Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D

Abstract

In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group $G_f$ of the ground state splits as $G_f=G_b\times {\mathbb{Z}}_2^f$, where ${\mathbb{Z}}_2^f$ is generated by fermion parity, $(-1{)}^F$. In contrast, (2+1)D topological superconductors (TSC) can host unpaired MZMs at defects even when $G_f$ is not of the form $G_b\times {\mathbb{Z}}_2^f$. In this paper we study how $G_f$ together with the chiral central charge $c_{-}$ strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a non-perturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as $\mathrm{U}(1{)}^f⋊H,\mathrm{SU}(2{)}^f\times H,\mathrm{U}(2{)}^f⋊H$, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr${}_2$RuO${}_4$, transition metal dichalcogenides and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.

28 + 23 pages, 3 figures. Added some examples to improve presentation in Sec II

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• Definition and Classification of Fermi Surface Anomalies
• Rotational Symmetry Protected Edge and Corner States in Abelian topological phases
• Classification of symmetry-enriched topological quantum spin liquids
• Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators
• Crystalline invariants of fractional Chern insulators
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Boundary topological orders of (4+1)d fermionic ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ SPT states
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