Fig. 4: Exciton-polaron magnetic interactions at different hole fractional fillings.

a σ⁻- (blue) and σ⁺-resolved (red) ΔR/R₀ spectra at representative hole ν values under a B = 1 T. b B-field-dependent Zeeman splitting of \({{\rm{AP}}}_{{{{\rm{W}}}}}^{+}\) from −1 to 1 T at representative hole ν values. c Evolution of the g-factor of \({{\rm{AP}}}_{{{{\rm{W}}}}}^{+}\) as a function of the hole ν extracted from linear fits of ΔE in the range ∣B∣ ≤ 1 T (solid lines in panel b). d Valley Zeeman splitting of the \({{\rm{AP}}}_{{{{\rm{W}}}}}^{+}\) resonance at ν = −1 for different temperatures. e Evolution of the measured g-factor of \({{\rm{AP}}}_{{{{\rm{W}}}}}^{+}\) as a function of temperature for ν = −1 in the temperature range for which the oscillator strength and linewidth of \({{\rm{AP}}}_{{{{\rm{W}}}}}^{+}\) are sufficient to enable a reliable estimate of the Zeeman splitting. The red solid line represents a fit of the experimental data (black dots) to a Curie–Weiss law from which the Weiss constant is estimated to be θ = −4.6 ± 0.9 K. The negative sign of the extracted Weiss constant reveals antiferromagnetic ordering of neighbouring hole spins. f Theoretical prediction of the ν dependence of the g-factor based on a model that solves the Heisenberg Hamiltonian for charge-ordered hole states with antiferromagnetic exchange interactions between nearest neighbour spins. For all panels the error bars represent 68% confidence intervals.