Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order

We construct an exactly soluble Hamiltonian on the $D=3$ cubic lattice, whose ground state is a topological phase of bosons protected by time-reversal symmetry, i.e., a symmetry-protected topological (SPT) phase. In this model, excitations with anyonic statistics are shown to exist at the surface but not in the bulk. The statistics of these surface anyons is explicitly computed and shown to be identical to the three-fermion ${\mathbb{Z}}_2$ model, a variant of ${\mathbb{Z}}_2$ topological order which cannot be realized in a purely $D=2$ system with time-reversal symmetry. Thus the model realizes a novel surface termination for three-dimensional (3D) SPT phases, that of a fully symmetric gapped surface with topological order. The 3D phase found here was previously proposed from a field theoretic analysis but is outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. Such phases may potentially be realized in spin-orbit-coupled magnetic insulators, which evade magnetic ordering. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons, which can be extended to other topological phases.

Figure 1

Choice of links on which $B_P^{(e/m)}$ act for the three different types of plaquettes in the lattice. In the chosen projection $O$ links (red) cross over the plaquette $P$, and $U$ links (blue) cross under it.

Figure 2

The plumber's nightmare geometry. Our model is defined on the links of the blue lattice; its dual lattice is displayed in red.

Figure 3

The blue arrows describe the nucleation, transport, and fusion and splitting process performed to construct a basis state in the 2D chiral theory with specified, well-defined link quantum numbers. The purple arrows describe the process corresponding to a plaquette term. Expressing this plaquette term in the aforementioned basis amounts to fusing the purple and blue processes using associativity and braid phases, exactly as in [27]. These phases lead precisely to the extra signs associated with the $O$ and $U$ links in Eqs. (3) and (4).

Figure 4

Excitations in the bulk are confined. The path $C_{12}$ is shown in red; the displaced path used to determine ${}^{*}{C}_{12}$ is indicated with a dashed blue line. Links that cross under (over) this path are colored green (purple). The violated plaquettes (shaded blue) are those that are threaded by the dashed blue line.

Figure 5

Mutual statistics. The operation of braiding a pair of anyons (say, the $e$ and $m$ particles) is captured by first creating a pair of $e$ particles (red) followed by a pair of $m$ particles (blue) in the manner shown. We now annihilate first the $e$ and then the $m$ particles to return to the vacuum and examine the resulting phase.

Figure 6

Fermionic statistics: the sequence of operations described in the text to exchange a pair of fermions. The positions of the fermions are indicated by the red dots. [The open red circles in (b)–(d) indicate the original positions of the fermions.] Solid red arrows indicate the link acted on by a string operator to move the fermion at this step; the solid red lines show where the string operators have acted at previous steps. The links crossed by the dashed blue line are in ${}^{*}C$: there is a phase of $-1$ every time a dashed blue line crosses a solid red line.

Figure 7

The low-energy Hilbert space of the lattice model consists of loops of three colors that satisfy fusion rules at the vertices; that is, either they are closed loops of a single color, or segments of three colors can meet at a vertex.

Figure 8

Braiding rules for strings in the ground-state wave function. (a) applies to strings of the same color, while (b) applies to strings of different colors.

Figure 9

Open strings in the bulk change the quantum fluctuation phase factors of loops along its length, which costs finite energy. Therefore, the ends of strings are confined in the bulk.

Figure 10

The anyonic excitations on the surface are created by open strings. At the ends of the strings are three species of fermions (corresponding to three colors of open strings) which have mutual semionic statistics. This can be seen from the braiding statistics of the strings generating (a) the exchange and (b) the braiding of the ends of the strings.