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.
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.
We demonstrate a general gauging procedure of a pure matter theory on a lattice with a global symmetry, a subsystem symmetry, or a semidirect product of them. By applying the general gauging procedure to a semidirect product of subsystem and global symmetries, we obtain the non-Abelian fracton order. We demonstrate this gauging procedure on a cubic lattice in three dimensions with two symmetries, (Z2sub×Z2sub)⋊Z2glo and Z3sub⋊Z2glo. The former case reproduces the non-Abelian fracton by gauging the bilayer X-Cube code, and the latter case is a new non-Abelian fracton order. We further identify the electric charges (including the non-Abelian fractons) with the irreducible representations of the symmetry. The corresponding magnetic fluxes can be obtained from the conjugacy classes of the symmetry.
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) pointlike topological excitations with restricted mobility, with nontrivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore they serve as parent orders for both long-range-entangled quantum liquid and nonliquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional Z2 topological order with (i) the X-cube fracton order or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry G and subsystem symmetries of a normal subgroup N along lower-dimensional subregions.
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.
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 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.
Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.