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

Motivated by recent examples of three-dimensional lattice Hamiltonians with massless Dirac fermions in their (bulk) spectrum, I revisit the problem of fermion doubling on bipartite lattices. The number of components of the Dirac fermion in a time-reversal and parity invariant d-dimensional lattice system is determined by the minimal representation of the Clifford algebra of $d+1$ Hermitian Dirac matrices that allows a construction of the time-reversal operator with the square...

This review is based on lectures given by M. J. Duff summarising the far reaching contributions of Ettore Majorana to fundamental physics, with special focus on Majorana fermions in all their guises. The theoretical discovery of the eponymous fermion in 1937 has since had profound implications for particle physics, solid state and quantum computation. The breadth of these disciplines is testimony to Majorana's genius, which continues to permeate physics today. These lectures ...

Lorentz invariant quantum field theories (QFTs) in four spacetime dimensions (4D) have a $\mathbb{Z}_4$ symmetry provided there exists a basis of operators in the QFT where all operators have even operator dimension, $d$, including those with $d > 4$. The $\mathbb{Z}_4$ symmetry is the extension of operator dimension parity by fermion number parity. If the $\mathbb{Z}_4$ is anomaly-free, such QFTs can be related to 3D topological superconductors. Additionally, imposing the $\...

One purpose of this proceedings-contribution is to show that at least for free massless particles it is possible to construct an explicit boson theory which is exactly equivalent in terms of momenta and energy to a fermion theory. The fermions come as $2^{d/2-1}$ families and the to this whole system of fermions corresponding bosons come as a whole series of the Kalb-Ramond fields, one set of components for each number of indexes on the tensor fields. Since Kalb-Ramond fiel...

We constructed all possible kinematically allowed three-point interactions of two massless Dirac spinors with massive higher-spin bosons. In any $D$ spacetime, the interactions have been constructed using the projections of the massive higher spin representations of $Spin(D-1)$ over the massless complex spinor representations of $Spin(D-2)\times Spin(D-2)$. Based on this analysis, we have further classified the space of theories involving one massless Dirac spinor and a singl...

Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with the time reversal (TR) symmetry satisfying $T^{2}=1$. Without extra symmetry, it has been found that $\mathbb{Z}_{p}$ parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labelled by the divisors $n$ of $p$. In the presence of ...

In spite of the breakthrough in non-perturbative chiral gauge theories during the last decade, the present formulation has stubborn artefacts. Independently of the fermion representation one is confronted with unwanted CP violation and infinitely many undetermined weight factors. Renormalization group identifies the culprit. We demonstrate the procedure on Weyl fermions in a real representation.

A pseudoclassical theories of Majorana, Weyl and Majorana--Weyl particles in the space--time dimensions $D=2n$ are constructed. The canonical quantization of these theories is carried out and as a result we obtain the quantum mechanical description of neutral particle in $D=2n$ , Weyl particle in $D=2n$ and neutral Weyl particle in $D=4n+2$. In $D=2,4({\rm mod}8)$ dimensional space--time the description of the neutral particle coincides with the field theoretical description ...

We discuss fermions for arbitrary dimensions and signature of the metric, with special emphasis on euclidean space. Generalized Majorana spinors are defined for $d=2,3,4,8,9$ mod 8, independently of the signature. These objects permit a consistent analytic continuation of Majorana spinors in Min-kowski space to euclidean signature. Compatibility of charge conjugation with complex conjugation requires for euclidean signature a new complex structure which involves a reflection ...

We present the domain-wall fermion operator which is reflection symmetric in the fifth dimension, with the approximate sign function $ S(H) $ of the effective 4-dimensional Dirac operator satisfying the bound $ |1-S(\lambda)| \le 2 d_Z $ for $ \lambda^2 \in [\lambda_{min}^2, \lambda_{max}^2] $, where $ d_Z $ is the maximum deviation $ | 1- \sqrt{x} R_Z(x) |_{\rm max} $ of the Zolotarev optimal rational polynomial $ R_Z(x) $ of $ 1/\sqrt{x} $ for $ x \in [1, \lambda_{max}^2/\l...