Symmetry Fractionalization and Projective Quantum Numbers

Symmetry fractionalization means that an isolated anyon can transform projectively even though every local physical state carries a linear representation. The phase ambiguity is possible because symmetry can be implemented on separated quasiparticles only up to gauge transformations and topological charge. Fusion consistency and anomaly constraints decide which assignments are physical in the declared dimension.

Required background. Parton constraints supplies gauge redundancy; projective representations supplies factor sets; anyon quasiparticles supplies fusion sectors.

Helpful background. Higher-form symmetry operators supplies extended-operator viewpoints.

Projective action on an anyon

If symmetry does not permute anyon types, its localized action on sector $a$ can satisfy

$$U_g^{(a)}U_h^{(a)}=\eta_a(g,h)U_{gh}^{(a)}.$$

Redefining $U_g^{(a)}\mapsto\lambda_a(g)U_g^{(a)}$ changes $\eta_a$ by a coboundary, so only its cohomology class is meaningful. For Abelian topological order, all sectors together are organized by $H^2(G,\mathcal A)$ under standard assumptions, where $\mathcal A$ is the Abelian anyon fusion group Essin and Hermele 2013.

Fusion enforces compatibility. If $a\times b=c$ in a unique channel, then the fractional phases multiply, schematically $\eta_a\eta_b=\eta_c$ after the appropriate braiding conventions are included. A local excitation has trivial topological charge and must carry an ordinary allowed microscopic representation.

Examples include a spinon with $T^2=-1$ even when the microscopic unit cell contains integer total spin, or translations acting as

$$T_x^{(a)}T_y^{(a)}=\eta^{(a)}_{xy}T_y^{(a)}T_x^{(a)}, \qquad \eta^{(a)}_{xy}=\pm1.$$

The minus sign means the anyon experiences a background $\pi$ flux per unit cell. It can enforce enhanced periodicity in a two-anyon continuum, but boundary conditions and other sectors must be accounted for.

Projective symmetry groups and gauge equivalence

In a parton ansatz, a microscopic symmetry may preserve the links only after a gauge transformation. The combined operations form a projective symmetry group. Gauge-related ansätze share the same physical symmetry action; raw signs on bonds are not invariant. After projection, one extracts gauge-invariant symmetry fractionalization from quasiparticle quantum numbers, symmetry defects, or entanglement.

Not every algebraically consistent fractionalization pattern is realizable in a strictly two-dimensional system. Some are anomalous and can occur only at a boundary of a three-dimensional SPT. If symmetry permutes anyons, ordinary $H^2$ data are insufficient and twisted categorical consistency is required Barkeshli et al. 2019, §§ III–V.

Exercise

Two identical spinons each have $T^2=-1$ and fuse to a local spin singlet. What is $T^2$ on the pair?

Solution

The projective signs multiply, giving $(-1)(-1)=+1$ for the local pair, consistent with a linear time-reversal representation. Additional exchange signs must be handled when a specified fusion basis is used, but the local total sector cannot retain an isolated projective obstruction.

References

• Maissam Barkeshli, Parsa Bonderson, Meng Cheng, and Zhenghan Wang, “Symmetry Fractionalization, Defects, and Gauging of Topological Phases,” Physical Review B 100 (2019) 115147, doi:10.1103/PhysRevB.100.115147.
• Andrew M. Essin and Michael Hermele, “Classifying Fractionalization: Symmetry Classification of Gapped Z2 Spin Liquids in Two Dimensions,” Physical Review B 87 (2013) 104406, doi:10.1103/PhysRevB.87.104406.