basics
Examples
Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In metamaterials:
topological phononics (sound waves?)
For quantum computation:
Graphene is one of the solid phases of carbon, appearing as a single-layer honeycomb lattice. This may be planar or tubular, etc.
Graphene is a prime example (and among the first to be discovered) of a topological phase of matter. Specifically:
at coarse resolution (currently accessible to experiment) it appears as a (time-reversal and space inversion-symmetric) topological semi-metal, due to a gap between its valence band and conduction band which closes only over two Dirac points in its Brillouin torus;
at finer resolution – namely when the spin-orbit coupling of the electrons is resolved (thought to be $\sim10^{-3}\,$meV $[$MHSSKM06$]$ and hence too small for experimental observation, currently), which reveals a small energy gap opening at the two would-be Dirac points – graphene appears as a (time-reversal symmetric) topological insulator (Kane & Mele 05a), whose non-trivial topological phase is witnessed by the non-trivial Kane-Mele invariant in $\mathbb{Z}/2$ (Kane & Mele 05b).
The electronic band structure of graphene (reviewed in WZLLJHD 12) was predicted (long before the term was coined) already in
The synthesis/detection of graphene is due to
(The procedure, won a Nobel Prize and the authors made it freely available without patenting.)
Further discussion of the electron band structure of graphene:
Computation of the (tiny) spin-orbit coupling in graphene:
Observation that the spin-orbit coupling in graphene should open the gap at the Dirac points revealing a quantum spin Hall effect in graphene:
Charles Kane, Eugene Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95, 226801 (2005) (arXiv:cond-mat/0411737, doi:10.1103/PhysRevLett.95.226801)
Charles Kane, Eugene Mele, $Z_2$ Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95 (2005) 146802 (doi:10.1103/PhysRevLett.95.146802)
Review:
Nathan Weiss, Hailong Zhou, Lei Liao, Yuan Liu, Shan Jiang, Yu Huang, Xiangfeng Duan, Graphene: an emerging electronic material, Adv Mater. 24 (43) (2012) 5782-825 (doi:10.1002/adma.201201482)
Wikipedia, Graphene
Review in the context of topological phases of matter and specifically topological semi-metals:
The 2+1 dim Dirac equation is used in modeling graphene:
Konstantin Novoselov, Andre Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Two-dimensional gas of massless Dirac fermions in graphene, Nature 438, 197-200 (2005) doi
María A. H. Vozmediano, Renormalization group aspects of graphene, pdf
On non-perturbative effects in graphene:
Discussion via AdS-CFT in condensed matter physics:
On “fictitious” contributions to the Berry connection on the Brillouin torus of graphene:
Moving the nodal points in graphene(-variants) by changing external parameters such as lattice anisotropy or strain (see also further discussion of external manipulation by strain here, and see the references on momentum-space braiding of band nodes):
Cui-Lian Lia, New position of Dirac points in the strained graphene reciprocal lattice, AIP Advances 4 (2014) 087119 [doi:10.1063/1.4893239]
Marc Dvoraka, Zhigang Wu, Dirac point movement and topological phase transition in patterned graphene, Nanoscale 7 (2015) 3645-3650 [doi:10.1039/C4NR06454B]
Zhenzhu Li, Zhongfan Liu, Zhirong Liu: Movement of Dirac points and band gaps in graphyne under rotating strain, Nano Research 10 (2017) 2005–2020 [doi:10.1007/s12274-016-1388-z]
Jian Kang, Oskar Vafek, Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene, Phys. Rev. B 102 (2020) 035161 [arXiv:2002.10360, doi:10.1103/PhysRevB.102.035161]
Movement of Dirac points in a cousin of the Haldane model:
and in photonic crystal-analogs:
Last revised on September 12, 2022 at 14:32:47. See the history of this page for a list of all contributions to it.