Singlet-Triplet Transition of Rotating Moiré Atoms in a Magnetic Field

Abstract

Moiré superlattices can bind several electrons to each potential minimum, forming few-electron "moiré atoms." Using neural quantum states that obey magnetic periodic boundary conditions, include full Landau-level mixing, and treat both spin sectors, we map the ground state of two-electron moiré atoms in a superlattice as a function of moiré depth and magnetic field strength. At strong field each atom is a spin-triplet maximum-density droplet with canonical angular momentum $\langle L_z\rangle=-1$ and a persistent circulating current: the rotating counterpart of the "Wigner molecular crystal" recently imaged in moiré superlattices. Deepening the wells drives a singlet-triplet transition to non-rotating spin-singlet atoms whose only current is a weak diamagnetic counter-circulation. The transition moves to deeper wells as the field grows, so depth and field together open a window for the formation of rotors. This is the collective analog of the singlet-triplet transition of two-electron quantum dots. At the transition, the orbital magnetic moments of the atoms reverse sign as their spin polarization switches, a joint signature accessible to magnetometry. If the atoms are instead polarized by other means, such as magnetic proximity coupling, the rotors lock at weak field into a spontaneously ordered crystal of dimers in a pinwheel pattern.

The paper asks whether magnetic-field-induced singlet–triplet transitions of isolated two-electron quantum dots persist in interacting moiré arrays, and how orbital circulation affects collective orientational order. Variational Monte Carlo uses neural quantum states satisfying magnetic periodic boundary conditions and retaining Landau-level mixing. Calculations compare fully polarized and $S_z=0$ sectors for 24 electrons in 12 moiré cells, identifying the localized unpolarized state as a singlet. The MoSe$_2$/WSe$_2$ continuum model fixes two electrons per cell, $a_M=21.4$ nm and $r_s=15$, while varying potential depth over $0$–$15$ meV and field over $1.7$–$20.9$ T. At $B=20.9$ T ($\nu=1$), localized atoms form near $V_M=4.3$ meV and remain polarized throughout the computed depth range. Each carries canonical angular momentum approximately $-\hbar$ and a persistent circulating current. Deepening confinement favors singlets; the triplet window narrows at $10.4$ T and disappears by $5.2$ T for the chosen band $g$-factor, $g_0=2.5$. Energy crossings establish the transition, while sampled currents predict an accompanying reversal of orbital magnetic moment. Within the polarized sector at $1.7$ T, two-lobed dimers instead develop a $\sqrt3\times\sqrt3$ pinwheel pattern, attributed to interatomic Coulomb interactions competing with field-driven rotation. This state is not the ground state for the default parameters and requires additional polarization, such as magnetic proximity coupling. Results concern one density, moiré potential and finite simulation cell; phase boundaries include uncertainty from $g_0=2.5\pm0.4$. The predicted circulating states and pinwheel order remain experimentally unmeasured.