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

We study faithful representations of the discrete Lorentz symmetry operations of parity $\mathbf P$ and time reversal $\mathbf T$, which involve complex phases when acting on fermions. If the phase of $\mathbf P$ is a rational multiple of $\pi$ then $\mathbf P^{2 n}=1$ for some positive integer $n$ and it is shown that, when this is the case, $\mathbf P$ and $\mathbf T$ generate a discrete group, a dicyclic group (also known as a generalised quaternion group) which are genera...

Electron fractionalization is intimately related to topology. In one-dimensional systems, fractionally charged states exist at domain walls between degenerate vacua. In two-dimensional systems, fractionalization exists in quantum Hall fluids, where time-reversal symmetry is broken by a large external magnetic field. Recently, there has been a tremendous effort in the search for examples of fractionalization in two-dimensional systems with time-reversal symmetry. In this lette...

I show how the symmetry-breaking of a recently proposed embedding of the standard model of particle physics in $E_8$ can be explained in terms of the representation theory of the binary tetrahedral group. This finite group provides a link between various types of spin and isospin that can be exploited to `explain' the chirality of the weak interaction, and the existence of three generations of fermions. Two apparently small technical differences between the finite group model...

It is known that the $2+1$d single Majorana fermion theory has an anomaly of the reflection, which is canceled out when 16 copies of the theory are combined. Therefore, it is expected that the reflection symmetric boundary condition is impossible for one Majorana fermion, but possible for 16 Majorana fermions. In this paper, we consider a reflection symmetric boundary condition that varies at a single point, and find that there is a problem with one Majorana fermion. The prob...

A self-contained derivation of the formalism describing Weyl, Majorana and Dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous Lorentz group. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory which are commonly not included in introductory courses when treating fermionic fields via the Dirac equation in the first p...

The construction of CP-invariant lattice chiral gauge theories and the construction of lattice Majorana fermions with chiral Yukawa couplings is subject to topological obstructions. In the present work we suggest lattice extensions of charge and parity transformation for Weyl fermions. This enables us to construct lattice chiral gauge theories that are CP invariant. For the construction of Majorana-Yukawa couplings, we discuss two models with symplectic Majorana fermions: a m...

In this work, we study the universal total and symmetry-resolved entanglement spectra for a single interval of some $2$d Fermionic CFTs using the Boundary Conformal Field theory (BCFT) approach. In this approach, the partition of Hilbert space is achieved by cutting out discs around the entangling boundary points and imposing boundary conditions preserving the extended symmetry under scrutiny. The reduced density moments are then related to the BCFT partition functions and ar...

It is shown that certain fractionally-charged quasiparticles can be modeled on \(D-\)dimensional lattices in terms of unconventional yet simple Fock algebras of creation and annihilation operators. These unconventional Fock algebras are derived from the usual fermionic algebra by taking roots (the square root, cubic root, etc.) of the usual fermionic creation and annihilation operators. If the fermions carry non-Abelian charges, then this approach fractionalizes the Abelian c...

Recent researches show that by breaking inversion symmetry Dirac fermions can split into new fermions with 3-component. In this article, we demonstrate that Dirac fermions can also split into 3-component fermions with time reversal symmetry (TRS) breaking while inversion symmetry is preserved. Firstly, we conduct a symmetry analysis with the commutation relations among all symmetry operators of a Dirac semimetal and find out the symmetry conditions of Dirac fermions splitting...

We describe a novel class of quantum mechanical particle oscillations in both relativistic and nonrelativistic systems based on $PT$ symmetry and $T^2=-1$, where $P$ is parity and $T$ is time reversal. The Hamiltonians are chosen at the outset to be self-adjoint with respect to a PT inner product. The quantum mechanical time evolution is based on a modified CPT inner product constructed in terms of a suitable C operator. The resulting quantum mechanical evolution is shown to ...