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

TL;DR

This paper develops a comprehensive framework for C-R-T fractionalization, treating it as a group extension of the internal and spacetime symmetry group of Dirac fermions by fermion parity. For Dirac fermions with canonical CRT, the authors classify the extension by cohomology and reveal an eightfold periodic structure in spacetime dimension modulo 8, yielding two distinct order-16 nonabelian realizations. They extend the analysis to Majorana, Weyl, and Majorana-Weyl fermions, obtaining an order-8 R-T fractionalization for Majorana fermions in d=0,1,2,3,4 mod 8, an order-4 C or T fractionalization for Weyl fermions in even dimensions, and a trivial C for Majorana-Weyl fermions at d=2 mod 8. The work also derives the maximal sets of Dirac and Majorana mass terms, analyzes how conventional mass terms break C, R, or T, and studies domain-wall dimensional reduction, showing consistent reductions of the fractionalizations from d to d−1 dimensions. The results provide a robust, dimension-aware map between bulk and domain-wall symmetry structures with potential implications for topological phases and high-energy phenomenology.

Abstract

Charge conjugation (C), mirror reflection (R), time reversal (T), and fermion parity $(-1)^{\rm F}$ are basic discrete spacetime and internal symmetries of the Dirac fermions. In this article, we determine the group, called the C-R-T fractionalization, which is a group extension of $\mathbb{Z}_2^{\rm C}\times\mathbb{Z}_2^{\rm R}\times\mathbb{Z}_2^{\rm T}$ by the fermion parity $\mathbb{Z}_2^{\rm F}$, and its extension class in all spacetime dimensions $d$, for a single-particle fermion theory. For Dirac fermions, with the canonical CRT symmetry $\mathbb{Z}_2^{\rm CRT}$, the C-R-T fractionalization has two possibilities that only depend on spacetime dimensions $d$ modulo 8, which are order-16 nonabelian groups, including the famous Pauli group. For Majorana fermions, we determine the R-T fractionalization in all spacetime dimensions $d=0,1,2,3,4\mod8$, an order-8 abelian or nonabelian group. For Weyl fermions, we determine the C or T fractionalization in all even spacetime dimensions $d$, which is an order-4 abelian group. We only have an order-2 $\mathbb{Z}_2^{\rm F}$ group for Majorana-Weyl fermions. We determine the maximal number of linearly independent Dirac and Majorana mass terms and construct them explicitly. We also discuss how the conventional Dirac and Majorana mass terms break the symmetries C, R, or T. We study the domain wall dimensional reduction of the fermions and their C-R-T fractionalization: from $d$-dim Dirac to $(d-1)$-dim Dirac or Weyl and from $d$-dim Majorana to $(d-1)$-dim Majorana or Majorana-Weyl.

Figures (4)

• Figure 1: Mass profile. The $d$d fermion obeys the Lagrangian $\bar{\psi}^{d} (\space\mathrm{i}\space \Gamma^\mu \partial_\mu - m(x)) \psi^{d}$, then there is an effective massless $(d-1)$d domain wall fermion theory $\psi^{d-1}$ at $x_{d-1}=0$, with its time and spatial coordinates $(t,x_1,x_2,\dots,x_{d-2})$, which is obtained by the projection ${\rm P}_{\pm}= \frac{1 \pm \space\mathrm{i}\space \Gamma^{d-1}}{2}$, $\psi^{d-1}_{\pm} = {\rm P}_{\pm} \psi^{d}= \frac{1 \pm \space\mathrm{i}\space \Gamma^{d-1}}{2} \psi^{d}$ where ${\rm P}_{\pm}^2={\rm P}_{\pm}$, ${\rm P}_{+}{\rm P}_{-}={\rm P}_{-}{\rm P}_{+}=0$, and $\space\mathrm{i}\space\Gamma^{d-1}{\rm P}_{\pm}=\pm {\rm P}_{\pm}$. For odd $d$, $\Gamma^{d-1}=\space\mathrm{i}\space{\rm diag}(-I,I)$ in the chiral representation (see App. \ref{['app:Weyl']}). The Gamma matrices $\Gamma^{\mu}$ in $(d-1)$d are the same as those in $d$d for odd $d$ and $\mu=0,1,\dots,d-2$. Therefore, for odd $d$, the projection maps a Dirac (Majorana) fermion in $d$d to a Weyl (Majorana-Weyl) fermion in $(d-1)$d. For even $d$, the projection maps a Dirac (Majorana) fermion in $d$d to a Dirac (Majorana) fermion in $(d-1)$d. See also Fig. \ref{['fig:domain-wall-fermion']}.
• Figure 2: The distance between two domain walls is $x$, we can take $x \to0$, which becomes massive $-m$ bulk fermion, and $x \to\infty$, which becomes massive $+m$ bulk fermion, but degree of freedom matches. So 2 domain wall degree of freedom in $(d-1)$-dim = bulk degree of freedom in $d$-dim.
• Figure 3: Domain wall dimensional reduction of fermions. The target of the arrow is the domain wall of the source of the arrow. The number indicates the dimension of the representation of the fermion, the upper index indicates that the representation is real or complex, and the lower index indicates the chirality of the fermion.
• Figure 4: Domain wall dimensional reduction. Suppose $\text{S}_b$ is (spontaneously or explicitly) broken by mass, $\text{S}_b$ switches the two degenerate ground states, and assume that $\text{S}_b$ does not flip the spatial coordinate $x_{d-1}$ (which excludes ${\rm R}_{d-1}$, but $\text{S}_b$ can be ${\rm C}$, ${\rm R}_i$ for $i\ne d-1$, and ${\rm T}$, or other internal symmetries), we need to combine it with the space-orientation-reversing symmetry ${\rm C}{\rm R}_{d-1}{\rm T}$ to obtain a new symmetry on the domain wall HasonKomargodskiThorngren1910.140391910.14046Wang:2019obe1910.14664. However, some of the domain wall symmetries may not be directly induced from $\text{S}_b\cdot ({\rm C} {\rm R}_{d-1} {\rm T})$, the domain wall symmetries may be induced from the $\text{S}_b= {\rm R}_{d-1}$ itself or other symmetries.

Theorems & Definitions (21)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Proposition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6