Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding

TL;DR

This work generalizes modular transformations to 3+1D topologically ordered phases by formulating a cohomological gauge theory on the three-torus and constructing a minimum-entropy state (MES) basis. It shows that the 3D S and T transformations are encoded by the braiding of three flux-loops, with the triple linking number (TLN) of their world sheets serving as the fundamental invariant underpinning these processes. The authors explicitly demonstrate this in Abelian $G=Z_2\times Z_2$ cohomological models, deriving both the S/T data and the TLN values for three-loop braidings, and showing that membrane operator algebras reproduce the same braiding information as MES overlaps. The results provide a concrete, geometrical nonlocal order parameter for 3+1D topological order, with potential implications for SPT/SET classifications and extensions to non-Abelian loop statistics.

Abstract

In topologically ordered quantum states of matter in 2+1D (space-time dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property which is directly related to global transformations of the ground-state wavefunctions on a torus (the modular transformations). On the other hand, there are theoretical descriptions of various topologically ordered states in 3+1D, which exhibit both point-like and loop-like 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 loop-like 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 non-trivial 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+1D 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.

Figures (13)

• 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+1D. (a) Matrix element equals an overlap of two MES, 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 space-time process of (a) in three dimensions shows that the two worldlines are linked. Time grows in the radial direction as shown, so the space-time 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. Due to taking the expectation value of this operator product, the initial and final state are the same, in contrast to panel (a). (d) Connecting the worldlines from (c) one obtains two worldlines that are linked.
• Figure 3: (color online) Movie for three-flux-loop braiding. This process has nontrivial triple linking number of three worldsheets. Firstly, 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 $\omega^{\varepsilon}(g_{54},g_{43},g_{32},g_{21})$ to a 4-simplex, where $\varepsilon$ is the chirality of the 4-simplex, defined as $\varepsilon=\mathrm{sgn}[\mathrm{det}(\vec{12},\vec{23},\vec{34},\vec{45})]$. 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 [$1'2'3'4'$].