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Superconducting Quantum Interference Devices (SQUIDs) and othersuperconducting circuits are limited by intrinsic flux noise with spectraldensity $1/f^{\alpha}$ with $\alpha<1$ whose origin is believed to be due tospin impurities. Here we present a theory of flux noise that takes into accountthe vectorial nature of the coupling of spins to superconducting wires. Wepresent explicit numerical calculations of the flux noise power (spectraldensity integrated over all frequencies) for electron impurities and latticenuclear spins under several different assumptions. The noise power is shown tobe dominated by surface electron spins near the wire edges, with bulk latticenuclear spins contributing $\sim 5$% of the noise power in aluminum and niobiumwires. We consider the role of electron spin phase transitions, showing thatthe spin-spin correlation length (describing e.g. the average size offerromagnetic spin clusters) greatly impacts the scaling of flux noise withwire geometry. Remarkably, flux noise power is exactly equal to zero when thespins are polarized along the flux vector direction, forming what we call apoloidal state. Flux noise is non-zero for other spin textures, but getsreduced in the presence of correlated ferromagnetic fluctuations between thetop and bottom wire surfaces, where the flux vectors are antiparallel. Thisdemonstrates that engineering spin textures and/or inter-surface correlationprovides a method to reduce flux noise in superconducting devices.

Flux-vector model of spin noise in superconducting circuits: Electron versus nuclear spins and role of phase transition