Quantum Disordered State of Magnetic Charges in Nanoengineered Honeycomb Lattice

Abstract A quantum magnetic state due to magnetic charges is never observed, even though they are treated as quantum mechanical variables in theoretical calculations. Here, the occurrence of a novel quantum disordered state of magnetic charges in a nanoengineered magnetic honeycomb lattice of ultra‐small connecting elements is demonstrated. The experimental research, performed using spin resolved neutron scattering, reveals a massively degenerate ground state, comprised of low integer and energetically forbidden high integer magnetic charges, that manifests cooperative paramagnetism at low temperature. The system tends to preserve the degenerate configuration even under large magnetic field application. It exemplifies the robustness of disordered correlation of magnetic charges in a 2D honeycomb lattice. The realization of quantum disordered ground state elucidates the dominance of exchange energy, which is enabled due to the nanoscopic magnetic element size in nanoengineered honeycomb. Consequently, an archetypal platform is envisaged to study quantum mechanical phenomena due to emergent magnetic charges.

cludes all three terms, eg. nearest neighbor (J 1 ) and next-nearest neighbor (J 2 ) exchange interaction as well as magnetic dipolar (D) interaction. As we can see in Figure S1, J 2 is much larger than D in our honeycomb lattice (element length ∼ 12 nm) of moderate thickness, ∼ about 10 nm.
Thus, J 2 term is more important than dipolar interaction in our honeycomb lattice. Our calculation is most likely not valid in large element size (about a micrometer element size) honeycomb where dipolar interaction is much stronger. In our case, the dipolar interaction is ∼12 K.
2 Exchange energy of nearest and next-nearest neighbor type at different layer thickness The nearest neighbor and the next-nearest neighbor exchange interactions were estimated as a function of permalloy layer thickness by considering the magnetostatic energy of magnetic charges located at the vertices of honeycomb lattice. The system's energy for a well-converged 30×30 kagome superlattice was calculated using Monte-Carlo method under the Heisenberg spinmodel. Here, the nearest-neighbor exchange interaction as antiferromagnetic (J 1 < 0) and the next-nearest neighbor exchange interaction is ferromagnetic (J 2 > 0). Fig. S2 shows the plot of J 2 /J 1 as a function of permalloy thickness for different values of J 1 . The length of honeycomb element is 12 nm.  Figure S3: The plot of magnetization with respect to temperature, obtained using ZFC/FC protocol.
Inset: Scanning electron micrograph of the magnetic honeycomb lattice.
We have performed magnetization measurements of our honeycomb lattice using Quantum Design SQUID magnetometer. Magnetic measurement was performed in 50 Oe magnetic field, applied in plane to the sample. As we can see in Fig. S3, the system exhibits paramagnetic behavior throughout the measurement range. This is consistent with estimated magnetization from PNR measurements, as shown in Fig. 2e, where magnetization nearly doubles between 200K and 30K.
But no signature of long range order or blocking characteristics was detected. It further suggests that the system is behaving as a cooperative paramagnet even at low temperature.

X-ray reflectivity
The X-ray reflectivity (XRR) of the Permalloy honeycomb lattice was measured using Cu-Kα 1 with wavelength of 1.5406Å. The XRR is well-fitted with an X-ray scattering length density (SLD) profile as shown in Fig. S4. The depth-profile estimated from XRR is consistent with neutron reflectivity fitting results. The off-specular reflectivity at low temperature was simulated within the Distorted-Wave Born Approximation (DWBA). For the off-specular profile. We observe that the off-specular reflectivity profile is indistinguishable for two degenerate spin configurations, with an equal number of ±3Q and ±Q magnetic charges distributed across the lattice.
For completeness, we have also simulated the off-specular reflectivity for pure states of spin ice and spin solid configurations. As shown in Fig. S7, the off-specular patterns due to pure states are very different from the mixed charge configurations, as shown in Fig. S6. 7 Spin configurations with an applied field at low-temperature The polarized neutron reflectivity measurement at T = 5 K was repeated with an in-plane external magnetic field of H = 0.5 T. The specular reflectivity, spin-asymmetry (SA), and the estimated scattering length density (SLD) of the PNR measurement are shown in Fig. S8.

Roughness analysis
The roughness analysis from atomic force micrograph is shown in Fig. S9. The RMS height difference of the top surface is less than 0.5 nm along any travel direction of connecting honeycomb elements.