Loops in 4+1d Topological Phases

TL;DR

This work analyzes 4+1d topological phases with only loop excitations, showing that all loop self-statistics are trivial while e, m, and dyons exhibit nontrivial mutual braiding. It presents a lattice and field-theory description via a Z2 2-form gauge theory, demonstrates invertible domain walls that map m to ψ, and reveals an obstruction to gauging loop-permutation symmetry for even N through a 3-group structure. The authors generalize to Z_N theories, classify gapped boundaries through Lagrangian subgroups and Higgsing, and construct an explicit unitary circuit implementing m↔ψ exchange, including a circuit-based low-energy realization. They also discuss gapless (fractional Maxwell) boundaries and the Higgsing pathways to gapped boundaries, tying these to higher-form and higher-group symmetries and identifying an H^4 obstruction that governs gauging consistency. Overall, the paper deepens understanding of higher-form topological order, boundary phenomena, and symmetry structures in 4+1d, with implications for higher-dimensional quantum information and topological phases.

Abstract

2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist? We study the 4+1d gauge theory with 2-form $\mathbb{Z}_2$ gauge field (the loop only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self `exchange' statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The $SL(2,\mathbb{Z}_2)$ symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the `fractional Maxwell theory' and show how it can be Higgsed into gapped boundary conditions. We also discuss the generalization of these results from the $\mathbb{Z}_2$ gauge group to $\mathbb{Z}_N$.

Figures (13)

• Figure 1: Hamiltonian of the 4+1d toric code model on 4D hypercubic lattice. (a) Coordinates $x$, $y$, $z$ and $w$ to represent 4 spatial dimensions (b) a representation of the 4-cell using two 3-cubes shifted along $w$ direction. $w=0$ is shown in black and $w=1$ is shown in blue and the $w$ edge between $w=0$ and $w=1$ is shown in pink. (c) a $2 \times 2 \times 2 \times 2$ lattice using this 4-cell (d) a $2 \times 2 \times 2 \times 3$ lattice; $w=-1$ is added in maroon to the lattice in (c) with the $w$ edge from $w=-1$ to $w=0$ shown in green. The qubits in the $(2,2)$ 4+1d toric code live on the 2-cells (e) $Z$ stabilizers supported on the 3-cubes $xzw$, $yzw$, $xyz$ and $xyw$. (f) $X$ stabilizers associated with the edges $w$, $y$, $z$ and $x$.
• Figure 2: Braiding $e$ and $m$ loop excitations.
• Figure 3: a) Smooth and b) Rough boundary conditions. The terms on the left are the flux terms ($Z$ stabilizers) which are 6-body for smooth boundary conditions shown in (a) and 5-body for rough boundary conditions shown in (b). The terms on the right are the charge terms ($X$ stabilizers) which are 5-body for smooth boundary conditions shown in (a) and 6-body for rough boundary conditions shown in (b).
• Figure 4: $e$ loop and $m$ loop condensation. $m$ loop condenses on the smooth boundary and the $e$ loop condenses on the rough boundary.
• Figure 5: Decorated smooth boundary. Left figure shows the coupling between smooth boundary and WW domain wall. Right figure shows the condensation of $\psi$ loop on the decorated smooth boundary.