Layer construction of 3D topological states and string braiding statistics

TL;DR

The paper introduces a layer-construction framework to realize 3D topological order by stacking 2D Abelian layers and condensing interlayer composites, yielding phases with purely surface order or with bulk and surface order and enabling string-string braiding. It provides a concrete K-matrix formalism and null-condensation criteria to determine bulk versus surface excitations, and demonstrates explicit examples including $Z_p$ toric code stacks (and twists) that reproduce conventional and twisted 3D lattice gauge theories. A BF-type topological field theory with a space-time dependent theta term captures both particle-string and string-string braiding, and a general Abelian identity for three-string braiding is proven, with a non-Abelian string-braiding example discussed in a chiral vortex-string setup. The framework connects to Walker-Wang constructions, offers guidance for lattice realizations, and points to future work on non-Abelian layer structures and symmetry-protected/ enriched 3D topological orders, highlighting a versatile route to understanding 3D topological phases.

Abstract

While the topological order in two dimensions has been studied extensively since the discover of the integer and fractional quantum Hall systems, topological states in 3 spatial dimensions are much less understood. In this paper, we propose a general formalism for constructing a large class of three-dimensional topological states by stacking layers of 2D topological states and introducing coupling between them. Using this construction, different types of topological states can be obtained, including those with only surface topological order and no bulk topological quasiparticles, and those with topological order both in the bulk and at the surface. For both classes of states we study its generic properties and present several explicit examples. As an interesting consequence of this construction, we obtain example systems with nontrivial braiding statistics between string excitations. In addition to studying the string-string braiding in the example system, we propose a generic topological field theory description which can capture both string-particle and string-string braiding statistics. Lastly, we provide a proof of a general identity for Abelian string statistics, and discuss an example system with non-Abelian strings.

Figures (17)

• Figure 1: (a) Each line represents a layer of 2D Abelian topological state, and each brown circle represents a composite particle, the condensation of which will introduce the coupling between layers. Here each composite particle lives only in two consecutive layers, and the component of the particle in upper and lower layers is given by $p_i$'s and $q_i$, respectively. (b) For open boundary system in the $z$ direction, $q_i$'s ($p_i$'s) are deconfined particles on the top (bottom) surface which will form surface topological order. Depending on the choice of $\{p_i\}$ and $\{q_i\}$, there can be deconfined particles (represented by "$X$") in the 3D bulk that organize themselves to form 3D topological orders.
• Figure 2: (a) Illustration of the condensation of the composite particle of $e+m$, $e$ and $e-m$ in three consecutive layers, with each layer a $Z_p$ toric code state. (b) For $p\equiv 1,2 \text{ mod }3$, there is no deconfined particle in the 3D bulk. The deconfined particles only stay on the surface, which are generated by two elementary particles shown in the green circles. (The case of $p\equiv 0 \text{ mod }3$ is discussed later in Fig. \ref{['coexTopo']}.)
• Figure 3: In the discussion of situation ii) (see text), we consider the braiding between a deconfined particle $X_{1,m2}$ (green circle) with a condensed null particle $n_i^{(m_2)}$ (brown circle).
• Figure 4: (a) The condensation in the 3D bulk induces a coupling between the edge states of each layer. The side surface is effectively a coupled wire system. In the strong coupling limit, the vacuum expectation values $\langle \Lambda_i^{m+1/2}\rangle_0$ are pinned to the minima determined by the $\cos$ terms in Eq. (\ref{['SurfCoupling']}). The coloring on the right hand side indicates non-trivial, but uniform $\langle \Lambda_i^{m\pm 1/2}\rangle_0$ on the side surface. (b) The operator $e^{i p^T_j{\Phi^{m}(x)}}$ creates a collection of kinks (depicted as the change in color) in $\langle \Lambda_i^{m+1/2}(x)\rangle$, which will be identified as a topological quasi-particle $w_j$ on the $m+1/2^\text{th}$ layer of the side surface. (c) The operator $\chi_j^m$ creates a kink-anti-kink pair that effectively tunnels the quasi-particle $w_j$ from the $m-1/2\text{th}$ layer to the $m+1/2\text{th}$ layer.
• Figure 5: This figure illustrates the braiding between two quasi-particles of the surface topological order on the side surface. The blue dot represent the quasi-particle $w_j$ created by the operator $e^{ip_j^T \Phi^(m)}$. The red line represents the trajectory of the second quasi-particle $w_k$ that the first one braids with. The operators that tunnel $w_k$ along the trajectory is explained in the main text. Especially, the vertical tunnel operator from the ${m'}^\text{th}$ layer to the ${m'+1}^\text{th}$ is given by $\chi_k^{m'}$.