Layer Construction of 3D Topological States and String Braiding Statistics

While the topological order in two dimensions has been studied extensively since the discovery of the integer and fractional quantum Hall systems, topological states in three 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 topological field-theory description for the layer-constructed systems, which captures not only the string-particle braiding statistics but also the string-string braiding statistics when the coupling is twisted. Last, we provide a proof of a general identity for Abelian string statistics and discuss an example system with non-Abelian strings.

Popular Summary

Under certain circumstances, unexpected collective behaviors can “emerge” in a sea of electrons or a jungle of spins that bear no resemblance to the individual behavior of their fundamental building blocks. Physicists have been searching for the universal concepts that lie at the core of all emergent phenomena. Symmetry, which has long been considered to be such a concept, enables scientists to understand many emergent behaviors, albeit collective, through local descriptions. After the discovery of fractional quantum Hall states in the 1980s, the concept of topology emerged, which is intimately connected to the underlying long-range entanglement that is insensitive to local behaviors of a system. We propose a general scheme of constructing three-dimensional topological states by stacking layers of two-dimensional topological states and coupling them.

The majority of topological phases that have been studied thus far reside in two spatial dimensions. These phases exhibit very intriguing properties based upon which quantum qubits and quantum gates are designed, at least theoretically. Some of these phases have been realized in experiments and might become the building blocks of quantum computers in the future. In contrast to the two-dimensional case, scientists know very little about the topological properties of three-dimensional phases. We propose how to engineer interlayer couplings between two-dimensional Abelian sheets to produce three-dimensional topological structures. We find that we are able to selectively induce topological order in the surface and/or bulk of the three-dimensional materials and we propose a field-theory description to explain our findings. Furthermore, we discuss the string-string and particle-string braiding statistics of the three-dimensional materials, which are thermodynamically stable.

We anticipate that our work will prompt new studies exploring and characterizing three-dimensional topological states.

Figure 1

(a) Each line represents a layer of a 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 the upper and lower layers is given by $p_i$’s and $q_i$, respectively. (b) For an open boundary system in the $z$ direction, $q_i$’s ($p_i$’s) are deconfined particles on the top (bottom) surface that 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 $p\equiv 1,2\mathrm{mod}3$, there is no deconfined particle in the 3D bulk. The deconfined particles only stay on the surface, which is generated by two elementary particles shown in the green regions. (The case of $p\equiv 0\mathrm{mod}3$ is discussed later in Fig. 7.)

Figure 3

In the discussion of situation (ii) (see the text), we consider the braiding between a deconfined particle $X_{1,m2}$ (green region) with a condensed null particle $n_i^{(m_2)}$ (brown region).

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 $⟨{\mathrm{\Lambda }}_i^{m+1/2}{⟩}_0$ are pinned to the minima determined by the cosine terms in Eq. (28). The coloring on the right-hand side indicates nontrivial but uniform $⟨{\mathrm{\Lambda }}_i^{m\pm 1/2}{⟩}_0$ on the side surface. (b) The operator $e^{i{p}_j^T{\mathrm{\Phi }}^m(x)}$ creates a collection of kinks (depicted as the change in color) in $⟨{\mathrm{\Lambda }}_i^{m+1/2}(x)⟩$, which will be identified as a topological quasiparticle $w_j$ on the ($m+1/2$)th layer of the side surface. (c) The operator ${\chi }_j^m$ creates a kink-antikink pair that effectively tunnels the quasiparticle $w_j$ from the ($m-1/2$)th layer to the ($m+1/2$)th layer.

Figure 5

This figure illustrates the braiding between two quasiparticles of the surface topological order on the side surface. The blue dot represents the quasiparticle $w_j$ created by the operator $e^{i{p}_j^T{\mathrm{\Phi }}^{(m)}}$. The red line represents the trajectory of the second quasiparticle $w_k$ that the first one braids with. The operators that tunnel $w_k$ along the trajectory are explained in the main text. Especially, the vertical tunnel operator from the $m^{\prime }$th layer to the ($m^{\prime }+1$)th layer is given by ${\chi }_k^{m^{\prime }}$.

Figure 6

(a) The condensation of $e$ and $-e$ composite in the two consecutive layers in the system of stacked layers of $Z_p$ toric codes. (b) The electric particle in each layer is a deconfined particle that should be identified as the electric particle in the 3D $Z_p$ lattice gauge theory. The string of $m$ particles in every layer is a deconfined string excitation that can be viewed as the flux string in the $Z_p$ gauge theory. The flux strings do not have to align strictly along the $z$ direction. A contractible flux string of finite size, as is shown in the figure, can be considered. (c) A different set of condensed particles in layers of $Z_p$ toric code, the condensation of which leads to the twisted lattice gauge theory (see the text). (d) The deconfined particles in the twisted theory, including an electric particle and the “twisted” flux string.

Figure 7

(a) The condensed particle in the coupled $Z_p$ toric code system. This case is the same as the case studied in Sec. 3a, but here, we focus on the cases with $p\equiv 0(\mathrm{mod}3)$. (b) The bulk deconfined excitations in this system. The one on the left is a point particle, while the one on the right is a string excitation. They have nontrivial mutual statistics.

Figure 8

(a) The condensed particles for a more generic layer-constructed 3D state. Each layer contains two identical copies of $Z_p$ toric code, with $p$ a multiple of 4. $c_{1,2}$ and $f_{1,2}$ are the charge and flux particles in the two $Z_p$ toric code systems, respectively. (b) The set of deconfined excitations, including two types of point particles and two types of string excitations with nontrivial mutual braiding.

Figure 9

(a) The definition of a $c$ string (blue line) and an $f$ string (orange line). Each plane represents a layer of the 2D topological state. The convention on the orientation of the string is explained in the main text. (b1) The braiding process of the two strings. (b2)–(b4) The braiding process shown in decomposed steps.

Figure 10

(a) The configuration of the $c$ string and the $f$ string that link with an edge dislocation (or a $d$ string) which is indicated by the purple line. (b1) The braiding process of the $c$ string and the $f$ string that are both linked with the $d$ string. (b2)–(b4) The decomposed steps in this braiding process.

Figure 11

One $c$ string can be deformed and split into two $c$ strings through the fusion of $c_2$’s within the layers. The reverse process can be viewed as the fusion between two $c$ strings.

Figure 12

(a)–(d) An alternative string braiding process that is topologically equivalent to that shown in Fig. 10. Upon fusion and splitting of the strings, the process in (b) can be deformed to the combined process shown in (e)–(h). (i) The local identification of string configuration under splitting and fusion, which plays a central role in the reinterpretation of the process shown in (b). The total Berry’s phase in the string braiding process equals the Berry’s phase in the process induced by the braiding of the link $L_{c,f}$ around the $d$ string, as is shown in (j).

Figure 13

(a) The configuration of the link $L_{c,f}$ with the quasiparticles associated with it in each layer. The layers are horizontal and are not drawn in this figure. In this link $L_{c,f}$, the $c$ string and the $f$ string have the linking number 1. Their orientation is indicated by the colored arrow (blue for the $c$ string and orange for the $f$ string). (b) The operation that brings the linked part of the two strings down by one layer along the $-z$ direction is a $\pi$ rotation of the part of the link indicated by the dashed gray box.

Figure 14

(a) The configuration of two strings ${\mathrm{\Sigma }}_I$ and ${\mathrm{\Sigma }}_J$ that are aligned along the $z$ direction and penetrate the $z=0$ plane (green region) where $\mathrm{\Theta }$ jumps by $2\pi$. The strings ${\mathrm{\Sigma }}_I$ and ${\mathrm{\Sigma }}_J$ have nontrivial braiding. (b) The configuration of a finite open surface $\gamma$ across which $\mathrm{\Theta }$ jumps by $2\pi$ together with strings ${\mathrm{\Sigma }}_I$ and ${\mathrm{\Sigma }}_J$ that are linked with the vortex loop $\partial \gamma$ (purple border).

Figure 15

Three different string braiding processes, named (a) the string-string braiding, (b) the link-string braiding, and (c) the linked braiding. As is explained in the main text, the Berry’s phases associated with these three processes are identical. (d) The decomposed steps that relate processes in (c) and (b).

Figure 16

The configuration of three mutually linked strings. This configuration is topologically equivalent to the configuration shown in Fig. 15 but is deformed to a shape with threefold rotation symmetry according to an axis perpendicular to the plane at the point $o$.

Figure 17

(a),(b) Two different configurations of chiral vortex strings.