Geometry, topology, and physics of non-Abelian lattices

Conclusion  After reviewing in some detail the notion of non-Euclidean lattices, whose domain of physical realization lies mostly in the novel carbon structures of the family offullerenes, we have discussed a number of physical problems denned over such lattices. We have shown that the group-theoretical definition of these lattices leads to “designing” new tubular regular structures, endowed with symmetries unheard of in the frame of customary crystallography, which combine features of extreme complexity and, at the same time, of great regularity. We have compared the role of the non-Abelian symmetries which these super-lattices are characterized by, with that of (discrete) harmonic (Fourier) lattice symmetry typical of customary crystallographic lattices. Many novel features enter into play, due to thenon-flatness of the related lattice geometry, which led us to a novel—sometimes unexpected—insight into the dynamical and/or thermodynamical properties of various physical systems which have these lattices as ambient space. We have analyzed how lattice topology bears on the complex combinatorics (related to loop-counting) of the classical Ising model. These lattices, even though finite, are, of course, much closer to being three-dimensional than regular 2D lattices simply equipped with periodic boundary conditions. We have shown, on the other hand, how the relation between the lattice symmetry (for example, in the case of fullerene, the discrete subgroup ofSU(2) that we have denotedg ₆₀ and the symmetry proper to the Hamiltonian of quantum systems of many itinerant interacting electrons (Hubbard-like models) allows us to reduce the calculation of the system spectral properties to a “size” that can be dealt with numerically with present-day numerical exact diagonalization techniques much more easily than a regular 3D cluster with a quite smaller number of sites.