We introduce a stabilizer code model with a qutrit at every edge on a square lattice and with noninvertible plaquette operators. The degeneracy of the ground state is topological as in the toric code, and it also has the usual deconfined excitations consisting of pairs of electric and magnetic charges. However, there are novel types of confined fractonic excitations composed of a cluster of adjacent faces with vanishing flux. They manifest confinement, and even larger configurations of these fractons are fully immobile although they acquire emergent internal degrees of freedom. Deconfined excitations change their nature in presence of these fractonic defects. As for instance, fractonic defects can absorb magnetic charges making magnetic monopoles exist while electric charges acquire restricted mobility. Furthermore, some generalized symmetries can annihilate any ground state and also the full sector of fully mobile excitations. All these properties can be captured via a novel type of and fusion category in which the product is associative but does not commute, and can be expressed as a sum of (operator) equivalence classes. Generalized noninvertible symmetries give rise to the feature that the fusion products form a nonunital category without a proper identity. We show that a variant of this model features a deconfined fracton liquid phase and a phase where the dual (magnetic) strings have condensed. Published by the American Physical Society 2024
We introduce a stabilizer code model with a qutrit at every edge on a square lattice and with noninvertible plaquette operators. The degeneracy of the ground state is topological as in the toric code, and it also has the usual deconfined excitations consisting of pairs of electric and magnetic charges. However, there are novel types of confined fractonic excitations composed of a cluster of adjacent faces with vanishing flux. They manifest confinement, and even larger configurations of these fractons are fully immobile although they acquire emergent internal degrees of freedom. Deconfined excitations change their nature in presence of these fractonic defects. As for instance, fractonic defects can absorb magnetic charges making magnetic monopoles exist while electric charges acquire restricted mobility. Furthermore, some generalized symmetries can annihilate any ground state and also the full sector of fully mobile excitations. All these properties can be captured via a novel type of and fusion category in which the product is associative but does not commute, and can be expressed as a sum of (operator) equivalence classes. Generalized noninvertible symmetries give rise to the feature that the fusion products form a nonunital category without a proper identity. We show that a variant of this model features a deconfined fracton liquid phase and a phase where the dual (magnetic) strings have condensed. Published by the American Physical Society 2024
We propose a lattice spin model on a cubic lattice that shares many of the properties of the 3D toric code and the X-cube fracton model. The model, made of Z3 degrees of freedom at the links, has the vertex, the cube, and the plaquette terms. Being a stabilizer code the ground states are exactly solved. With only the vertex and the cube terms present, we show that the ground state degeneracy (GSD) is ${3}^{{L}^{3}+3L\ensuremath{-}1}$, where L is the linear dimension of the cubic lattice. In addition to fractons, there are free vertex excitations we call the freeons. With the addition of the plaquette terms, GSD is vastly reduced to 33, with fracton, fluxon, and freeon excitations, among which only the freeons are deconfined. The model is called the AB model if only the vertex (Av) and the cube (Bc) terms are present and the ABC model if in addition the plaquette terms (Cp) are included. The AC model consisting of vertex and plaquette terms is the Z3 3D toric code. The extensive GSD of the AB model derives from the existence of both local and nonlocal logical operators that connect different ground states. The latter operators are identical to the logical operators of the Z3 X-cube model. Fracton excitations are immobile and accompanied by the creation of fluxons---plaquettes having nonzero flux. In the ABC model, such fluxon creation costs energy and ends up confining the fractons. Unlike past models of fractons, vertex excitations are free to move in any direction and pick up a nontrivial statistical phase when passing through a fluxon or a fracton cluster.
The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an inverse. We show that a consistent system can be constructed for an arbitrary groupoid and analyze the simplest example that can be seen as the analog of the Abelian Z2 toric code. We find several exactly solvable models that have fracton-like features which include an extensive ground state degeneracy and excitations that are either immobile or have restricted mobility. Among the possibilities we study in detail the one where the ground state degeneracy scales as 2×2Nv, where Nv is the number of vertices in the lattice. The origin of this degeneracy can be traced to loop operators supported on both contractible and non-contractible loops. In particular, different non-contractible loops, along the same direction on a torus, result in different ground states. This is an exponential increase in the number of logical qubits that can be encoded in this code. Moreover the face excitations in this system are deconfined, free to move without an energy cost along certain directions of the lattice, whereas in certain other directions their movement incurs an energy cost. This places a restriction on the types of loop operators that contribute to the ground state degeneracy. The vertex excitations are immobile. The results are also extended to the groupoid analogs of Abelian ZN toric codes.
The authors generalize here three-dimensional \ensuremath{\mathbb{Z}}2 stabilizer models---the X-cube, toric code, and Haah's code---to their \ensuremath{\mathbb{Z}}N counterparts, revealing size-dependent ground-state degeneracy and novel excitations. They demonstrate that the generalized X-cube hosts quasi-fractons that remain immobile under local operations but become mobile via global ones. In the toric code, both closed-loop and open-string excitations emerge, while Haah's code features new fracton tripole and monopole excitations. These findings uncover unconventional topological orders and broaden the understanding of fracton phases beyond the conventional \ensuremath{\mathbb{Z}}2 framework.
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.
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.
Product code construction is a powerful tool for constructing quantum stabilizer codes, which serve as a promising paradigm for realizing fault-tolerant quantum computation. Furthermore, the natural mapping between stabilizer codes and the ground states of exactly solvable spin models also motivates the exploration of many-body orders in the stabilizer codes. In this work, we investigate the fracton topological orders in a family of codes obtained by a recently proposed general construction. More specifically, this code family can be regarded as a class of generalized hypergraph product (HGP) codes. We term the corresponding exactly solvable spin models \textit{orthoplex models}, based on the geometry of the stabilizers. In the 3D orthoplex model, we identify a series of intriguing properties within this model family, including non-monotonic ground state degeneracy (GSD) as a function of system size and non-Abelian lattice defects. Most remarkably, in 4D we discover \textit{fragmented topological excitations}: while such excitations manifest as discrete, isolated points in real space, their projections onto lower-dimensional subsystems form connected objects such as loops, revealing the intrinsic topological nature of these excitations. Therefore, fragmented excitations constitute an intriguing intermediate class between point-like and spatially extended topological excitations. In addition, these rich features establish the generalized HGP codes as a versatile and analytically tractable platform for studying the physics of fracton orders.