Weakly coupled non-Abelian anyons in three dimensions

We introduce a Hamiltonian coupling Majorana fermion degrees of freedom to a quantum dimer model. We argue that, in three dimensions, this model has deconfined quasiparticles supporting Majorana zero modes obeying nontrivial statistics. We introduce two effective field theory descriptions of this deconfined phase in which the excitations have Coulomb interactions. A key feature of this system is the existence of topologically nontrivial fermionic excitations, called “Hopfions” because, in a suitable continuum limit of the dimer model, such excitations correspond to the Hopf map and are related to excitations identified by Ran et al. [Phys. Rev. B 84, 184501 (2011)]. We identify corresponding topological invariants of the quantum dimer model (with or without fermions) that are present even on lattices with trivial topology. The Hopfion energy gap depends upon the phase of the model. We briefly comment on the possibility of a phase with a gapped, deconfined ${\mathbb{Z}}_2$ gauge field, as may arise on the stacked triangular lattice.

Figure 1

Illustration of signs in matrix $M$ in two-dimensions. Arrows between sites show the signs. An arrow pointing from site $i$ to site $j$ implies that $M_{ij}=+i$ and $M_{ji}=-i$. There is $\pi$ flux around each plaquette.

Figure 2

The basic plaquette flip move depicted above does not change $n_H$.

Figure 3

(a) A trivial dimer configuration (thick blue lines) with Hopfion number $n_H=0$. (The blue pattern is assumed to be repeated outside of this 18-site section of the lattice.) (b) The simplest nontrivial dimer configuration (thick red lines) with $n_H=1$. We call such a configuration a “Hopfion,” following Ref.3. The arrows are determined according to the $\pi$-flux rule.

Figure 4

The octahedron is viewed as a discrete approximation to the sphere.

Figure 5

Examples of dimer configurations on a plaquette and the corresponding simplices of the octahedron.

Figure 6

An example of the refinement of the dimer configuration of a cube. A configuration, which is dimerized in the $|x_1|$ direction, is inserted for the central cube.

Figure 7

A plaquette flip is depicted on the left. A cyclic sequence of cube flips is depicted on the right. Each arrow corresponds to a single cube flip.

Figure 8

The 2-simplex is associated to “cube flip.”

Figure 9

A three-dimensional rendering of two possible height functions defining the top and bottom halves of the body.

Figure 10

The height function of an $x_1$ columnar configuration of dimers.

Figure 11

The octahedron is viewed as a discrete approximation to the sphere.

Figure 12

A unit vector field $\vec{n}$ with $N_H=1$.

Figure 13

Dimerization pattern with two defects with zero modes on each defect.

Figure 14

Labelling of points in a plaquette.

Figure 15

(a) and (b) Illustration of possible processes moving a defect and a dimer. The dashed arrow is used to indicate “before” and “after” configurations. In (a), the defect moves along a diagonal, while in (b) it moves along a straight line. (c) and (d) How to compute the signs in the defect motion. The arrows on the dimers indicate the arrows as in Fig. 1, while the dashed arrow again indicates “before” and “after.” Solid circles are used to indicate the defect site, while an open circle indicates that the defect site changes sign.

Figure 16

(a) Illustration of the $(4+1)$-d theory, which consists of lattice Dirac fermions coupled with an SU(2) gauge field described by the action (72). For $-2<M/B<0$, the bulk is gapped and topologically nontrivial, with one doublet of Weyl fermions on each surface. (b) When the Higgs field $\vec{n}$ is in the condensed phase, the surface Weyl fermions become gapped Majorana fermions, and the monopoles on the 3D surface are end points of one-dimensional monopole lines in the 4D bulk.

Figure 17

Illustration of the creation of a Hopfion in the bulk by creating a monopole-antimonopole pair, rotate the monopole by $2\pi$ and annihilate the pair. The Hopfion is a pointlike defect of the order parameter $\vec{n}$ in the bulk 4D space.

Figure 18

Illustration of the configuration with the order parameter in the Higgs phase in the S (“superconducting”) and vanishes in the N (“normal”) regions. Each perpendicular monopole line traps a single Majorana zero mode confined at the top endpoint, and two monopole lines are coupled by Coulomb interaction. The bottom surface state is a SU(2) doublet of gapless Weyl fermion.