle$ (in blue) and $n_W(\mathbf{k})=\langle c^\dag_W(\mathbf{k}) c_W(\mathbf{k})\rangle$ (in red) of the occupation number matrix $n_{ij}(\mathbf{k})$ for the three quasi-degenerate ground states in Fig. \ref{['fig:2']}(g). The circle, square and star shaped markers correspond to the ground states at $k=0$ (zone center), $k=6$ and $k=8$ (zone corners), respectively. (b-d) show the absolute value of off-diagonal component $n_{CW}(\mathbf{k})=|\langle c^\dag_C(\mathbf{k}) c_W(\mathbf{k})\rangle|$ (in blue) and the determinant (in red) of the occupation number matrix $n_{ij}(\mathbf{k})$ for the ground-states at momenta $k=0$, $k=6$ and $k=8$ in Fig. \ref{['fig:2']}(g), respectively.