Generalized Modular Transformations in $(3+1)\mathrm{D}$ Topologically Ordered Phases and Triple Linking Invariant of Loop Braiding

In topologically ordered quantum states of matter in $(2+1)\mathrm{D}$ (spacetime dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property that is directly related to global transformations of the ground-state wave functions on a torus (the modular transformations). On the other hand, there are theoretical descriptions of various topologically ordered states in $(3+1)\mathrm{D}$, which exhibit both pointlike and looplike excitations, but systematic understanding of the fundamental physical distinctions between phases, and how these distinctions are connected to quantum statistics of excitations, is still lacking. One main result of this work is that the three-dimensional generalization of modular transformations, when applied to topologically ordered ground states, is directly related to a certain braiding process of looplike excitations. This specific braiding surprisingly involves three loops simultaneously, and can distinguish different topologically ordered states. Our second main result is the identification of the three-loop braiding as a process in which the worldsheets of the three loops have a nontrivial triple linking number, which is a topological invariant characterizing closed two-dimensional surfaces in four dimensions. In this work, we consider realizations of topological order in $(3+1)\mathrm{D}$ using cohomological gauge theory in which the loops have Abelian statistics and explicitly demonstrate our results on examples with $Z_2\times Z_2$ topological order.

Popular Summary

Systems with topological order cannot be described by conventional symmetry breaking and local ordering (for example, ordered magnetic moments in magnets) but are instead defined by their long-range quantum entanglement. These systems, including fractional quantum Hall states and quantum spin liquids, have been studied extensively in two spatial dimensions. A famous and experimentally interesting signature of topological order is that quasiparticle excitations are fractionalized, meaning that the excitations, upon encircling each other (i.e., by being braided), pick up a quantum phase that is somewhere between the usual values $+1$ (bosons) and $-1$ (fermions). We study the case of three spatial dimensions and find that when the braiding is Abelian (i.e., braiding changes the state by only an overall quantum phase), modular transformations—certain global twists that one performs on the periodic system—contain two pieces of information: one that describes the result of braiding particles through loops and another that describes a certain type of loop braiding.

Because of the long-range quantum entanglement in the state, it seems appropriate that global twists can extract the signature information about the topological order. Although the modular transformations are not feasible in experiments, they have recently been successively utilized in numerical measurements, confirming the expected behavior in two spatial dimensions. The specific type of loop braiding involved in three dimensions surprisingly involves three loops, instead of the simplest case of two. To universally characterize this three-loop braiding process, we study the loops’ worldsheets, which are two-dimensional surfaces (one dimension along the loop and one along time) living in four space-time dimensions. It turns out that in three-loop braiding, the worldsheets intertwine in a way that gives them a nonzero triple linking number, which is an integer analogous to the well-known “linking number” that characterizes the linking of closed strings (one-dimensional objects) in three dimensions and is commonly used in the study of knots.

Our results will be applicable in future numerical studies of three-dimensional topologically ordered states, which have recently also become experimentally relevant. These results furthermore open the door for understanding the physical interplay of looplike excitations in such states and their quantification in various models.

Figure 1

$S$ (left) and $T$ (right) transformations on the (a) two-torus and (b) three-torus, which are defined by periodic boundary conditions.

Figure 2

An $\mathcal{S}$-matrix element and braiding in $(2+1)\mathrm{D}$. (a) Matrix element equals an overlap of two MESs, shown as a time sequence. Any MES is defined by action of particle tunneling operator along $x$ on the appropriate reference state defined by $x$ direction. The MES at later time (blue) has been acted on by $S$ modular transformation, so the tunneling is along $y$, and the final reference state is defined by $y$ direction. (b) Embedding the spacetime process of (a) in three dimensions shows that the two worldlines are linked. Time grows in the radial direction as shown, so the spacetime of (a) spans the volume of a toroidal slab. (c) Alternatively, the matrix element equals a product of tunneling operators, also presented as a time sequence. Arrows mark the action of tunneling operator and its inverse. Because of taking the expectation value of this operator product, the initial and final state are the same, in contrast to (a). (d) Connecting the worldlines from (c), one obtains two worldlines that are linked.

Figure 3

Time sequence for three-flux-loop braiding. This process has nontrivial triple linking number of three worldsheets. First, loop $G$ (red) is created and grows, forming the $G$ worldsheet. Then, loop $H$ (black) emerges, encircling loop $G$ halfway. Then, loop $F$ (blue) completely encircles loop $H$. After this, loop $H$ finishes the route around loop $G$. Finally, loop $G$ is annihilated.

Figure 4

The 4-cocycle $\omega$ assigns a $U(1)$ complex number ${\omega }^{ϵ}(g_{54},g_{43},g_{32},g_{21})$ to a 4-simplex, where $ϵ$ is the chirality of the 4-simplex, defined as $ϵ=\mathrm{sgn}[\det (\overrightarrow{12},\overrightarrow{23},\overrightarrow{34},\overrightarrow{45})]$. The dashed lines represent that the vertex 5 has a different coordinate in the fourth dimension (time) with respect to the other vertices.

Figure 5

Geometric meaning of 3-cocycle ${\beta }_s(c,b,a)$ corresponds to evolution (along fourth dimension) of tetrahedron [1234] to [$1^{\prime }{2}^{\prime }{3}^{\prime }{4}^{\prime }$].

Figure 6

Evolution from a triangle to a 4D manifold. Phase associated with this colored manifold is ${\gamma }_{a,b}^{ϵ}(c,d)$, where $ϵ=\mathrm{sgn}[\det (\overrightarrow{d},\overrightarrow{c},\overrightarrow{b},\overrightarrow{a})]$. This phase can also be written as ${\gamma }_{b,a}^{{ϵ}^{\prime }}(c,d)$, where ${ϵ}^{\prime }=\mathrm{sgn}[\det (\overrightarrow{d},\overrightarrow{c},\overrightarrow{a},\overrightarrow{b})]=-ϵ$. So, we conclude that ${\gamma }_{a,b}(c,d)={\gamma }_{b,a}^{-1}(c,d)$.

Figure 7

The simplest triangulation of three-torus has a single vertex and three independent edges. Periodic boundary conditions are imposed on the cube.

Figure 8

Evolution from single vertex to two vertices.

Figure 9

The action of membrane operators.

Figure 10

A positive triple point (left) and a negative triple point (right), where we denote the orientations of sheets by their normals.

Figure 11

Time sequence for process $⟨G^{-1}{F}^{-1}{H}^{-1}FHG⟩$. The worldsheets in this process share the same topological properties as the three-flux-loop braiding process in Fig. 3.

Figure 12

Projection of the membrane process time sequence in Fig. 11 to three-dimensional space at $t=-\infty$. Lines show the pairwise intersections of projected worldsheets. Purple lines, $F$ and $G$ worldsheets; orange lines, $F$ and $H$; green lines, $G$ and $H$. Although there are eight triple points here, the triple linking is still the same as for the three-flux-loop braiding process, Fig. 13. The directions $t_{1,2,3}$ show the time ordering of contributions to projection from worldsheets $G$, $H$, $F$, and so clarify to which $Tl{k}_{IJK}$ some triple point contributes [see after Eq. (33)]. For example, at point $a$, direction of $t_1$, $t_2$, $t_3$ shows that worldsheet projection at this point comes from $G$ rather than $G^{-1}$, $H^{-1}$ rather than $H$, $F^{-1}$ rather than $F$, respectively. Therefore, point $a$ contributes to $Tl{k}_{FHG}$.

Figure 13

Time sequence in Fig. 3 projected to three-dimensional space at $t=-\infty$. Triple points are marked $a$, $b$, $c$, $d$. Lines show the pairwise intersections of projected worldsheets. Purple lines, $F$ and $G$ worldsheets; orange lines, $F$ and $H$; green lines, $G$ and $H$. The projected $G$ worldsheet in this figure takes the form of the sphere; $H$ and $F$ take the form of tori (not shown). The directions $t_{1,2}$ show the time ordering of contributions to projection from worldsheets $G$, $H$, and so clarify to which $Tl{k}_{IJK}$ some triple point contributes [see after Eq. (33)]. For example, at point $c$, $t_2$ shows that projection of $H$ at this point comes at later times, so after $F$, while $t_1$ shows that $G$ comes from earlier times, so before both $F$ and $H$, altogether contributing to $Tl{k}_{HFG}$. This process has the same triple linking number as the one in Fig. 12.