Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions

We study Abelian braiding statistics of loop excitations in three-dimensional gauge theories with fermionic particles and the closely related problem of classifying 3D fermionic symmetry-protected topological (FSPT) phases with unitary symmetries. It is known that the two problems are related by turning FSPT phases into gauge theories through gauging the global symmetry of the former. We show that there exist certain types of Abelian loop braiding statistics that are allowed only in the presence of fermionic particles, which correspond to 3D “intrinsic” FSPT phases, i.e., those that do not stem from bosonic SPT phases. While such intrinsic FSPT phases are ubiquitous in 2D systems and in 3D systems with antiunitary symmetries, their existence in 3D systems with unitary symmetries was not confirmed previously due to the fact that strong interaction is necessary to realize them. We show that the simplest unitary symmetry to support 3D intrinsic FSPT phases is ${\mathbb{Z}}_2\times {\mathbb{Z}}_4$. To establish the results, we first derive a complete set of physical constraints on Abelian loop braiding statistics. Solving the constraints, we obtain all possible Abelian loop braiding statistics in 3D gauge theories, including those that correspond to intrinsic FSPT phases. Then, we construct exactly soluble state-sum models to realize the loop braiding statistics. These state-sum models generalize the well-known Crane-Yetter and Dijkgraaf-Witten models.

Popular Summary

Topological phases are an exotic state where certain properties are independent of the details of the material involved and remain stable even as the material is deformed. There are huge differences though when comparing topological phases of fermions (particles such as electrons, which cannot occupy the same quantum state) and bosons (particles such as helium-4 atoms, which can overlap). For example, while there are many topological phases for noninteracting electrons, all topological phases of bosons require strong interactions. This raises the question as to whether interactions among fermions can lead to new topological phases. Using mathematical arguments, we uncover such a new class of fermionic interacting topological phases in three dimensions. Discovery of these phases of matter might help us to understand strongly correlated electronic materials.

In three dimensions, topological phases of matter are characterized by properties of both particlelike and looplike excitations. Particlelike excitations can be either bosonic or fermionic, depending on whether a minus sign appears in the wave function when one exchanges the positions of two identical particles. Exchanging properties of looplike excitations are even richer and depend on whether the excitations are linked or not. We show that there exists a class of interacting fermionic topological phases of matter, where certain properties of looplike excitations have an intrinsic relation to the existence of fermionic particles. To find these phases, we perform a systematic analysis on the relation between properties of particlelike and looplike excitations and then construct a family of exactly solvable lattice models to realize them.

Our work has focused on those fermionic topological phases that are associated with Abelian looplike excitations. Future work should explore new three-dimensional fermionic topological phases that come with non-Abelian loop-braiding statistics.

Figure 1

A ${\mathbb{Z}}_4$ symmetry defect in the ${\mathbb{Z}}_2^f\times {\mathbb{Z}}_2\times {\mathbb{Z}}_4$ intrinsic FSPT phase. There lives a 1D helical Dirac fermion (denoted by red and blue arrows) on the defect. The shaded region represents a branch surface associated with the defect.

Figure 2

Three-loop braiding process involving vortex loops $\alpha$, $\beta$, and $\gamma$. The blue lines are trajectories swept out by two points on $\alpha$.

Figure 3

Two ways of fusing loops. (a) Fusing ${\beta }_1$ and ${\beta }_2$, that are linked to the same base $\gamma$, into a new loop, denoted as ${\beta }_1\times {\beta }_2$. (b) Fusing ${\beta }_1$ and ${\beta }_2$, that are linked to different bases ${\gamma }_1$ and ${\gamma }_2$ and that carry the same amount of flux ${\varphi }_{{\beta }_1}={\varphi }_{{\beta }_2}$, into a new loop, denoted as ${\beta }_1\circ {\beta }_2$.

Figure 4

The first thought experiment.

Figure 5

The second thought experiment.

Figure 6

Illustration of a prism.