Higher-Rank Tensor Field Theory of Non-Abelian Fracton and Embeddon

TL;DR

The authors present a novel class of non-Abelian higher-rank tensor gauge theories that fuse symmetric rank-2 tensor gauge dynamics with antisymmetric tensor TQFTs, enabling a unified framework for mixed gapless and gapped phases in any dimension. By gauging a higher-moment vector symmetry $ ext{U}(1)_{x_{(d+1)}}$ together with a discrete $ ext{Z}_2^C$ charge-conjugation symmetry, they realize a gauge-structure of the form $[ ext{Z}_2^C times ext{U}(1)_{x_{(d+1)}}]$ and couple to twisted Dijkgraaf–Witten topological terms $oldsymbol{ extomega}_{d+1} eq 0$, yielding a continuum theory with both gapless modes and gapped topological sectors. The work develops both Euclidean/Lorentz-invariant and anisotropic variants, analyzes the independent components of the field strengths, and discusses the fracton-like mobility restrictions, embeddon concepts, and foliation as organizing principles, while outlining potential lattice realizations and implications for dark matter. Overall, the framework broadens the landscape of fracton-like theories and provides a path toward realizing non-Abelian tensor gauge dynamics in higher dimensions with rich topological structure.

Abstract

We formulate a new class of tensor gauge field theories in any dimension that is a hybrid class between symmetric higher-rank tensor gauge theory (i.e., higher-spin gauge theory) and anti-symmetric tensor topological field theory. Our theory describes a mixed unitary phase interplaying between gapless and gapped topological order phases (which can live with or without Euclidean, Poincaré or anisotropic symmetry, at least in ultraviolet high or intermediate energy field theory, but not yet to a lattice cutoff scale). The "gauge structure" can be compact, continuous, abelian or non-abelian. Our theory sits outside the paradigm of Maxwell electromagnetic theory in 1865 and Yang-Mills isospin/color theory in 1954. We discuss its local gauge transformation in terms of the ungauged vector-like or tensor-like higher-moment global symmetry. The non-abelian gauge structure is caused by gauging the non-commutative symmetries: a higher-moment symmetry and a charge conjugation (particle-hole) symmetry. Vector global symmetries along time direction may exhibit time crystals. We explore the relation of these long-range entangled matters to a non-abelian generalization of Fracton order in condensed matter, a field theory formulation of foliation, the spacetime embedding and Embeddon that we newly introduce, and possible fundamental physics applications to dark matter or dark energy.

Figures (6)

• Figure 1: The ordinary (0-form) U(1) global symmetry transformation acts on the complex charged matter $\Phi(x) \in \mathbb{C}$, e.g. the rotor field, distributed on the spacetime coordinate $x$. For demonstration, here we show various $\Phi(x)$ fields sitting on discretized lattice points on $x$. Thus we get Eq. (\ref{['eq:oglobal-1']}): $\Phi \to e^{\space\mathrm{i}\space \eta} \Phi$.
• Figure 2: The vector global symmetry belongs to a generalized class of higher-moment symmetry. The vector U(1) global symmetry transformation acts on the complex charged matter $\Phi(x)$, e.g. the rotor field, following the set-up in Fig. \ref{['clock-ordinary-global-1']}. Thus we get Eq. (\ref{['eq:vglobal-1']}): $\Phi \to e^{\space\mathrm{i}\space \eta_v(x)} \Phi := e^{\space\mathrm{i}\space \Lambda \cdot x} \Phi$. The angle $\Lambda \cdot x$ depends on a reference point (say $x=0$) and the distance $x$ away from the reference point.
• Figure 3: A demonstration of the gauge fluctuation (the dark gray color) deviates away from the 0-form global symmetry transformation of Fig. \ref{['clock-ordinary-global-1']} (the light gray color). See Eq. (\ref{['eq:0-gauge']}).
• Figure 4: A demonstration of the gauge fluctuation (the dark gray color) deviates away from the vector global symmetry of Fig. \ref{['clock-vector-global-3']} (the light gray color, which belongs to a generalized class of higher-moment symmetry). See Eq. (\ref{['eq:vector-gauge']}).
• Figure 5: Interpretations of the embeddon: (1) Quantization of spacetime embedding JWang2018Embeddon. (2) The anyonic objects (e.g. particles/strings/branes) live in the embedded manifold ${M^{n}_{\text{sub-$M$}}} \subset {M^{d+1}}$ in $n$d, thus the anyonic objects are embedded inside the sub-dimensions.