Skyrmions was rediscovered in 1983 in the context of Quantum Chromodynamics (QCD). A skyrmion contains 'knots' in its magnetization; the vector character of the magnetization leads to more complex topological states. The term skyrmion derives from the analogy with the Skyrme model for nucleons in pion (electron, muon, tau) field theory, and is used in the modelling of atomic nuclei as quantized skyrmions. Skyrmions are topological soliton solutions whose conserved topological charge B is identified with the baryon number of a nucleus. Apart from an energy and length scale, the Skyrme model has just one dimensionless parameter m proportional to the pion mass. It has been found that a good fit to experimental nuclear data requires m to be of order 1 (reals?). Modelling of atomic nuclei here.
Roaming through a skyrmion. When an electron moves through a special type of magnetic texture called a skyrmion, its magnetic moment (spin) twists to adjust to the skyrmion's local spin structure (ribbon-like pattern). This twisting changes the electron's direction of travel and pushes the electron and the skyrmion in opposite directions (not shown). From Nature (1).
It is interesting to observe skyrmions in the more conventional, 3-D magnetic system MnSi at higher temperatures. Chlorophyll resembles much MnSi basic unit. Chlorophyll support assembly and accumulation of light-harvesting complexes, only its central atom is Mg and its 'arms' are C. Modifications that introduce electronegative groups on the periphery of the chlorophyll molecule withdraw electrons from the pyrrole nitrogens and thus reduce their basicity. Consequently, the tendency of the central Mg to form coordination bonds with electron pairs in exogenous ligands, a reflection of its Lewis acid properties, is increased, here.
The Dirac world.
There are a force emerging perpendicular to this electron movement, and also consequently a repelling force (antimatter?). This will create a cone of different magnetism (and gravity?) expending forth and back. A non-material Dirac cone creating topology and hierarchy and matter condensation? Gravity emergense in the 4-D? Dirac electrons can only exist on the honeycomb lattice. About the Carbon world:
Graphene's band dispersion and low energy Dirac cone. From EV Castro.
Castros research group:
Graphene is a semi-metal, with a linearly vanishing density of states (DOS) at zero energy (the Fermi level for the undoped material). Near the Fermi energy, the DOS fills up to a constant plateau (also predicted in a CPA) but in addition has a sharp peak closer to zero energy, reminiscent of the delta function of the quasi-localized state for a single impurity. A study of the inverse participation ratio showed that the states contributing to this feature have a real space decay similar to the single impurity state (~1/r, where r is the distance to the vacancy). This work in currently being extended to study long range (Coulomb) impurity potentials.What say Castro Neto? He looks at quantum phase transitions and graphene.
We also developed a tight-binding study of the graphene bilayer, subject to a perpendicular electric field. This turns out to be a very interesting system for applications, since the bilayer opens a gap, tunable via the external electric field, and managed to obtain a consistent description of the measured cyclotron mass and energy gap as a function of the external field.
We showed that a relative rotation of the layers (a rather common stacking defect in graphite, often observed in STM studies of its surface) has profound effects on the electronic structure near the Fermi level:
* the low energy dispersion remains linear, as in a single layer
* the Fermi velocity can be significantly reduced with respect to the single layer value
* a perpendicular electric field does not open a gap.
Gap opening in a biased graphene bilayer. An energy gap is created if the material is placed between positive and negative electrodes. The gap arises because the transverse voltage causes an excess of negatively charged electrons in one layer and an excess of positively charged “holes” in the other layer. These electrons and holes are believed to pair up to create quasiparticles, which behave differently than their constituent particles.
A magnetic field was also applied to the bilayer, which caused the quasiparticles to move in circular orbits – an effect called cyclotron resonance. The team measured the period of this resonance, which depends on the mass of the quasiparticles. The team discovered that this cyclotron mass increased as the applied voltage increased from zero to about 100 V, allowing them to conclude that the energy gap was also changing from zero to about 150 meV (2).
A peculiar feature of electrons and holes in graphene is that they move through the material as if they have no rest mass – something that makes the material a very good conductor. However, the quasiparticles have a rest mass and according to Castro Neto, this mass leads to an energy gap that must be overcome before current can flow.
Many phases of matter are characterized by order of some electronic degree of freedom such as spin (as in the case of spin density waves - SDW), or charge (as in the case of charge density waves - CDW). Other phases are characterized by more complicated order parameters that are directly related to many-body effects (as in the casea of superconductivity). It is possible to tune between phases with order and without order by tunning some parameter such as the temperature of the system. These finite temperature phase transititons have been well studied in the context of classical critical phenomena. Much less is known by phase transitions that occur at zero temperature in which quantum fluctuations play an important role. These quantum phase transitions (QPT) are responsible for very unusual behavior that cannot be described within traditional theoretical frameworks. QPT transitions can be tuned by the application of pressure, magnetic field, or even chemical modification, that unavoidably introduces disorder in the material. Disorder can strongly affect QPT because quantum fluctuations are very sensitive to geometrical constraints. Cold atom systems, high temperature superconductors, transition metal dichalcogenides, and heavy fermion materials are example of systems where QPT are believed to play an important role.
Example of a Quantum Phase Transition: T is temperature and Fm is a tunning parameter.
Spin and charge densities as opposed to many-body systems, which only them can have superconductions and superpositions. Of course, it takes two to tangos. The Scrödinger cat phenomen...
This is said to be usable for molecular sensors. Surprise? I think at once of sensors in biology. Carbon is central in biology, as also cyclotrone resonances (Liboff, TGD).
Is this 'gap opening' the same as the A-phase? Could this be the quantum window in biology, giving rise to paranormal experiences? And also diseases like MS, arthritis, pain...? The meridian functionings? The beginning of the entangled qubit-thing, used by Rakovic. Some kind of makrostate qubit? A chrystalline living oscillating phase that can be managed by small changes in temp.,pressure, pH? This list sound familiar from the nerve pulse. But how learn to dance?.
Chrystals of different kinds. The A-phase.
The abstract to another Nature article 17.6 (3).
Crystal order is not restricted to the periodic atomic array, but can also be found in electronic systems such as the Wigner crystal or in the form of orbital order, stripe order and magnetic order. In the case of magnetic order, spins align parallel to each other in ferromagnets and antiparallel in antiferromagnets (= diamagnetic and paramagnetic order?) In other, less conventional, cases, spins can sometimes form highly nontrivial structures called spin textures. Among them is the unusual, topologically stable skyrmion spin texture, in which the spins point in all the directions wrapping a sphere, the skyrmion. The skyrmion configuration in a magnetic solid is anticipated to produce unconventional spin–electronic phenomena such as the topological Hall effect. The crystallization of skyrmions as driven by thermal fluctuations has recently been confirmed in a narrow region of the temperature/magnetic field (T–B) phase diagram in neutron scattering studies of the three-dimensional helical magnets MnSi and Fe1−xCoxSi. With a magnetic field of 50–70 mT applied normal to the film, we observe skyrmions in the form of a hexagonal arrangement of swirling spin textures, with a lattice spacing of 90 nm. The related T–B phase diagram is found to be in good agreement with Monte Carlo simulations. In this two-dimensional case, the skyrmion crystal seems very stable and appears over a wide range of the phase diagram, including near zero temperature. Such a controlled nanometre-scale spin topology in a thin film may be useful in observing unconventional magneto-transport effects.
a, b, Helical (a) and skyrmion (b) structures predicted by Monte Carlo simulation. c, Schematic of the spin configuration in a skyrmion. d–f, The experimentally observed real-space images of the spin texture, represented by the lateral magnetization distribution as obtained by TIE analysis of the Lorentz TEM data: helical structure at zero magnetic field (d), the skyrmion crystal (SkX) structure for a weak magnetic field (50 mT) applied normal to the thin plate (e) and a magnified view of e (f). The colour map and white arrows represent the magnetization direction at each point.
a–c, Spin textures observed using Lorentz TEM obtained by Monte Carlo simulation. e–g, Spin textures after TEM. H, helical structure; SkX, skyrmion crystal structure; FM, ferromagnetic structure. d, h, Observed (d) and calculated (h) phase diagrams in the B–T plane. The colour bars in the phase diagrams indicate the skyrmion density per 10−12 m2 (d) and per d2 (h), d being the helical spin wavelength. Dashed lines show the phase boundaries between the SkX, H and FM phases. Stars in d and h indicate (T, B) conditions for the images shown in a–c and e–g, respectively.
Molecular orbital theory (4):
A molecular orbital is the volume within which a high percentage of the negative charge generated by the electron is found. The molecular orbital volume encompasses the whole molecule. The electrons would fill the molecular orbitals of molecules like electrons fill atomic orbitals in atoms.
- The molecular orbitals are filled in a way that yields the lowest potential energy for the molecule.
- The maximum number of electrons in each molecular orbital is two (Pauli exclusion principle.)
- Orbitals of equal energy are half filled with parallel spin before they begin to pair up (Hund's Rule.)
- Molecular orbitals are formed from the overlap (superpositions?) of atomic orbitals.
- Only atomic orbitals of about the same energy interact to a significant degree.
- When two atomic orbitals overlap, they interact in two extreme ways to form two molecular orbitals, a bonding molecular orbital and an antibonding molecular orbital.
Light waves can interact in-phase, which leads to an increase in the intensity of the light (brighter) and out-of-phase, which leads to a decrease in the intensity of the light (less bright). Electron waves can also interact in-phase and out-of-phase. In-phase interaction leads to an increase in the intensity of the negative charge. Out-of-phase interaction leads to a decrease in the intensity of the negative charge.
In-phase leads to wave enhancement similar to the enhancement of two in-phase light waves. Where the atomic orbitals overlap, the in-phase interaction leads to an increase in the intensity of the negative charge in the region where they overlap. This creates an increase in negative charge between the nuclei and an increase in the plus-minus attraction between the electron charge and the nuclei for the atoms in the bond. The greater attraction leads to lower potential energy.
Because electrons in the molecular orbital are lower potential energy than in separate atomic orbitals, energy would be required to shift the electrons back into the 1s orbitals of separate atoms. This keeps the atoms together in the molecule, so we call this orbital a bonding molecular orbital. The molecular orbital formed is symmetrical about the axis of the bond. Symmetrical molecular orbitals are called sigma, σ, molecular orbitals.
The second way that two atomic orbitals interact is out-of-phase. Where the atomic orbitals overlap, the out-of-phase interaction leads to a decrease in the intensity of the negative charge. This creates a decrease in negative charge between the nuclei and a decrease in the plus-minus attraction between the electron charge and the nuclei for the atoms in the bond. The lesser attraction leads to higher potential energy. The electrons are more stable in the 1s atomic orbitals of separate atoms, so electrons in this type of molecular orbital destabilize the bond between atoms. We call molecular orbitals of this type antibonding molecular orbitals. The molecular orbital formed is symmetrical about the axis of the bond, so it is a sigma molecular orbital with a symbol of σ*1s.
This is an simple box, like a Feynman diagram.
When two larger atoms atoms combine to form a diatomic molecule (like O2, F2, or Ne2), more atomic orbitals interact. The LCAO approximation assumes that only the atomic orbitals of about the same energy interact. For O2, F2, or Ne2, the orbital energies are different enough so only orbitals of the same energy interact to a significant degree.
The two 2py atomic orbitals overlap in parallel and form two pi molecular orbitals. Pi molecular orbitals are asymmetrical about the axis of the bond.
bond order = 1/2 (#e- in bonding MO's - #e- in antibonding MO's)
Oxygen, the negentropic element.
The molecular orbital diagram for a diatomic oxygen molecule, O2.
- O2 has a bond order of 2. Bond Order = 1/2(10 - 6) = 2
- The bond order of two suggests that the oxygen molecule is stable.
- The two unpaired electrons show that O2 is paramagnetic.
The bond between the carbon and oxygen in carbon monoxide is very strong despite what looks like a strange and perhaps unstable Lewis Structure. The plus formal charge on the more electronegative oxygen and the minus formal charge on the less electronegative carbon would suggest instability. The molecular orbital diagram predicts CO to be very stable with a bond order of three. About the geometry of molecular orbits here.
Other odd features.
Coexistence of both magnetic order and cooperative paramagnetic spin liquid behavior is possible, seen in spin glass phenomens, as instance. The temperature Tc is joined to a point in the phase diagram where a transition between two paramagnetic solutions happens. This gives liquid crystal properties, called a mesophase? "The existence of a long range intermolecular exchange interaction at all the temperatures is established by the collapse of the hyperfine structure in the spectra of the condensed samples. The comparison of the principal effective g- values in the solid phase with those of the isolated molecule obtained from the toluene solution spectra indicates that in the solid, the molecules pack keeping their molecular axes parallel. The drastic changes observed in the spectra when the sample reaches the smectic phase are interpreted as a consequence of the structure of this mesophase".
Long-range polarization in an antiferrimagnetic order has also shown a window of ferromagnetic order. An emergence of 2-D? An temp. dependent interference between the electric and magnetic properties? It is shown that there exists a transition between a Néel state and a quantum paramagnetic phase, characterized by broken translational invariance. The non-magnetic phase preserves the lattice rotational symmetry, and has a correlated plaquette nature.
A mirror symmetry is identified to characterize both electron and impurity magnetizations (spin polarizations) when the impurities are symmetrically (with respect to this mirror) positioned and when pinning fields are absent.
Fluorescence polarization and electron paramagnetic resonanse spectroscopy are used to study order and orientation of protein elements in biology.
Life is these multiphases? As Rakovic said, Life is not trying to eliminate the noise, but to use it to create oscillations and a new order. That is exactly some kind of superpositions of different em-properties (liquid chrystals?).
In oxides the O is shown to be important for the topology. It brings in the 'islands'. Here.
...homolog but also that grain boundaries are the preferential sites for oxide nucleation and the oxide islands formed along the grain boundaries show a faster growth rate than that on flat Cu surface. The oxidation on the faceted Cu(110) surface results in heterogeneous nucleation of oxide islands in the facet valleys and one-dimensional growth along the intersection direction of the facets. Self-organised nanostructures, obtained by oxidation of III–VI compounds It was established that self oxide surfaces possess numerous geometrical morphologies of nanostructures (wires, trees, self assembled networks).
The spin states are analogs to solitons? The charged excitations are very light, and form a degenerate Bose gas even at high temperatures. -> Oxide superconductors.
Complex oxide materials exhibit a tremendous diversity of behavior encompassing a broad range of functional properties, such as magnetism, ferroelectricity, and superconductivity. As diverse as this behavior is, an even richer spectrum of possibilities becomes available if one starts to combine different complex oxides together with atomic-scale precision to create new artificially structured, heterogeneous systems.
Quantum spin Hall effect in a transition metal oxide Na2IrO3 an antiferromagnetic order first develops at the edge, and later inside the bulk at low temperatures. The nontrivial topology is more ubiquitous in solids than expected.
A recent breakthrough in this field is the theoretical and experimental discoveries of the quantum spin Hall (QSH) effect in time-reversal symmetric insulators. Intuitively it can be regarded as two copies of QH systems with up and down spins, but is driven by the spin-orbit interaction (SOI) instead of the external magnetic field. The topological insulator is closely related to a Kramers doublet protected by the time-reversal symmetry, and corresponds to the presence or absence of gapless helical edge modes in the semi-infinite system.
The theoretical design of a topological insulator using HgTe/CdTe quantum wells, and it is shown that the lowtemp superconduction relies on large SOI and the fine tuning of the band structure. Therefore one important development is to realize more robust topological insulators at higher temperature by the larger SOI. Try to design topological insulators in 5d transition metal oxides by using the complex transfer integrals and the lattice geometry, which was shown.
Another interesting development is to study the interplay between the non-trivial topology and the electron correlation. Generally the electron correlation is stronger in d- and f-electrons than in s- and p-electrons. When we look at transition metal ions in the periodic table, the electron correlation is the strongest in 3d elements and decreases to 4d and to 5d elements because d-orbitals are more and more extended, while the SOI increases as the atomic number.
5d-orbitals are rather extended and subject to the large crystalline field. Under the octahedral crystalline field, d-orbitals are split into e(g) (x2 − y2, 3z2 − r2)- and t(2g) (xy, yz, zx)-orbitals by 10Dq of the order of 3eV. The SOI is quenched in e(g)-orbitals but remains effective in t(2g)-orbitals, which form effectively the triplet with ℓ(eff) = 1. Including the SOI, we obtain the states with the total angular momentum j(eff) = 3/2 and 1/2. The central idea is that the transfer integrals between these complex orbitals and oxygen orbitals become complex. This complex transfer integral is responsible for topological states in iridates ( - look also at the molecular orbital model).
Na2IrO3 whose layered crystal structure contains the honeycomb lattice, with each Ir atom surrounded by an octahedron of six O atoms, which leads to the similar energy level scheme as Sr2IrO4, i.e. one electron in j(eff) = 1/2-states, we can construct the effective single-band model on the honeycomb lattice. Since the O p-level ǫp are around 3eV lower than the Ir d-level ǫd we can integrate out p-orbitals to obtain the following effective Hamiltonian where [ij] and [[ij]] denote the nearest-neighbor (NN) and next-nearest-neighbor (NNN) pairs, respectively. The transfer integral t between a NN pair is real and spin-independent. The transfer integral between a NNN pair depends on spin, leading to a topological insulator, perpendicular to the honeycomb plane. With this convention, the transfer integral is a 2 × 2 matrix in the spin space. The key to these complex transfer integrals is the asymmetry between two paths connecting a NNN pair. The real transfer integral t′0 can be produced by the direct dd hopping and breaks the particle-hole symmetry.
To summarize these results, the transfer integrals are real and spin-independent for a NN pair, while complex and spin-dependent for a NNN pair, where ψ(~r) is the eight-component spinor field operator, and kX and kY are measured from K or K′ points. η’s, τ’s, and σ’s are the Pauli matrices for the valley (K or K′), sublattice (1 or 2), and spin (+ or −) degrees of freedom, respectively. Diagonally is a zig-zag geometry. The crossing point is at k = π for the zig-zag geometry, while at k = 0 for the armchair geometry, where k is the wavenumber along the edge. Anyway such crossing is protected by the Kramers theorem, and can get gapped only if the time-reversal symmetry is broken.
The circular movement in the skyrmions give rise to an helix. Carbon is an asymmetric molecule giving chirality.
Carbon is also one possible way Dirac electrons will enter. And Carbon will form a superposition upon the hexameric lattice as a trimeric sublattice. Oxygenal secondary lattice is hexagonal. I think this is very important in biology.
The intensity pattern in the A-phase is characterized by a six-fold symmetry. These patterns are exclusively seen in the plane perpendicular to the magnetic field. The A-Phase can be identfii ed as a hexagonal lattice of skyrmion tubes. The spin structure in the A-phase of MnSi is a so-called triple-Q state, i.e., a superposition of three helices under 120 degrees, and similar to the vortex lattice in superconductors.
Twenty years ago it has been predicted that skyrmions exist in anisotropic spin systems with chiral spin-orbit interactions, where they are expected to form crystalline structures. The origin of the topological Hall effect is a Berry phase collected by the conduction electrons when following adiabatically the spin polarization of topologically stable knots in the spin structure. Thus the Berry phase reflects the chirality and winding number of the knots. The topological Hall effect arises besides the normal Hall effect, which is proportional to the applied magnetic field, and the anomalous Hall effect that scales with ferromagnetic components of the magnetization.
This provides clear experimental evidence that the magnetic structure observed in neutron scattering has indeed the topological properties (chirality and winding number). The helimagnetic state may be understood as the result of a hierarchy of energy scales where ferromagnetic exchange is on the strongest scale and the isotropic spin-orbit interactions due to the lack of inversion symmetry give rise to a long-wavelength helimagnetic modulation, where wavelength of h is about 190Å. The propagation vector ~Q of the helix is pinned to the cubic spacediagonal by higher order spin-orbit coupling terms, which represent the weakest scale.
1. Ordered states with helical arrangement of the magnetic moments are described by a chiral order parameter which yields the left- or right-handed rotation of neighboring spins along the pitch of the helix.
2. Spins on a frustrated lattice form another class of systems, where simultaneous ordering of chiral and spin parameters can be found. Ex. triangular lattice with antiferromagnetic nearest neighbor interaction, the classical ground-state is given by a non-collinear arrangement with the spin vectors forming a 120◦ structure. In this case, the ground state is highly degenerate as a continuous rotation of the spins in the hexagonal plane leaves the energy of the system unchanged. In addition, it is possible to obtain two equivalent ground states which differ only by the sense of rotation (left or right) of the magnetic moments from sublattice to sub-lattice, hence yielding an example of chiral degeneracy.
3. CP breaking is the basic prediction of TGD and relates to the CP breaking of Chern-Simons action inducing CP breaking in the modified Dirac action defining the fermionic propagator. Chiral selection requires parity breaking, and the parity breaking of the molecule itself would induce the symmetry breaking if molecule possesses dark magnetic body., says Matti Pitkänen. I take my freedom to translate this magnetic body to the Dirac cone, (and it can also be seen as a wormhole?). DM-interaction depends on the polarization with respect to the magnetic vectors. The required asymmetry is a consequence of the cone, with one part dark, 'taking away' about half of the interacting particles, so that the effeciency of the reaction pathways can be much higher.
Water can be an interacting, reducing medium in this process. Conformal changes in proteins are also important as environmental influences. As are oxygen, of course, and other biomolecules.
4. Chiral fluctuations can be directly observed in non-centrosymmetric crystals without disturbing the sample by a magnetic field. polarized inelastic neutron scattering experiments performed in the paramagnetic phase of the itinerant ferromagnet MnSi that confirm the chiral character due to spin-orbit coupling. Antisymmetric (in forming the magnetic groundstate) and asymmetric interactions are possible.
Time can also be seen as a chiral structure?
5. There are a possibility for more than one mirror image, an hierarchy of particles can be created, but usually the other mirrorimages are dark (too low energy). TGD talks of an fractional quantum Hall effect.
Life as Islands.
Life is 'emergent'. It use the same atoms and molecules as rest of the matter, but maybe in one aspect Life is different. Life is built on hexagons. Life interact with the quantum world in many ways. It is important to keep that interaction at moderate levels, said Rakovic and Pitkänen, for organizational reasons.
One of the most fascinating aspects of physics is the 'emergence' of complex structures out of simple laws, an interesting case of which are structures protected by topological invariants. To illustrate an example of this, consider vortex lines in superconductors. In such systems, the phase of the Cooper-pair wave function rotates by 2πn as one circles a vortex. Continuity then requires the vorticity n to be an integer. For this reason, a vortex with n=±1 cannot decay.
Stabile topological structures lies behind biology. The stability are rationals, the oscillations are p-adics? The glue are entanglement in electon-sheets? Dirac-electrons are p-adics? One important task is to clearly define this interaction of stabilizating chaos. Life is complex.
TGD: The recent claim could be explained if antineutrinos and neutrinos in the experimental situation favor different p-adic primes p≈ 2k. If one has k(ν)=k(νbar)+1 for all families and the topological mixing matrices are otherwise the same (recall that fermion families correspond in the absence of mixing to different topologies for wormhole throats: sphere, torus, sphere with two handles, perhaps also higher families although there is good argument supporting the view that three lowest genera are "light").
Another kind of fraction is the cyclotrone resonance giving rise to molecular sensing?
The third one is the topological changes with distance and time? Then also the cognition, memory and the consciousness follows as a natural aspect.
Matti Pitkänen writes: A good guess is that algebraic entanglement is essential for quantum computation, which therefore might correspond to a conscious process. Hence cognition could be seen as a quantum computation like process, a more approriate term being quantum problem solving. Living-dead dichotomy could correspond to rational-irrational or to algebraic-transcendental dichotomy: this at least when life is interpreted as intelligent life. Life would in a well defined sense correspond to islands of rationality/algebraicity in the seas of real and p-adic continua.
Algebraic numbers are related to rationals by a finite number of algebraic operations (Gaussian distribution?) and are intermediate between periodic and chaotic orbits* allowing an interpretation as an element in an algebraic extension of any p-adic number field. The projections of the orbit to various coordinate directions of the algebraic extension represent now periodic orbits.
*) Chaotic orbits should be interpreted as potential possibilities?
The special nature of the Golden Mean, which involves 5^1/2, conforms the view that algebraic numbers rather than only rationals are essential for life. And may I fill in, spheres? One especially interesting aspect with this skyrmionic atomic model is that the Platonic solids are found directly from it.
The new secondary p-adic time scales assigned with elementary particles, in particular the .1 second time scale assignable with electron as time scale of corresponding "causal diamond" and defining fundamental biorhythm will mean a direct connection between living and dead matter. Time and entropy are so important for life to give the coherence. The chrystal phases is entanglement, or SOL-phases in biology.
Water, temp., pressure, pH etc. increases the 'noise' and hence the reactivity. Together with inhibition noise gives coherence in the different systems, so they can behave as entangled quantum dots, at least for a while. As instance when the chromophore is in excited state as opposite electronic ground state.
The basic prediction of zero energy ontology is that second law does not make sense in the time scale below the time scale defined by the temporal size of CD for physics associated with the field body of system in question - say electron, say Matti.
Complex structures are function of entropic/negentropic forces as amplified forces drive spatial expansion of matter. If this oscillation is a vortex the apex is also spiralling up. Stability = rationals, oscillation = complex numbers?
The gap opening is the window into the quantum world?
1. Condensed-matter physics: Single skyrmions spotted
Christian Pfleiderer & Achim Rosch: Nature Volume: 465,
Pages: 880–881 (17 June 2010) DOI: doi:10.1038/465880a
2. News. Dec 5, 2007 Tuneable gap semiconductor is a first Gapless semiconductor. http://physicsworld.com/cws/article/news/32085;jsessionid=34FB29F720E6613E9DC...
http://www.worldscibooks.com/physics/7397.html modelling of atomic nuclei.
http://physics.bu.edu/~neto/ quantum phase transition, theorists are being forced to revisit the conceptual basis for the theory of metals. http://arxiv.org/ftp/arxiv/papers/1003/1003.4520.pdf
3. Real-space observation of a two-dimensional skyrmion crystal
X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa & Y. Tokura
Nature 465, 901-904 (17 June 2010) | doi:10.1038/nature09124
4. MO-theory. Mark Bishop, http://www.mpcfaculty.net/mark_bishop/molecular_orbital_theory.htm http://arxiv.org/ftp/arxiv/papers/1001/1001.2021.pdf