Two-Body Solution and Instabilities along Streda Lines in Moire Flat Bands

Abstract

Moire minibands in twisted homobilayer semiconductors can, under suitable approximations, be modeled as a pair of Landau levels with opposite Chern numbers. This provides a minimal model for searching novel topological states in a time-reversal-symmetric Hamiltonian. In this work, we investigate the effects of an external magnetic field in this model. We study the many-body ground state in the density-magnetic-field (n-B) plane along the dn/dB = \pm1/Phi0 Streda line with Hartree-Fock approximation. Away from charge neutrality, we find the Chern-insulating (incompressible) state is very robust while towards charge neutrality, we find a transition from incompressible phase to compressible phase as the interaction strength kappa decreases. Using time-dependent mean-field theory, we further analyze spin-flip excitations and find that the incompressible state along Streda line toward charge neutrality becomes unstable even at large kappa when magnetic field is sufficiently large. Finally, we solve the two-body problem in a given Landau level exactly where the two particles experience unequal magnetic fields using a new basis called center-of-charge basis. This basis allows any isotropic interaction to be parameterized by a single quantum number, the relative angular momentum, thereby extending the Haldane pseudopotentials to the unequal-magnetic-fields case. As the difference of the two magnetic fields varies, these pseudopotentials show a sequence of level crossings, leading to non-monotonic structure of pseudopotentials that is absent in ordinary Landau level systems. Our formulation provides a useful starting point for studying weak-field physics in moire flat bands, where magnetic Bloch-state basis becomes computationally impossible due to the large basis sizes.

Figures (6)

• Figure 1: Single particle energy-level $\epsilon_{n\sigma}^0$ in Eq. \ref{['eq:single_particle_LL']} and \ref{['eq:single_particle_LL-2']}. The applied magnetic field is aligned with Skyrmion magnetic field in valley $K$ so the Landau-level degeneracy in valley $K$ is greater than in valley $K'$. The energy offset between the two valleys is determined by the competition of Zeeman splitting and the cyclotron energy of the applied field. For $g>e\hbar/m^*c\approx 3$, the lowest energy level in valley $K$ is lower than that in valley $K'$, as shown above.
• Figure 2: (a) Schematic illustration of the two Středa lines emerging from filling fraction $\nu=-1$ in the $(\nu,B)$ plane. The solid line denotes the branch where an incompressible gap (vanishing longitudinal resistance $R_{xx}$) is observed experimentally, while the dashed line denotes the branch where no incompressible feature is observed Cai2023Zeng2023Park2023PhysRevX.13.031037. (b–c) $\kappa-\lambda$ phase diagram along the Středa line toward charge neutrality (red dashed line in panel (a)) for g-factor $g=5.3$ and $g=9$. The horizontal axis denotes the interaction strength $\kappa\approx92/(\theta\epsilon)$, and the vertical axis denotes the ratio of external magnetic field to the skyrmion field, $\lambda= B/B_{\mathrm{sky}}$. The red region corresponds to the parameter regime where the compressible state $|\Psi_{CO}^T\rangle$ is the ground state. The blue region corresponds to the regime where the incompressible state $|\Psi_{IC}^T\rangle$ is the ground state. The green region indicates that the lowest spin-reversal excitation energy of $|\Psi_{IC}^T\rangle$ becomes negative. The star on the $x$-axis indicates our estimated $\kappa\sim2.6$ for twisted MoTe$_2$ using dielectric constant $\epsilon=10$ and twisted angle $\theta=3.5^\circ$.
• Figure 3: Spin-flip exciton excitation energy (the lowest eigenvalue of the RPA equation) of the incompressible Chern insulator along the Středa branch pointing toward charge neutrality, shown as a function of $\lambda$ for fixed $\kappa$ and $g$. Including Landau-level mixing (increasing $N_{\mathrm{cut}}$) shifts the instability to smaller values of $\lambda$.
• Figure 4: Schematic diagram illustrating the magnetic-field parameter space for the two-body problem. $\mathbf{B}_1$ and $\mathbf{B}_2$ denote the magnetic fields acting on particles 1 and 2. The blue region indicates the sector in which the two fields are aligned, whereas the red region corresponds to anti-aligned field orientations. The red dashed line marks the Kallin--Halperin line, $B_1=-B_2$, where the two-body problem leads to a propagating exciton PhysRevB.30.5655. The blue dashed line corresponds to the conventional quantum Hall regime $B_1=B_2$. Our results show that all eigenstates are localized except along the Kallin--Halperin line. The black arrows indicate the direction in the $B_1$--$B_2$ parameter space along which the generalized Haldane pseudopotentials evolve, as shown in Figs. 5 and 6
• Figure 5: Generalized Haldane pseudopotentials projected onto the $n=0$ (a) and $n=1$ (b) Landau levels as a function of $\lambda$. Increasing $\lambda$ corresponds to moving perpendicular to the blue dashed line that marks the conventional quantum Hall limit in Fig. \ref{['fig:magnetic_field_phase_space']}.