Extended Data Fig. 7: Possibilities for XY ferromagnetic ordering enhancement.

a, Simulated OAT amplification under dipolar XXZ Hamiltonians \({\widehat{H}}_{ij}\propto \frac{1}{{r}_{ij}^{3}}({\widehat{\sigma }}_{i}^{x}{\widehat{\sigma }}_{j}^{x}+{\widehat{\sigma }}_{i}^{y}{\widehat{\sigma }}_{j}^{y}+\Delta {\widehat{\sigma }}_{i}^{z}{\widehat{\sigma }}_{j}^{z})\), in the case of a (111)-oriented diamond. The solid (dashed) traces correspond to 100% (80%) spin polarization. The amplification is predicted to improve as one engineer the Hamiltonian towards the Heisenberg point (Δ = 1), as expected from the XY ferromagnetic ordering mechanism⁴¹. The simulation is done considering positional disorder and finite 2D layer thickness (9 nm). For simplicity of simulation, we neglected external decoherence, which has the potential to reduce significantly in the near future (H.G. et al., in preparation). The sensing angle simulated is δθ = 5^°, and the horizontal axis is rescaled by the distance from the Heisenberg point for ease of comparison. b, The same simulation, but for (100)-orientated diamond with the engineered quantization axis η_eff. (i.e. the configuration in this work). Here, the distance from the Heisenberg point is parameterized by ϵ (smaller ϵ indicates Hamiltonian closer to the Heisenberg point, see Supplement for exact definition), as Δ is no longer a well-defined number due to its angular dependence (see panel (d)). No improvement is observed as the Hamiltonian is engineered towards the Heisenberg point. c, Dipolar interaction for a (111)-oriented diamond, similar to main text Fig. 1(c,d). When the Hamiltonian is engineered towards the Heisenberg point from the easy-plane side, the negative A^Heis leads to a tendency for XY ferromagnetic ordering. d, Same as (c), but for (100)-orientated diamond with the engineered quantization axis η_eff. The green and yellow curves are the same as main text Fig. 1(d), repeated here for convenience. The Heisenberg term (blue) shows a mixed sign that leads to magnetic frustration⁵⁴, which explains the absence of the ordering enhancement in (b).