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

TL;DR

This work develops a microscopic theory of dynamic nuclear spin polarization (DNP) in quantum Hall systems, focusing on current-driven spin flips at domain walls between polarized and unpolarized fractional quantum Hall liquids near the ν=2/3 spin transition. By combining hyperfine-induced spin-flip tunneling with Kubo-formulated transport and bosonized edge theories, the authors derive steady-state nuclear polarization, the associated Overhauser field, and the resulting displacement and reconstruction of domain-wall paths, including an RG analysis of hyperfine couplings for the ν=2/3 edge. Key findings show that DNP can polarize most nuclei in a quantum point contact or along a domain wall, with the Overhauser field capable of reversing spin gaps, shifting phase boundaries, and producing asymmetric, current-driven domain-wall motion; under certain conditions this mechanism also provides a route to parafermion zero modes when proximitized by a superconductor. The results offer a comprehensive, quantitative framework linking nuclear spin dynamics to edge-state transport and topological phenomena in quantum Hall devices, with direct relevance to experiments reporting current-induced nuclear polarization, domain-wall displacement, and time-dependent resistance. Overall, the paper advances understanding of spin physics in the QHE and opens avenues for electrically controlling nuclear spins and emergent topological excitations.

Abstract

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

Figures (9)

• Figure 1: Quantum Hall effect spin transitions, formation of the domain wall between polarized P and unpolarized U regions, and charge carriers and nuclear spins flip-flop transitions. (a): The FQHE system is divided into two regions with different polarization states, due to two top electrostatic gates Wang2021, with a domain wall separating two regions. Charge carriers are shown as blue circles, edge modes with spin up are red-colored, the edge mode with spin down is blue-colored. Besides the spin-conserving current flowing between P and U regions along the edge of the sample, additional contribution to current emerges due to mutual spin flip of nuclei (green circles) and the spin of an interacting electron in the domain wall region; (b) The spectrum of composite fermions in the interior of the 2D system for electron filling factor $\nu=2/3$. At the transition magnetic field $B_t$, $\Lambda_{1,\downarrow}$ level crosses $\Lambda_{2,\uparrow}$ level. The two lower fully filled states have the same spin polarization at $B<B_t$ and opposite spin polarization at $B>B_t$, resulting in polarized and unpolarized phases, correspondingly; (c) the energy separation between $\Lambda_{1,\downarrow}$ and $\Lambda_{2,\uparrow}$ levels in the interior of $P$ and $U$ regions, chosen to be of equal value D on two sides. Within the domain wall between two regions the separation decreases and vanishes when the order of $\Lambda_{1,\downarrow}$ and $\Lambda_{2,\uparrow}$ levels inverts, for a given $B_t$, as in panel b; (d) nuclear spins leading to correlated electron and nuclear spin flip-flop transitions.
• Figure 2: The RG flow of hyperfine interaction between nuclear spins and electron spins for the $\nu=1/3$ FQH edge state. Only a positive part of coupling is considered, as it occurs in the GaAs system. Stable fixed points are $(0,0)$ to $(0, 2)$ on the $\rho_eA_z$ axis, and $(\infty,\infty)$. Unstable fixed points form the line from $(0,2)$ to $(0,\infty)$ on the $\rho_eA_z$ axis. In the case of $\rho_e A<2$, when energy scale approaching $T_\text{kondo}$, the RG flows to fixed points in the $(0,2)$ region on the $\rho_eA_z$ axis, resulting in an irrelevant spin flip tunneling process. However, as discussed in the text, hyperfine coupling remains effective on energy scales that are relevant experimentally. In the case $\rho_e A>2$, the RG flows to a fixed point at $(\infty,\infty)$.
• Figure 3: The dependence of the nuclear spin polarization on temperature and bias at $1/K=3$, $B=4T$, $u=3.5*10^7$cm/s, $\tau=60$s, $\delta=10^{-6}$cm and $A_0=90\mu$eV at different temperatures in the interval between $20\text{mK}$ to $100\text{mK}$. The horizontal line is the equilibrium nuclear spin polarization under $B=4$T at $20$mK.
• Figure 4: The propagation of dynamic nuclear spin polarization and a domain wall shift. (a) (right panel) The displacement of the level crossing due to current-induced nuclear spin polarization within the domain wall towards the boundary of the original domain wall with unpolarized nuclei. In the domain wall with unpolarized nuclei (left panel), energy separation $D$ between filled and unfilled composite fermion level is assumed equal. Current-induced spin polarization produces energy separation $D$ within the domain wall, compensating the separation in the part of the domain wall adjacent to the injector, increasing the separation in the part adjacent to collector and leading to inverted order of filled levels at the original crossing point compared to the left half of the device. Box approximation is assumed; no nuclei are polarized outside the domain wall strip; (b) The displacement of the domain wall beyond the box approximation. Solid lines show crossing levels in the domain walls with unpolarized nuclei. Current-induced nuclear spin polarization produces maximal spin splitting of levels at the original level crossing. Due to the Overhauser field leading to decrease in energy separation on the injector side, the domain wall is displaced, shown by displaced dashed lines for composite fermion levels; (c,d,e,f) Stages of the domain wall displacement. (c) Nuclear spins polarized in the domain wall; (d) The current path follows the shift of the level crossing and shifts towards the boundary of the initial domain wall;(f) Nuclear spins are polarized in the adjacent area of the injector (beyond the box approximation) reducing energy separation between levels or inverting order of filled levels due to the Overhauser field. (e) the domain wall moves further towards the left edge of the injector region.
• Figure 5: Maximum edge length with respect to initial applied voltage $V(0)$, edge channel width $\delta$, spin gap D and edge velocity $u$. For the length dependence on any of the parameters, values of other parameters are taken from the set $D=60\mu$eV, $\delta=100$Å, $V(0)=20\mu$V, $u=3.5*10^7\text{cm/s}$ and $1/\beta=2\mu$eV.