Effects of Berry curvature on ideal fractional Chern insulator many-body gaps

Abstract

We investigate the many-body ground states in a family of fractionally-filled bands where the Berry curvature fluctuations can be tuned while maintaining ideal quantum geometry. We numerically find that the neutral gap of the fractional Chern insulator (FCI) ground state decreases as the Berry curvature becomes less homogeneous, ultimately driving an instability to a charge density wave. We further extend our analysis to bands perturbed away from the ideal limit and give examples where a less ideal band geometry results in a more stable FCI phase. To explain our findings, we apply the single mode approximation to the ground state wave functions of the ideal band, from which we obtain analytic expressions for the magnetoroton minimum. Finally, we make a connection between our results and experimentally relevant systems where FCIs have been observed.

Figures (14)

• Figure 1: (a)-(b) Many-body spectra at filling $\nu=1/3$ for an AC band with $B_1=-1$ and $B_1=+1$, respectively. (c) Dependence of the magnetoroton gap $\Delta$ at filling $\nu=1/3$ (red) and Berry curvature standard deviation $F_{\Omega}$ (see Eq. \ref{['eq:BC_Stdev']}) of the AC band (blue) as a function of $B_1$. (d) Berry curvature distribution of the AC band as a function of $B_1$. Calculations were performed for systems with 27 unit cells.
• Figure 2: (a) Magnetoroton gap $\Delta$ as a function of $U_1$ and $B_1$; $\Delta\le 0$ indicates that the FCI is not the ground state. The red line indicates the parameters obtained from applying the adiabatic approximation to a continuum model of twisted MoTe$_2$FCI_DiXiao for twist angles $\theta\sim 2.5^{\circ}-5.0^{\circ}$. The black line traces the minimum value of $F_{\Omega}$. (b) Band width $W$, (c) quantum weight $\mathcal{K}$ (see main text) and (d) Berry curvature standard deviation for adiabatic bands, as a function of $U_1$ and $B_1$. The white line in (d) indicates the trajectory of the minimum value of $F_{\Omega}$ and is also plotted as a black solid line in (a). The optimal lines in (b)-(c) correspond to the axis $U_1=0$.
• Figure 3: Band structures including the top 4 bands of (a) AC model with $B_1 = 1/6$, and (b) adiabatic model with $B_1 = 1/6$, $U_1 = 0.2$. $E_{\bm k}$ is in units of $\omega_c$.
• Figure 4: Band structure and Berry curvature distribution for (a) a continuum model for MoTe$_2$FCI_DiXiao, (b) the adiabatic approximation morales-duran2024magicAharonovCasher_TMD to the continuum model including 25 harmonics in the periodic functions $B(\bm r)$ and $U(\bm r)$, and (c) the adiabatic approximation where both $B(\bm r)$ and $U(\bm r)$ are truncated to only the first harmonic, which corresponds to the model studied in the present work. We have taken $\omega=23.8$ meV FCI_DiXiao as the unit of energy and present plots as a function of twist angle. The relation between the cyclotron gap and the twist angle is $\hbar\omega_c=(4\pi^2\hbar^2/\sqrt{3}ma_0^2)\theta^2$morales-duran2024magicAharonovCasher_TMD, with $a_0$ the MoTe$_2$ monolayer lattice constant and $m$ the effective mass.
• Figure 5: Many-body gap obtained from ED calculations for (a) the continuum model of MoTe$_2$FCI_DiXiao, (b) the adiabatic approximation to the same continuum model including 25 harmonics in the periodic functions $B({\bm r})$ and $U({\bm r})$, and (c) the adiabatic approximation with $B({\bm r})$ and $U({\bm r})$ truncated to the first harmonic. Both filling fractions $\nu=1/3$ and $\nu=2/3$ are shown and we have used $\varepsilon=20$ for all calculations.