Exotic Invertible Phases with Higher-Group Symmetries

TL;DR

The paper introduces a new class of invertible phases in even spacetime dimensions, focusing on a 3+1d exotic loop topological order (iELTO) protected by a spacetime two-group symmetry that fuses a $\mathbb{Z}_2$ one-form symmetry with time-reversal. It provides both continuum and UV realizations, notably describing iELTO via a twisted $\mathbb{Z}_2$ two-form gauge theory and via an $SO(3)_-$ gauge theory with $\theta=2\pi$, whose low-energy limit matches the iELTO phase and exhibits a boundary anti-semion with a unit thermal Hall response. The work establishes a $\mathbb{Z}_8$ classification for iELTO, computes partition functions and correlation functions, and analyzes boundary anomalies, chiral central charge, and potential gapless boundary states with extended symmetry. It also outlines a higher-dimensional generalization to invertible exotic higher topological orders and develops a generalized fermionization framework that connects bosonic theories with higher-form symmetries to these invertible phases. Overall, the results reveal a broad landscape of topological phases protected by spacetime higher-group symmetries, with concrete UV realizations, robust bulk-boundary structure, and implications for lattice models and deconfined critical points.

Abstract

We investigate a family of invertible phases of matter with higher-dimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev's chain in 1+1d. The excitation has $\mathbb{Z}_2$ higher-form symmetry that mixes with the spacetime Lorentz symmetry to form a higher group spacetime symmetry. We focus on the invertible exotic loop topological phase in 3+1d. This invertible phase is protected by the $\mathbb{Z}_2$ one-form symmetry and the time-reversal symmetry, and has surface thermal Hall conductance not realized in conventional time-reversal symmetric ordinary bosonic systems without local fermion particles and the exotic loops. We describe a UV realization of the invertible exotic loop topological order using the $SO(3)_-$ gauge theory with unit discrete theta parameter, which enjoys the same spacetime two-group symmetry. We discuss several applications including the analogue of "fermionization" for ordinary bosonic theories with $\mathbb{Z}_2$ non-anomalous internal higher-form symmetry and time-reversal symmetry.

Figures (4)

• Figure 1: Slices of the two open surfaces that intersect in 3+1d bulk and end on the boundary ($z=0$) by two linked loops. The slices are taken at different bulk coordinate $z\geq 0$. The loops on the boundary ($z=0$) represent creating and annihilating two pairs of particles and braiding two of them in the time evolution. The intersection point in 3+1d bulk is denoted by the green dot at the slice $z=1$.
• Figure 2: The genuine line operators of the $SO(3)_+$ and the $SO(3)_-$ theories (both with $\theta=0$) are marked as the green dots in the integer lattice $(q_e,q_m)$. The letters "$b$" and "$f$" indicates the bosonic and fermionic statistics of the corresponding line.
• Figure 3: The upper diagram describes the phase transition of massless Majorana fermion in 1+1d with the fermion parity symmetry $\mathbb{Z}_2^{(0)}$ embedded in the full spacetime $Spin(2)$ symmetry. This phase transition separates the trivial phase and the Kitaev's chain invertible fermionic phase. The free Majorana fermion in 1+1d is also dual to the $\mathbb{Z}_2$ gauge theory with the discrete theta angle (\ref{['eqn:Z21+1d']}), and the gauge field couples to a real scalar that has a quartic potential (for a review, see e.g. (2.9) of Karch:2019lnn, where the $\mathbb{Z}_2$ gauge field is dentoed by $s$, and the discrete theta angle is denoted by $\frac{\pi}{2}\int q_{\rho}(s)=\pi\int \text{Arf}(s\cdot \rho)+\text{Arf}(\rho)$ with spin structure $\rho$), which is the 1+1d analogue of the action (\ref{['eqn:Elooptransition']}). The mass square of the real scalar is identified with the fermion mass $m$ with appropriate sign. The lower diagram describes the phase transition with exotic loops, which has $\mathbb{Z}_2^{(1)}$ one-form symmetry and the (modified) time-reversal symmetry $\mathcal{T}$ that are both embedded in the spacetime two-group symmetry $\mathbb{Z}_{2}^{(1)}\times_{w_1w_2} O(4)$). The two sides of the phase transition are given by the trivial phase and the iELTO phase respectively. A proposed theory of this transition is given in \ref{['eqn:Elooptransition']}. In both phase diagrams, the symmetry is unbroken on both sides of the phase transition, and thus the symmetry is also unbroken at the phase transition.
• Figure 4: The interplay between fermionization and gauging $\mathbb{Z}_2$ symmetry. The downward arrow on the left denotes gauging the $\mathbb{Z}_2$$(n-1)$-form symmetry in $T$, while the upward arrow on the left denotes gauging the dual $\mathbb{Z}_2$$(n-1)$-form symmetry in $F[T]$. On the right, the downward arrow and upward arrow denotes stacking with the invertible phase in Section \ref{['sec:higherdim']} or its complex conjugate, together with a change in the symmetry fractionalization as discussed in (\ref{['eqn:stackingiTO']}).