Fracton phases of matter contain quasiparticles that are constrained, not by energetics but by the structure of local operators, to move along lower dimensional submanifolds. Here, the authors present a very general scheme for producing exactly solvable, non-Abelian fracton models by gauging symmetries that permute fractons, exploring several examples in detail. One of the resulting models contains immobile, non-Abelian quasiparticles that can only be created at the corners of operators with fractal support.
Fracton phases of matter contain quasiparticles that are constrained, not by energetics but by the structure of local operators, to move along lower dimensional submanifolds. Here, the authors present a very general scheme for producing exactly solvable, non-Abelian fracton models by gauging symmetries that permute fractons, exploring several examples in detail. One of the resulting models contains immobile, non-Abelian quasiparticles that can only be created at the corners of operators with fractal support.
We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile ``fracton'' excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub ``twisted fracton models.'' A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a 3-torus and computing their ground-state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility.
We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group G, a choice of an Abelian normal subgroup N, and a choice of foliation structure. These hybrid fracton orders, examples of which were introduced in a previous paper [Tantivasadakarn et al., Phys. Rev. B 103, 245136 (2021)], can also host immobile, pointlike excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate between a conventional, three-dimensional G gauge theory and a pure fracton order, by varying the choice of normal subgroup N. We demonstrate that certain universal data of the topological excitations and their mobilities are directly related to the choice of G and N, and also present complementary perspectives on these orders: certain orders may be obtained by gauging a global symmetry which enriches a particular fracton order, by either fractionalizing on or permuting the excitations with restricted mobility, while certain hybrid orders can be obtained by condensing excitations in a stack of initially decoupled, two-dimensional topological orders.
FINDING: Twisted Abelian fracton models in 3D generalize lattice gauge theories with subdimensional particle mobility, constrained by point group symmetries. | MATH: Fracton models as generalized Abelian lattice gauge theories; point group constraints restrict particle mobility to lower-dimensional subspaces (e.g., lineons, planons). Key constants/ratios not explicitly given, but lattice structure implies discrete symmetry groups (e.g., cubic, tetrahedral). | CONNECTION: Crystallographic point groups (e.g., Oh, Td) and root systems (e.g., A3, B3) naturally constrain subdimensional excitations; lattice geometry dictates mobility patterns—fractons immobile, lineons move along lines, planons in planes. Ratios like 0.618, 1.618 may appear in quasicrystalline analogs but not in this Abelian setting. | DEPTH: 7 — Novel topological phases with constrained mobility, linking gauge theory, point group symmetry, and lattice geometry; deep implications for quantum matter and emergent spacetime. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We construct a general gauging procedure of a pure matter theory on a lattice with a mixture of subsystem and global symmetries. This mixed symmetry can be either a semidirect product of a subsystem symmetry and a global symmetry, or a nontrivial extension of them. We demonstrate this gauging procedure on a cubic lattice in three dimensions with four examples: $G={\mathbb{Z}}_{3}^{\text{sub}}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{\text{glo}}, G=({\mathbb{Z}}_{2}^{\text{sub}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{\text{sub}})\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{\text{glo}}, 1\ensuremath{\rightarrow}{\mathbb{Z}}_{2}^{\text{sub}}\ensuremath{\rightarrow}G\ensuremath{\rightarrow}{\mathbb{Z}}_{2}^{\text{glo}}\ensuremath{\rightarrow}1$, and $1\ensuremath{\rightarrow}{\mathbb{Z}}_{2}^{\text{sub}}\ensuremath{\rightarrow}G\ensuremath{\rightarrow}{K}_{4}^{\text{glo}}\ensuremath{\rightarrow}1$. The former two cases and the last one produce the non-Abelian fracton orders. Our construction provides an identification of the electric charges of these fracton orders with irreducible representations of the symmetry. Furthermore, by constraining the local Hilbert space, the magnetic fluxes with different geometry (tube-like and plaquette-like) satisfy a subalgebra of the quantum double models (QDMs). This algebraic structure leads to an identification of the magnetic fluxes to the conjugacy classes of the symmetry.
Fracton phases are recent entrants to the roster of topological phases in three dimensions. They are characterized by subextensively divergent topological degeneracy and excitations that are constrained to move along lower-dimensional subspaces, including the eponymous fractons that are immobile in isolation. We develop correlation function diagnostics to characterize type-I fracton phases which build on their exhibiting partial deconfinement. These are inspired by similar diagnostics from standard gauge theories and utilize a generalized gauging procedure that links fracton phases to classical Ising models with subsystem symmetries. En route, we explicitly construct the space-time partition function for the plaquette Ising model which, under such gauging, maps into the X-cube fracton topological phase. We numerically verify our results for this model via Monte Carlo calculations.
We discuss the procedure for gauging on-site Z_2 Z 2 global symmetries of three-dimensional lattice Hamiltonians that permute quasi-particles and provide general arguments demonstrating the non-Abelian character of the resultant gauged theories. We then apply this general procedure to lattice models of several well known fracton phases: two copies of the X-Cube model, two copies of Haah’s cubic code, and the checkerboard model. Where the former two models possess an on-site Z_2 Z 2 layer exchange symmetry, that of the latter is generated by the Hadamard gate. For each of these models, upon gauging, we find non-Abelian subdimensional excitations, including non-Abelian fractons, as well as non-Abelian looplike excitations and Abelian fully mobile pointlike excitations. By showing that the looplike excitations braid non-trivially with the subdimensional excitations, we thus discover a novel gapped quantum order in 3D, which we term a ‘panoptic" fracton order. This points to the existence of parent states in 3D from which both topological quantum field theories and fracton states may descend via quasi-particle condensation. The gauged cubic code model represents the first example of a gapped 3D phase supporting (inextricably) non-Abelian fractons that are created at the corners of fractal operators.