Incommensurate-Stabilized Fractional Chern Insulator in Alternating Twisted Trilayer Graphene

Abstract

Fractional Chern insulators (FCIs) typically emerge in topological flat bands and are regarded as lattice analogs of fractional quantum Hall states. Conventionally, the flat-band wavefunctions that support FCIs are expected to mimic the lowest Landau level, a condition that can be quantified by the quantum-geometric indicators. In realistic systems, however, FCIs often compete with lattice symmetry-breaking orders, especially when the hosting flat bands not ideal. In this work, we propose stabilizing FCIs by exploiting the intrinsic incommensurability of alternating twisted trilayer graphene, which naturally suppresses competing charge-density-wave (CDW) phase while FCIs are less effected. Within an adiabatic approximation at the supermoiré scale, the effect of incommensuration on local physics can be quantified as phase shifts of interlayer coupling. Using exact diagonalization, we compute ground states in different local patches and uncover a strikingly counterintuitive result: the FCI gap increases as the quantum-geometric indicators worsen. Within certain parameter ranges, we further identify mixed phases where FCIs coexist with CDWs, but with CDWs confined only to patches of weak incommensurability. Finally, we provide experimental protocols and discuss how incommensuration enrich the system's topology and quantum geometry. Not only do our results establish incommensuration as a robust stabilizer of FCIs, but also provide a general paradigm for exploring strong-correlation physics in incommensurate systems.

Figures (5)

• Figure 1: (a) Schematic of the A-TTG structure The (b,c) Moiré Brillouin zones (mBZs) for the two twist angles: each hexagon corresponds to the mini Brillouin zone of a single moiré lattice (red for $\theta_1$, blue for $\theta_2$). The lack of a common periodicity means there is no single unified mBZ. (d) Supermoiré scale scattering $\delta q_0$ with an expansion of the second layer graphene BZ (dashed dark red) (e) Validity of adiabatic Hamiltonian at $\mathbf{k}_\phi=\Gamma_\phi=(0,0)$ in supermoiré scale with $\Gamma_\phi$,$K_{\phi1}$ and $K_{\phi2}$ labels different patches.(f, g) Distribution of the incommensurate moiré potential at different scale SI.
• Figure 2: (a) Local band structure at $\kappa =0.6$, $\xi_v=-K$ for $\mathbf{k}_\phi=\Gamma_\phi$ and $\mathbf{k}_\phi= K_{\phi1}$ (solid/dashed) with hBN substrate $\Delta_\mathrm{hBN}=30~meV$, where supermoiré scale coordinate $\mathbf{k}_\phi$ with higher symmetries are $\Gamma_\phi=(0,0)$ and $K_{\phi 1,2}=\frac{2\pi}{3}(\pm1,\mp1)$ (See Fig. \ref{['fig:ttgdigram']}(b) and \ref{['fig:edkappa']} (a) insert). (b) Corresponding density of states (DOS). (c, d) Nearly uniform berry curvature for two valleys, the origin points is $K_2$. And we have $k_i=k_1g_{1i}+k_2g_{2i},~i=x,y$ (e, f) Trace condition for two locals where $\mathbf{g}_{i}=\mathbf{g}'_{i,1}$. (g) Wannier center (W.C.) one loop winding at $k_{\phi2}=0$ respect to both $k_2$ and $k_{\phi1}$ for valence flat bands.
• Figure 3: ED calculations for $N_x\times N_y = 4\times 6$ lattice with $8$ electron at $\kappa=0.6$. $\Delta E$ is relative energy to local ground states. $\Delta E_0$ is relative to minimum local ground states energy. $K_x$,$K_y$ are total momentum in the unit of $\mathbf{g}_{1,2}$. (a) FCI phase with 3-fold degenerated ground state at $\mathbf{k}_\phi=K_{\Phi1}$. (b) Corresponding PES where $1088$ total states under the gap for: $N_A= 3,N_B=4$. $e^{-\xi}$ labels eigenvalues of reduced density matrix. (c) Structural factor without CDW signature, where $k_x,k_y$ are momentum in reciprocal space. (d) Spectral flow of FCI phase along the loop pass high symmetry points in mBZ. (e) PES evolution with $\mathbf{k}_\phi$, showing FCI and CDW gap transition. (f) CDW ground state at $\mathbf{k}_\phi=\Gamma_\phi$. (g) CDW counting rule with $3C_8^3=168$ states under the gap. (h) Obvious CDW signature for Structural factor. (i) $\eta$ (red) and band gap towards remote band $\Delta_G$ (orange) variation with respect to Local ground states energy (purple). (j) Variation of FCI ground states (blue) and $\eta$ (red).
• Figure 4: (a) Many body energy gap $\Delta$, single particle trace condition $\eta$ and mean value of generalized trace condition $\braket{\boldsymbol{\eta}}$ at different $\kappa$. Insert: Mixture phases distribution in Wagner Seitz supermoiré unit cell at $\kappa=0.6$, $\Gamma_\phi$ and $K_{\phi 1,2}$ labels higher local symmetries points at supermoiré scale in the real space. The ED calculation is performed on the $10\times 10$ mesh in $\mathbf{k}_\phi$ space. (b) Many body gap distribution, with $R_x,R_y$ are supermoiré scale coordinates. (c) Distribution of $\eta$ (d) Ground state energy distribution. (e, f) Energy gap $\Delta$ and relative ground state energy $\Delta E_0$ distribution at different $\kappa$.
• Figure 5: (a) Hierarchy of Hilbert space using different basis to describe. A: Attach vortex in local flat bands, representing satisfaction of traditional trace condition. B: Attach vortex that violet traditional one but satisfied by our new proposed trace condition. C: Violation of both two trace conditions. (b) New trace condition numerical results in supermoire scale, representing leakage of vortices of C process.