Dynamic nuclear spin polarization in the fractional quantum Hall effect spin transitions

Experiments suggest that nuclear spins play a significant role in the quantum Hall effect (QHE) near integer and fractional QHE spin transitions, but many of these phenomena still remain to be understood. Here we study theoretically the dynamic nuclear polarization (DNP) in the two-dimensional electron liquid near a quantum point contact (QPC) or a domain wall between spin polarized and unpolarized phases induced by electrostatic gating in the fractional QHE at a filling factor 2/3 and analyze the dependence of the spin transition on temperature and the magnitude of the flowing current. We demonstrate that nearly all nuclear spins in the QPC or in the domain wall can be polarized by the electric current. The Overhauser effective magnetic field from the DNP can be strong enough to induce (or modify) a phase transition between polarized and unpolarized phases. This changes the gate voltages and magnetic fields required for the spin transitions, and leads to the reconstruction of the boundary between two phases and a domain wall and a current path displacement. The spread of nuclear spin polarization and the domain wall displacement are strongly asymmetric with respect to the direction of the current flow. Equilibration due to hyperfine interactions and its role on the nuclear spin polarization, domain wall displacements and spin transitions is studied. Back and forth oscillatory transitions between polarized and unpolarized phases near a source contact are discussed. Hyperfine interactions of nuclear spins provide a route for observation and control of the parafermion zero modes that can be induced when the domain wall between the polarized and unpolarized regions is placed in the proximity of a superconductor

💡 Research Summary

The paper presents a comprehensive theoretical study of dynamic nuclear spin polarization (DNP) in the fractional quantum Hall (FQH) regime at filling factor ν = 2/3, focusing on two experimentally relevant geometries: a quantum point contact (QPC) and a domain wall separating spin‑polarized and spin‑unpolarized regions created by electrostatic gating. The authors begin by outlining the importance of hyperfine coupling between electron and nuclear spins in quantum Hall systems, noting that near spin‑transition points the energy conservation condition allows simultaneous electron‑spin flips and nuclear‑spin flips (flip‑flop processes).

In the QPC model, left‑moving spin‑up edge states and right‑moving spin‑down edge states are spatially separated. Hyperfine interaction enables an electron to tunnel from one edge to the opposite‑spin edge while flipping a nearby nuclear spin, thereby linking the net tunneling current directly to the rate of nuclear polarization. By employing a second‑quantization formalism and assuming isotropic contact hyperfine coupling, the authors derive a rate equation for the average nuclear magnetization ⟨M_z⟩. The equation shows that a bias voltage (or a non‑zero temperature) creates an imbalance between forward and backward flip‑flop processes, leading to a steady‑state Overhauser field proportional to the current.

For the ν = 2/3 FQH state, the authors adopt a K‑matrix description of composite‑fermion edge modes. The Zeeman energy of composite fermions competes with the cyclotron energy, producing a crossing of the Λ₁↓ and Λ₂↑ levels at a critical magnetic field B_t. Near this crossing, the hyperfine‑mediated flip‑flop processes become resonant. A renormalization‑group analysis demonstrates that, at experimentally relevant temperatures (≈20 mK) and current densities (nA range), the hyperfine coupling remains strong and does not flow to zero. Consequently, the Overhauser field generated by DNP can reach several tesla, sufficient to shift the effective spin‑transition point dramatically.

The Overhauser field modifies the local energy gap between the two composite‑fermion levels, causing the domain wall to move in the direction of the current. Nuclear spin diffusion, driven by dipole‑dipole interactions, spreads the polarization away from the QPC or domain wall, further displacing the boundary between polarized and unpolarized regions. This displacement is highly asymmetric: the “injector” side (where current enters the domain wall) experiences a much larger shift than the “collector” side. As a result, the current path is reconstructed, explaining experimentally observed hysteresis and conductance anomalies.

When the moving domain wall reaches the source contact, the local spin configuration can oscillate between polarized and unpolarized states, producing back‑and‑forth transitions that manifest as time‑dependent conductance fluctuations. The frequency and amplitude of these oscillations depend on the nuclear spin relaxation time, the strength of hyperfine coupling, and the applied bias.

In the final sections, the authors discuss the implications of DNP‑induced Overhauser fields for topological quantum computation. If the domain wall is placed in proximity to a superconductor, the strong, locally controllable Zeeman field can stabilize parafermion zero modes—non‑Abelian excitations that generalize Majorana fermions. Although the calculations assume nuclear spin‑½, the authors argue that the qualitative picture remains valid for the spin‑3/2 nuclei of GaAs, with additional quadrupolar effects only modestly modifying the dynamics.

Overall, the work provides a microscopic, quantitative framework linking electric current, hyperfine‑mediated nuclear spin polarization, domain‑wall motion, and topological excitations in the ν = 2/3 fractional quantum Hall system. It predicts near‑complete nuclear polarization in the QPC or domain wall region, asymmetric spreading of the Overhauser field, current‑dependent reconstruction of edge pathways, and the possibility of electrically controlling parafermion modes via DNP—offering clear guidance for future experiments.