A Unifying Framework for Fractional Chern Insulator Stabilization

Abstract

We present a theory of fractional Chern insulator stabilization against charge-ordered states. We argue that the phase competition is captured by an effective interaction range, which depends on both the bare interaction range and quantum geometric properties. We argue that short effective interaction ranges stabilize fractional states while longer-range interactions favor charge-ordered states. To confirm our hypothesis, we conduct a numerical study of the generalized Hofstadter model using the density matrix renormalization group. Our theory offers a new interpretation of the geometric stability hypothesis and generalizes it, providing a unifying framework for several approaches to fractional phase stabilization. Finally, we propose a route towards experimental verification of the theory and possible implications for fractional states in bands with higher Chern numbers.

Figures (5)

• Figure 1: Proposed qualitative phase diagram. $V$ is the interaction energy scale, $W$ the bandwidth, and $L_{\textnormal{eff}}$ the effective interaction range. FCI and CDW stand for fractional Chern insulator and charge density wave, respectively. Phase boundaries present trends with $L_{\textnormal{eff}}$ and $V/W$ and do not necessarily run parallel to the axes. The transition length, $L_{\textnormal{eff}}^*$, is not universal.
• Figure 2: Effective interactions in real space for different LLs. The bare interaction is taken to be $V_q \propto \tanh{qd} / q$ with $d = 5\ell_B$, motivated by screening in double-gated devices. In the inset, $L_{\textnormal{eff}}$ vs. LL index $N$, defined such that $\ln \left(\tilde{V}\left(L_{\textnormal{eff}}\right) / \tilde{V}\left(0\right)\right) = -2$. The dashed gray line presents a linear fit of $L_{\textnormal{eff}}^2 \left(N\right)$, as suggested by the scaling of $\sqrt{\textnormal{Tr} g} \sim \sqrt{N}$.
• Figure 3: (a) Hall conductivities and maximal Bragg peaks in the ground state for different values of $t_2$ and $n_{\textnormal{NN}}$ obtained using iDMRG. Calculated with $V_0 \left(t_2 = 0\right) = 30$ and a unit cell of dimensions $\left(24, 9\right)$. (b) Charge pumping for $t_2 = 0$ and, and $n_{\textnormal{NN}} = 1, 3$. $\Delta P$ is the charge polarization relative to the $\phi = 0$ ground state, $\phi$ is the threaded flux and $\phi_0$ the flux quantum. In gray, slopes of $0$ and $1/3$ for comparison.
• Figure 4: Band properties of the generalized Hofstadter model as a function of the lattice momentum $\boldsymbol{k}$ for $t_2/t_1 = 0.0, 0.2, 0.4$, calculated in the Landau x gauge ($B_y = B$).
• Figure 5: Charge order indicators for $t_2 = 0.1, 0.2$ and $n_\textnormal{NN} = 3$ of Fig. \ref{['fig: main results']}. Left, Average density-density correlation, calculated over 3 DMRG unit cells. Right, corresponding normalized structure factors. Found Bragg peaks are circled.