Exceptional symmetries in condensed matter system?

Lubos commented an interesting abstract reporting evidence for a realization of the mathematically extremely interesting exceptional Lie group E₈ as symmetries of a condensed matter system known as Ising chain consisting of a chain of spins in a strong transversal magnetic field causing magnetization. At criticality for phase transition destroying the magnetization excitations appear and E₈ would appear as a symmetry of these excitations. Here is the abstract.

Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E₈ Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi–one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E₈ spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.

The relation of the results to string theory and TGD

Lubos gives a nice summary of E₈, which I recommend. Unfortunately Lubos takes a completely non-critical attitude accepting the experimental evidence as a proof and also creates the impression that this as a victory of super string model. The emergent dynamical E₈ symmetry is actually predicted by conformal field theory approach to 1-D critical systems alone and has nothing to with the fundamental E₈×E₈ symmetry of heterotic strings as Lubos actually admits. E₈ symmetry is predicted to be possible by conformal symmetry characterizing 2-dimensional criticality and the Kac-Moody representation is obtained once one has 8 complex scalar fields describing excitations of a conformally invariant system. The associated Kac-Moody symmetry predicts also a presence of a large number of other excitations created by the Kac-Moody generators obtained as normal ordered exponentials of complex scalar fields and their presence in the spectrum should be shown.

Of course, also string models as well as TGD are characterized by conformal symmetry. In TGD conformal symmetries have interpretation as a 3-D generalization of 2-D conformal symmetries acting at light-like boundaries of light-cone of M⁴ and also at light-like 3-surfaces of H=M⁴×CP₂ (because of their metric 2-dimensionality). Also string theories apply the exponentiation trick so that the 10-D target space of superstring models could be a purely formal construct in which case the notion of spontaneous compactification, which has led to the landscape catastrophe, would not make sense physically. In TGD framework compactication is replaced by number theoretical compactification, which is not a dynamical process but a duality stating the equivalence of formulations of quantum TGD based on the possibility to interpret 8-D imbedding space either as M⁸ or H=M⁴×CP₂ (M⁸-H duality).

Could E₈ emerge in TGD?

E₈ is interesting also from the TGD point of view. Of course, to say anything detailed about the finding in TGD framework would require hard work and in the following I can make just speculative general remarks.

1. The rank of E₈ group is 8, which means that the Cartan algebra of E₈ spanned by maximum number of commuting algebra elements has dimension 8. The eigenvalues of the Cartan algebra generators define the 8 quantum numbers of a physical state belonging to a representation of E₈.

   In TGD framework the quantum numbers of particle correspond to Cartan algebra of the product of Poincare group color group SU(3) and electroweak group SU(2)×U(1). The dimension of the corresponding Cartan algebra is also 8 corresponding to 4 components of four-momentum, 2 color quantum numbers and 2 electroweak quantum numbers.

   In conformal field theories Lie groups are extended to Kac-Moody algebras. One can construct rank 8 Kac-Moody algebras by starting from 8 complex scalar fields which could be interpreted in terms of coordinates of 8-D Minkowski space. One would obtain both the complex form of E₈ and the current algebra defined by symmetries of TGD (and of standard model).

2. Hyper-finite factors of type II₁ (HFFs) are a particular class of von Neumann algebras, which is very interesting from the point of view of quantum theories and the mathematics of quantum groups relates to them very closely. The spinors of world of classical worlds (the 4-surfaces in 8-D imbedding space) define a canonical representative for HFF. The inclusions of HFFs known as Jones inclusions are in one-one correspondence with finite discrete subgroups of SO(3) and these in turn are in one-one correspondence with simply laced Lie groups containing also E₈. E₆,E₇ and E₈ correspond to tedrahedon, octahedron, and dodecahedron, which are 3-D polygons. For other subgroups the minimal orbit is 2-D polygon. The conjecture is roughly that these Lie groups appear as dynamical symmetries of quantum TGD so that TGD Universe is like a universal computer able to emulate any other computer. Now the emulation is emulation of any gauge theory and also string model type system. These symmetries would not be fundamental but achieved by engineering.

3. Also the hierarchy of Planck constants realized in terms of the book like structure of the 8-D imbedding space could involve the mathematics of Jones inclusions. The pages of the big book are singular coverings and factor spaces of both CP₂ and what I call causal diamond (CD). CD is the intersection of future and past directed light-cones of 4-D Minkowski space M⁴. At least cyclic subgroups Z_n are involved. Also Z_n with reflection added and perhaps all finite discrete subgroups of the rotation group as symmetries permuting the copies of M⁸ or CP₂ of the covering or permuting the identified points of the singular factor space.

   E₈ gauge symmetry could emerge as a dynamical symmetry at corresponding pages. Even E₈×E₈ of heterotic strings models could appear. The two E₈:s would be associated with M⁴ and CP₂: maybe TGD Universe is able to emulate also E⁸×E⁸ and heterotic super string model. In the case of E₈ the symmetries of dodecahedron would identify equivalent points of M⁴ for singular factor space option.

   These symmetries would be engineering symmetries requiring quantum criticality. The system should be very near to the back of the big book so that the 3-surface describing the physical system can leak to the other pages of the book. The E₈ symmetry would appear only at the other side of criticality (E₈ page) and would correspond to a non-standard value of Planck constant. The change of the value of Planck constant would stabilize the phase unstable for the standard value of Planck constant. The claimed condensed matter E₈ symmetry is indeed assigned with quantum criticality rather than thermal criticality. Maybe the space-time sheets serving as correlates for the magnetic excitations of the system reside at the E₈ page and correspond to dark matter in TGD framework.

4. The fundamental representation of E₈ is identical with its adjoint representation and obtained by combining the rotation generators of SO(16) acting as rotations of points of 16-D Euclidian space E¹⁶ and the spinors of the same space to form a Lie-algebra in which E₈ acts. The question whether TGD could allow to identify some natural 16-D space inspires some reckless numerology.

   The definition of singular covering and factor spaces means a choice of two points of M⁴ in case of CD so that the moduli space for CDs is M⁴×M⁴₊, where M⁴₊ is 8-D light-cone: p-adic length scale hypothesis is obtained if M⁴₊ reduces to a union of hyperboloids for which proper time is quantized as powers of two. A possible interpretation is in terms of quantum cosmology with quantization of cosmological time. This procedure fixes quantization axes and means fixing of preferred time-like direction and spatial direction at either tip of CD (rest system and quantization axes of spin).

   In the case of CP₂ the selection of quantization axes should fix of point of CP₂ and a direction of geodesic line at that point. Therefore this part of the moduli space is CP₂×E⁴. Altogether the moduli space labeling CD×CP₂ with fixed quantization axes and thus sectors of the world of classical worlds is 16-D space M⁴×M⁴₊ ×CP₂×E⁴. Could the tangent space of this space provide a natural realization of the generators of the complex form of E⁸?