Moiré Folded Helical States at the Interfaces of Heterostructures

Technical Summary: Moiré Folded Helical States at the Interfaces of Heterostructures

Problem Statement
Materials with intrinsically weak spin-orbit coupling (SOC), such as graphene, lack spin-momentum locking and spin-driven collective states. While van der Waals heterostructures (e.g., graphene on topological insulators) can induce proximity-induced SOC, a central unresolved question remains: does this induced SOC merely reflect a passive inheritance of the substrate's spin texture, or can it be actively enhanced and reprogrammed via interfacial structural modulations? Specifically, it is unclear how moiré superlattices, which arise from lattice mismatch or twist angles, influence the interplay between proximity-induced SOC and electronic band structures to potentially drive collective instabilities.

Methodology
The authors introduce a minimal toy model of a graphene–topological insulator (TI) heterostructure, represented as a quasi-one-dimensional two-leg ladder system.

• Model Construction: The two legs represent the active electronic states of the graphene $\pi$-orbitals and the TI topological surface states. Leg 1 is uniform, while Leg 2 undergoes a slight dimerization parameterized by $\delta$. This dimerization creates a commensurate moiré superlattice with a period determined by the ratio of integers $p/q$ (where $\delta = p/q$).
• Hamiltonian: The system is described by a tight-binding Hamiltonian in the extended Brillouin zone (EBZ) containing eight bands (accounting for leg, orbital, and spin degrees of freedom). The Hamiltonian includes:
  • Intra-leg nearest-neighbor hopping (constant in Leg 1, dimerized in Leg 2).
  • Inter-leg (rung) tunneling modulated by a long-wavelength moiré envelope $t_\perp(n)$.
  • Proximity-induced Rashba SOC, which locks spin polarization along the $y$-direction. Crucially, the SOC strength is opposite in sign for the two legs ($+\sigma_y$ for Leg 1, $-\sigma_y$ for Leg 2) to mimic the opposite helicity of the two surfaces.
• Analysis: The authors solve the energy spectrum exactly and analyze the density of states (DOS). They define a helicity operator as the difference in spin density between the two legs ($\Gamma_{hel} = S_y^{(1)} - S_y^{(2)}$) to isolate the moiré-modulated spin-momentum locking. They compute the bare helicity spectral functions and static/dynamical susceptibilities ($\chi_{hel}$) to characterize fluctuation channels, focusing on helicity rather than conventional charge or spin order.

Key Contributions and Results

1. Spectral Reconstruction and Periodicity Reduction:

   • In the absence of SOC, the system exhibits spin-degenerate, folded moiré minibands.
   • The inclusion of Rashba SOC lifts spin degeneracy and, counter-intuitively, reduces the effective spectral periodicity in the EBZ by a factor of two. This arises because the momentum-odd Rashba term breaks the equivalence of states related by the original moiré translation. The system becomes invariant only under a composite operation of a moiré translation and a $\pi$ spin rotation about the $y$-axis.
   • This symmetry reduction leads to the emergence of Dirac-like miniband crossings at finite SOC. These crossings are symmetry-protected and remain gapless, indicating that moiré heterostructures can host relativistic quasiparticles through band reconstruction even if the underlying bands are non-relativistic.

2. Helicity Fragmentation:

   • Unlike uniform Rashba systems where helicity is confined to a few branches, the moiré potential entangles spin, sublattice, and leg degrees of freedom.
   • This results in helicity fragmentation, where the helicity weight is distributed across a dense manifold of moiré minibands. Every miniband branch in the reduced Brillouin zone carries a non-zero helicity weight, forming an extended network of helicity-carrying states.
   • The helicity spectral function reveals a dense mesh of oblique streaks with alternating signs, evidencing robust spin-momentum locking across the entire spectrum.

3. Enhanced Helicity Fluctuations:

   • The authors calculate the static helicity susceptibility $\chi_{hel}(\kappa)$. They find that while helicity spectral functions vanish without SOC, the static susceptibilities become strongly enhanced when SOC and moiré hybridization coexist.
   • For specific modulation strengths (e.g., $\delta = 0.85$), the susceptibility exhibits peaks at finite wave vectors ($\pm \kappa^*$), signaling a tendency toward a helical density-wave state rather than a uniform helical phase.
   • Dynamical susceptibility calculations show significant absorption peaks at low frequencies, indicating a large phase space for particle-hole excitations that amplify helicity fluctuations.

Significance and Claims
The paper claims to provide a microscopic mechanism by which proximity-induced spin-orbit coupling can be amplified via moiré engineering. The key findings suggest that:

• Moiré modulation does not simply perturb the SOC but fundamentally reshapes the miniband structure, entangling spatial periodicity with spin-momentum locking.
• The system supports emergent helicity spectral functions and Dirac-like crossings that are robust against the specific details of the material, provided the symmetry constraints are met.
• The coexistence of SOC and moiré hybridization creates a "bare-response" environment with strongly enhanced helicity fluctuations. This establishes a foundation for the potential development of helical and spin-orbit-driven phases in van der Waals heterostructures through controlled structural modulation.

The authors emphasize that while their analysis is based on a minimal model (a quasi-1D ladder), it captures the essential physics of 2D graphene/TI interfaces (such as graphene on $Sb_2Te_3$) and predicts robust topological features observable in spectroscopic measurements. The work opens a route toward designing collective instabilities driven by helicity rather than conventional charge or spin orders.