Symmetry analysis is a cornerstone of modern physics, with charge- and spacetime-orientation-reversal (CRT) symmetry being a subject of particular interest. Recent research has revealed that the CRT symmetry for fermions exhibits a fractionalization distinct from the ${\mathbb{Z}}_2^{\mathcal{C}}\times {\mathbb{Z}}_2^{\mathcal{R}}\times {\mathbb{Z}}_2^{\mathcal{T}}$ symmetry for scalar bosons. In fact, the CRT symmetry for fermions can be extended by internal symmetries such as fermion parity ${\mathbb{Z}}_2^F$, chiral symmetry ${\mathbb{Z}}_2^{\chi }$, and continuous symmetries, thereby forming a group extension of the aforementioned ${\mathbb{Z}}_2$ direct product, and suffices to rule out bilinear mass terms. In the conventional framework, a Majorana fermion is defined by a single Dirac fermion with trivial charge conjugation. However, this definition encounters a fundamental challenge when the spacetime dimension $d+1=5,6,7mod8$, where the real dimension of Majorana fermion (${\dim }_{\mathbb{R}}{\chi }_{\mathcal{C}\ell (d,0)}$) aligns with the real dimension of Dirac fermion (${\dim }_{\mathbb{R}}{\psi }_{\mathcal{C}\ell \left(d\right)}$), rather than being half as in other dimensions. This peculiarity necessitates the introduction of a symplectic Majorana fermion, defined by a pair of Dirac fermions with trivial charge conjugation, to account for the discrepancy. To include these two types of Majorana fermions, we embed the Majorana theory in $n_{\mathbb{R}}$ and define the Majorana fermion field as a representation of the real Clifford algebra, which exhibits an eightfold Bott periodicity. Within the Hamiltonian formalism, we identify the eightfold CRT-internal symmetry groups across general spatial dimensions. In the case of Dirac fermions, the fermion field is defined as a representation of the complex Clifford algebra, which has a twofold Bott periodicity. Interestingly, we discover that the CRT-internal symmetry groups exhibit an eightfold periodicity that is distinct from that of the complex Clifford algebra. In certain dimensions where distinct mass terms can span a mass manifold, the CRT-internal symmetries can act nontrivially upon this mass manifold. Employing domain wall reduction method, we are able to elucidate the relationships between symmetries across different dimensions.

Close

PrevNext

View figure in article

• Dirac fermions
• Fermions
• Fractionalization
• Majorana fermions
• Mathematical physics
• Quantum field theory
• Weyl fermions
• Strongly correlated systems
• C-symmetry
• CPT symmetry
• Discrete symmetries
• P-symmetry
• T-symmetry