C-R-T Fractionalization, Fermions, and Mod 8 Periodicity

We study the topological classification of parafermionic chains in the presence of a modified time reversal symmetry that satisfies ${\cal T}^2=1 $. Such chains can be realized in one dimensional structures embedded in fractionalized two dimensional states of matter, e.g. at the edges of a fractional quantum spin Hall system, where counter propagating modes may be gapped either by back-scattering or by coupling to a superconductor. In the absence of any additional symmetries,...

A systematic procedure is developed for constructing fermion systems in discrete space-time which have a given outer symmetry. The construction is illustrated by simple examples. For the symmetric group, we derive constraints for the number of particles. In the physically interesting case of many particles and even more space-time points, this result shows that the permutation symmetry of discrete space-time is always spontaneously broken by the fermionic projector.

Parafermions are fractional excitations which can be regarded as generalizations of Majorana bound states, but in contrast to the latter they require electron-electron interactions. Compared to Majorana bound states, they offer richer non-Abelian braiding statistics, and have thus been proposed as building blocks for topologically protected universal quantum computation. In this review, we provide a pedagogical introduction to the field of parafermion bound states in one-dime...

In quantum field theory, we learn that fermions come in three varieties: Majorana, Weyl, and Dirac. Here we show that in solid state systems this classification is incomplete and find several additional types of crystal symmetry-protected free fermionic excitations . We exhaustively classify linear and quadratic 3-, 6- and 8- band crossings stabilized by space group symmetries in solid state systems with spin-orbit coupling and time-reversal symmetry. Several distinct types o...

The existence of three generations of neutrinos(charged leptons/quarks) and their mass mixing are deep mysteries of our universe. Recently, Majorana's spirit returns in modern condensed matter physics -- in the context of topological Majorana zero modes in certain classes of topological superconductors(TSCs). In this paper, we attempt to investigate the topological nature of the neutrino by assuming that a relativistic Majorana fermion can be divided into four topological Maj...

We give a simple model to explain the origin of fermion families, and chirality through the use of a domain wall placed in a five dimensional space-time.

Fermion-number fractionalization without breaking of time-reversal symmetry was recently demonstrated for a field theory in $(2+1)$-dimensional space and time that describes the couplings between massive Dirac fermions, a complex-valued Higgs field carrying an axial gauge charge of 2, and a U(1) axial gauge field. Charge fractionalization occurs whenever the Higgs field either supports vortices by itself, or when these vortices are accompanied by half-vortices in the axial ga...

Majorana modes and fractional fermions are two types of edge zero modes appearing separately in topological superconductors and dimerized chains. Here we reveal how to harvest both types of edge modes simultaneously in an exotic chain. Such modes are naturally spin-charge separated, and are protected by the inversion and spin-parity symmetries. We construct a lattice model to illustrate the nature of these edge modes, utilizing fermionic functional renormalization group, mean...

Eight Majorana fermions in $d=1+1$ dimensions enjoy a triality that permutes the representation of the $SO(8)$ global symmetry in which the fermions transform. This triality plays an important role in the quantization of the superstring, and in the analysis of interacting topological insulators and the associated phenomenon of symmetric mass generation. The purpose of these notes is to provide an introduction to the triality and its applications, with careful attention paid t...

We describe the occurrence and physical role of zero-energy modes in the Dirac equation with a topologically non-trivial background.