Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order

TL;DR

The paper presents an exactly solvable 3D lattice realization of a bosonic SPT protected by time-reversal symmetry, with a unique gapped bulk and a surface that hosts a deconfined 3-fermion Z2 topological order. The construction employs a Walker-Wang framework culminating in a surface theory whose anyons e, m, and ε are all fermions with mutual semionic statistics, an arrangement that cannot occur in purely 2D systems with time-reversal symmetry. The bulk exhibits confinement of excitations, ensuring no deconfined bulk order, while the boundary supports anomalous topological order consistent with a nontrivial 3D SPT phase beyond group cohomology. This work connects lattice realizations to field-theoretic predictions (BTSc and E8-related phases) and provides a flexible route to extend the approach to other symmetry classes and topological orders.

Abstract

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 anyonic excitations 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 3-fermion Z2 model, a variant of Z2 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 SPT phases, which only becomes available in D=3: that of a fully symmetric gapped surface with topological order. The 3D phase found here is also outside the group cohomology classification that appears to capture all SPT phases in lower dimensions. It is identified with a phase previously predicted from a field theoretic analysis, whose surface is roughly `half' of a Kitaev E8 phase. Our construction utilizes the Walker-Wang prescription to create a 3D confined phase with surface anyons,which can be extended to other symmetry classes and topological orders.

Figures (10)

• 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 Walker2012. These phases lead precisely to the extra signs associated with the $O$ and $U$ links in eqs. \ref{['eq:bpe']} and \ref{['eq:bpm']}.
• 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.