New ideas about the transition to ferromagnetic phase

I received a link to a highly interesting article about ferromagnetism. Yi Li, a physicist working at John Hopkins University and his two graduate students, Eric Bobrow and Keaton Stubis seem to have made a considerable progress in understanding how the system of electron spins in lattice ends up to a ferromagnetic sate. This ferromagnetism is known as itinerant ferromagnetism and involves vacancies, sites without electron, which can be moved freely without affecting the energy of the state.

1. Reasons to get interested?

In TGD framework there are two good reasons of getting interested on these results.

1. The first reason is that the notion of magnetic flux tube central in TGD might allow application to spontaneous magnetization. TGD predicts two kinds of flux tubes: Maxwellian ones and monopole flux tubes with magnetic fields requiring no currents to generate them: they are not not allowed by Maxwell’s theory.

The preservation of the Earth’s magnetic field predicted to decay rather rapidly as currents generating it dissipate supports the view that it contains monopole flux part which from biological input would correspond to endogenous magnetic field B_end, which is a fraction 2/5 about the nominal value of B_E=.5 Gauss. The presence of magnetic fields in cosmological scales is also a mystery finding a solution in terms of monopole flux tubes.

In TGD inspired biology Hamiltonian cycles associated with icosahedron and tetrahedron provide a realization of the vertebrate genetic code (see this) but it is still somewhat of mystery why the points of icosahedron and tetrahedron, which are lattices (tesselations) at sphere, would be connected by a curve connecting neighboring points.

The following considerations are not much more than first impressions and might contain mistakes.

2. The ideas related to the work of Li et al

Returning to the problem of understanding how the ferromagnetic state could emerge from an arbitrary state with some numbers of spin up and spin down states at lattice sites connected by edges.

1. Permutation of electrons with same spin leave the ferromagnetic state invariant and does not cost energy while permutations in arbitrary configuration can do so.
2. Li et al considered a simple 4× 4 lattice with single vacancy and noticed a connection with so called 15-puzzle involving 15 tiles and single vacancy with neighboring tiles of vacancy able to move to its position. The observation is following. If one has spin lattice containing single vacancy, one can number the sites by a number running from 1 to N (now 15) in arbitrary manner. If so called connectedness condition holds true one can realize any permutation of these numbers. This means that 15-puzzle has always a solution. In particular, one can arrange the situation that the numbers form an ordered sequence from 1 to N so that numbers n and n+1 are nearest neighbors.

The result found by Li et al first for 2-D 4× 4 lattice with single vacancy generalizes to lattices, which are non-separable in the sense the removal of a lattice site does not separate any pair of spins – they are still connected by an edge-loop.

3. TGD based view

Flux tubes are the new element of condensed matter physics predicted by TGD. Could they provide insights into ferromagnetism?

1. In TGD framework one can ask whether there could be a physical 1-D structure – say flux tube – connecting all the lattice points? Could one assign connect spins labelled by n and n+1 for any permutation of lattice points and assign to this configuration an interaction energy which is sum of interaction energies between points n and n+1 with sum over.

Flux tube has magnetic energy increasing with flux tube length. On the other hand, the first guess is that the magnetic interaction energy of spins at the ends of the flux tube portion connecting decreases with the distance. This would suggest a competition between these two energies. Thermodynamics would bring in free energy and entropy and Gibbs energy E-TS would be mimimized. Entropy maximization would favor long random flux tubes and energy minimization short flux tubes.

Magnetic energy of the flux tube is minimal for Hamiltonian curve curve with points 1 and N as near to each other as possible. It would be absolute minimum for Hamiltonian cycle.

3.1 Connection with text book picture about ferromagnetism

How could the proposed picture relate to the text book model of ferromagnetism?

1. In the standard model of ferromagnetism one assumes the presence of field B identified as sum B= M+H of magnetization and field H equal to B outside the magnet. M is due to magnetic dipoles besides magnetic field B and the interaction of spins with H is important. B is certainly the fundamental field and H is an auxiliary notion and its relation to B requires a model for the system: typically H and B are assumed to be linearly related.

The test particle would experience the sum of the magnetic fields associated with the two sheets. Could M and H as its return flux correspond to these space-time sheets so that particle would experience their sum B=M+H? Spontaneous magnetization would be a direct signature of many-sheeted space-ime.

3.2 Could braiding describe transition to ferromagnetism?

In the work of Li et al discussed in the article, the permutations of lattice points are induced by moving the vacancy around. As already mentioned, this picture is too limited.

1. TGD picture brings in mind braid-knot connection. One can replace braid with a knot by connecting the N points at the opposite ends of the braid by parallel strands of trivial braid. This trivial braid would carry the return flux having interpretation in terms of field H.

It is also possible that the flux tubes fuse to single larger flux tube carrying the return flux. This would conform with the idea that H provides a description in longer length scale. H would be analogous to a smoothed out total magnetic field acting as self-consistent background.

The additional structure would be the closed flux tube connecting the lattice points in the initial and final state of the braiding. In the initial state the flux tube would not have minimal length and minimum energy but in the final state it would have and this state would correspond to ferromagnetic state.

3.3 Could also monopole flux tubes be important?

There is still one important aspect related to the TGD view about magnetic field which might play important role. TGD predicts two kind of flux tubes. The first kind of flux tubes could be called Maxwellian and the corresponding magnetic fields require current to generate them. There are also flux tubes having closed cross section and carrying monopole fluxes. No currents are required to generate corresponding magnetic fields. Could also these flux tubes having no current as sources be present? This would mean new physics.

1. The first thing to notice is that the interpretation of magnetization M is as a magnetic field generated by magnetic moments. The usual interpretation is that spins are analogous to magnetic moments created by currents consisting of rotating charge. Now there is no such rotating charge. Second interpretation is as magnetic moments identifiable as infinitesimal monopole pairs.
2. Could one think that the flux tubes containing sequence magnetic moments correspond to monopole flux and that closing this loop could give rise to monopole magnetic field? Ordinary Maxwellian part could be also present and have current as source. How M and H would relate to these. Could M correspond to the monopole part and H the Maxwellian part? But are the spins necessary for the monopole flux tube? Could one think that quantum classical correspondence is realized. Could dark matter assigned with the monopole flux tubes give rise to the magnetic moments?

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.