Half state at $\nu_{tot}$ = -1/2 and its transition in Decoupled Twisted Double Bilayer Graphene

Technical Summary: Half State at $\nu_{tot} = -1/2$ and its Transition in Decoupled Twisted Double Bilayer Graphene

Problem Statement
The origin of fractional quantum Hall states at even-denominator fillings, particularly at $\nu = 1/2$, in double-layer quantum Hall systems has been a subject of decades-long debate. The nature of these states is theorized to depend on the competition between intralayer and interlayer Coulomb interactions, which is governed by the interlayer distance ($d$) and the magnetic length ($l_B$). In the strong coupling regime ($d \ll l_B$), the half-filling state is predicted to be either a two-component (2C) Abelian Halperin-Laughlin state (e.g., $\Psi_{331}$) or a one-component (1C) non-Abelian state (e.g., Moore-Read/Pfaffian). Previous experimental efforts in double-layer systems (such as GaAs quantum wells or double-layer graphene separated by thin dielectrics) have faced challenges in achieving the necessary strong coupling without inevitable interlayer tunneling or in controlling the charge distribution symmetry between layers.

Methodology
The authors performed magnetotransport measurements on a device fabricated from decoupled twisted double bilayer graphene (TDBG). The device consists of two Bernal-stacked bilayer graphene (BLG) layers twisted at a large angle ($\approx 8^\circ$). This specific geometry is chosen because the large twist angle ensures a mismatch of Dirac cones in reciprocal space, rendering interlayer tunneling negligible while maintaining a sub-nanometer interlayer distance. This setup provides a "decoupled" double-layer system where the interlayer distance is small enough to support strong Coulomb interactions without the complications of tunneling.

The device was encapsulated in hexagonal boron nitride (hBN) and equipped with top and bottom gates. The researchers measured longitudinal resistance ($R_{xx}$) and Hall resistance as a function of:

• Total carrier density ($n$) and displacement field ($D$).
• Total filling factor ($\nu_{tot}$) and displacement field ($D$).
• Temperature (ranging from 230 mK to 1 K).
• Magnetic field (up to 10.5 T).

Key Results

1. Observation of Fractional States:
   The study observed fractional quantum Hall states at both odd and even denominator fillings on the hole-doped side.

   • Jain Sequences: Near $\nu_{tot} = -2$ and $-3$, the authors observed states following Jain's sequence ($\nu = -n/(2n+1)$), consistent with a single-component composite fermion model.
   • Flux Series: Hierarchies of 2-flux and 4-flux composite fermion states were identified under non-zero displacement fields, consistent with previous graphene studies.
   • Layer Combinations: By analyzing the integer Landau level crossings, the authors determined the specific filling factor combinations in the top and bottom BLG layers for various total filling factors (e.g., $\nu_{tot} = -5/2$ corresponds to $\nu = -3/2$ in one layer and $\nu = -1$ in the other).

2. The Half-Filling State at $\nu_{tot} = -1/2$:

   • Zero Displacement Field: A robust fractional state was observed at $\nu_{tot} = -1/2$ specifically at zero displacement field ($D=0$). The authors attribute this to a two-component (2C) interlayer coherent Halperin-Laughlin ($\Psi_{331}$) state. This state is sensitive to the symmetry of charge distribution between the layers; at $D=0$, the layers are balanced, favoring the 2C state.
   • Finite Displacement Field: When a finite displacement field is applied, the charge distribution becomes imbalanced (polarized to either the top or bottom layer). Under these conditions, the system transitions to a one-component (1C) non-Abelian state.
   • Temperature Dependence: The state at $D=0$ shows strong dependence on the displacement field and temperature. As temperature increases, the $D=0$ state transitions from a fractional Hall state to an insulating state (characteristic of Landau level crossings), whereas the state at non-zero $D$ remains robust at higher temperatures, consistent with the larger energy gaps expected for non-Abelian states.

3. Other Fractional States:

   • At $\nu_{tot} = -1/3$ and $D=0$, an excitonic ground state (analogous to the $(333)$ state) was identified, attributed to interlayer coherent excitons formed by fractional quasiparticles.
   • Transitions between different fractional states (e.g., from Jain states to even-denominator states) were observed as the displacement field was tuned.

Significance and Claims
The paper claims that the decoupled twisted double bilayer graphene system serves as a highly tunable platform for studying double quantum Hall physics. The primary contribution is the experimental observation of the transition from a two-component Abelian state to a one-component non-Abelian state driven specifically by the displacement field.

• Tunability: Unlike previous methods that utilized in-plane magnetic fields or carrier density tuning in GaAs quantum wells, this work demonstrates that the displacement field in TDBG can effectively control the charge distribution symmetry, thereby switching the ground state nature of the $\nu = -1/2$ filling.
• Ground State Identification: The observation supports the theoretical prediction that the $\nu = -1/2$ state in a balanced double-layer system is the 2C Halperin state, which evolves into a 1C non-Abelian state upon layer polarization.
• Platform Potential: The results suggest that decoupled twisted graphene systems are promising candidates for constructing and investigating non-Abelian coherent states, which are of significant interest for topological quantum computing.

The authors remain modest regarding the direct application of these states, framing the work as a fundamental step toward understanding the ground states at half-filling in double quantum Hall systems and providing a new pathway for controlling topological phase transitions.