(Anti-)de Sitter, Poincaré, Super symmetries, and the two Dirac points of graphene

Abstract

We propose two different high-energy-theory correspondences with graphene (and related materials) scenarios, associated to grain boundaries, that are topological defects for which both Dirac points are necessary. The first correspondence points to a (3 + 1)-dimensional theory, with nonzero torsion, with spatiotemporal gauge group SO(3, 1), locally isomorphic to the Lorentz group in (3 + 1) dimensions, or to the de Sitter group in (2 + 1) dimensions. The other correspondence treats the two Dirac fields as an internal symmetry doublet, and it is linked here with unconventional supersymmetry with SU (2) internal symmetry. Our results are suggestive, rather than conclusive, and pave the way to the inclusion of grain boundaries in the emergent field theory picture associated with these materials, whereas disclinations and dislocations have been already well explored.

$$ \mathcal{N} $$ -extended D = 4 supergravity, unconventional SUSY and graphene

“…We emphasize that our construction follows a top-down approach, in that the effective D = 3 theory that we derive at the boundary of AdS 4 originates from a well defined supersymmetric effective supergravity in the bulk. Nevertheless it bears similarities with the D = 3 models considered in [17], whose formulation also features an underlying AdS 3 symmetry. We shall elaborate on this at the end of section 4.…”

“…3, are then consistently expressed in terms of the torsion parameters of the model. In the last subsection we consider a different model within our general construction, defined by p = 4 and q = 0, which allows us to make contact with the analysis in [17].…”

“…In [17] the doublet index A refers to the K and K valleys and the SU(2) internal symmetry group is naturally gauged by construction. This allows the authors to describe topological features of graphene such as grain boundaries.…”

“…We also emphasize that the approach pursued here is a top-down one. Although a supergroup of the form SU(2|1, 1) + × SO(2, 1) − is not comprised in our class of models, the closest we can get to the construction of [17] corresponds to choosing p = 4 and q = 0. In this case the supergroup we start from is OSp(4|2) + × SO(2, 1) − , whose bosonic subgroup is SO(4) × SL(2, R) + × SL(2, R) − .…”

“…Consequently, upon implementing the AVZ Ansatz, the index structure of the resulting spinors is χ I Aα , formally the same as in the model of [17] described above, with the difference that now the index I is no longer a flavour one but it is acted on by the gauge group SU(2) 1 which was absent in the construction of [17], while SU(2) 2 , acting on the A index, should be identified with the R-symmetry group of [17]. Therefore we expect that most of the applications of the model of [17] to the study of the topological features of graphene should also hold for our OSp(4|2) + × SO(2, 1) − model. The main difference is the presence in the latter of additional gauge vectors A I J µ = A J I µ , associated with SU(2) 1 .…”