Yi Li et al have proposed an interesting exact result, which might help to understand the transition to ferromagnetism. This finding involves as an auxiliary concept Hamiltonian curve in a lattice of N points with single vacant point. The observation was that by moving the vacancy around one can induce arbitrary permutation of the non-vacant lattice points carrying spins and leading to the Hamiltonian curve containing the spins ordered from 1 to N . The problem is that the result holds true only for single vacancy.
The finding however inspired the question about how ferromagnetism could be described in TGD framework. The notion of flux tube is key concept and besides Maxwellian flux tubes there are also monopole flux tubes. The observation that magnetic moments having formal identification as infinitesimal pairs of magnetic monopole and anti-monopole leads to ask whether their sequences parallel to local magnetization M could correspond to magnetic moments residing at possibly dark (in TGD sense) monopole flux tubes as classical counterparts by quantum-classical correspondences. heff=nh0>h would give rise to quantum coherence at these flux tubes in the scale large than atomic length scale.
The monopole flux lines must be closed, which suggests that the return fluxes for the strands of a braid formed by N flux tubes parallel to M and containing the magnetic moments along it, fuse to a monopole return flux at larger parallel space-time sheet identifiable as the carrier of the field H so that one would have B= M+H for the magnetic field experienced by test particle touching both space-time sheets. General magnetized state would correspond to a non-trivial braid with non-parallel flux tubes providing a representation for the Maxwellian counterpart of M. For ferromagnetism the braid would be trivial. Also closed flux tubes without return flux and thus defining Hamiltonian cycle can be considered and could give rise to spin glass phase when there the lattice (graph) allows a large number of Hamiltonian cycles.
The transition to ferromagnetism could be described as time-like braid for the space-like braid defined by the flux tubes of M. It would correspond to so-called 2-braid. Quantum groups and also 2-braid groups could be natural notion in their mathematical description. The transition to ferromagnetism would correspond to braid opening and if return flux arrives along single sheet, is possible for arbitrary braid. If the return flux is by separate flux tubes, braid opening must involve re-connections.
For a summary of earlier postings see Latest progress in TGD.