Generally, ``fracton'' topological orders are referred to as gapped phases that support pointlike topological excitations whose mobility is, to some extent, restricted. In our previous work [Phys. Rev. B 101, 245134 (2020)], a large class of exactly solvable models on hypercubic lattices are constructed. In these models, spatially extended excitations possess generalized fractonlike properties: not only mobility but also deformability is restricted. As a series work, in this paper, we proceed further to compute ground state degeneracy (GSD) in both isotropic and anisotropic lattices. We decompose and reconstruct ground states through a consistent collection of subsystem ground state sectors, in which mathematical game ``coloring method'' is applied. Finally, we are able to systematically obtain GSD formulas (expressed as ${log}_{2}\text{GSD}$) which exhibit diverse kinds of polynomial dependence on system sizes. For example, the GSD of the model labeled as [0,1,2,4] in four-dimensional isotropic hypercubic lattice shows $12{L}^{2}\ensuremath{-}12L+4$ dependence on the linear size $L$ of the lattice. Inspired by existing results [W. Shirley et al., Phys. Rev. X 8, 031051 (2018)], we expect that the polynomial formulas encode geometrical and topological fingerprints of higher-dimensional manifolds beyond toric manifolds used in this work. This is left to future investigation.
Generally, fracton'' topological orders are referred to as gapped phases that support pointlike topological excitations whose mobility is, to some extent, restricted. In our previous work [Phys. Rev. B 101, 245134 (2020)], a large class of exactly solvable models on hypercubic lattices are constructed. In these models, spatially extended excitations possess generalized fractonlike properties: not only mobility but also deformability is restricted. As a series work, in this paper, we proceed further to compute ground state degeneracy (GSD) in both isotropic and anisotropic lattices. We decompose and reconstruct ground states through a consistent collection of subsystem ground state sectors, in which mathematical game coloring method'' is applied. Finally, we are able to systematically obtain GSD formulas (expressed as log2GSD) which exhibit diverse kinds of polynomial dependence on system sizes. For example, the GSD of the model labeled as [0,1,2,4] in four-dimensional isotropic hypercubic lattice shows 12L2\ensuremath−12L+4 dependence on the linear size L of the lattice. Inspired by existing results [W. Shirley et al., Phys. Rev. X 8, 031051 (2018)], we expect that the polynomial formulas encode geometrical and topological fingerprints of higher-dimensional manifolds beyond toric manifolds used in this work. This is left to future investigation.
Fracton topological order hosts fractionalized pointlike excitations (e.g., fractons) that have restricted mobility. In this paper, we explore even more bizarre realization of fracton phases that admit spatially extended excitations with restriction on both mobility and deformability. First, we present exactly solvable lattice quantum frustrated spin models and study their ground states and excited states analytically. We construct a family tree in which parent models and descendent models share excitation DNA. Second, with the help of solvability and novel excitation spectrum of these models, we initiate the first step of general discussions on quantitative and qualitative properties of spatially extended excitations whose mobility and deformability are restricted to some extent. Especially, as a useful viewpoint for understanding such fracton physics, all excitations are divided into four mutually distinct sectors, namely, simple excitations, complex excitations, intrinsically disconnected excitations, and trivial excitations. Several implications in, e.g., condensed matter physics and gravity are briefly discussed.
Fracton topological order hosts fractionalized point-like excitations (e.g., fractons) that have restricted mobility. In this paper, we explore even more bizarre realization of fracton phases that admit spatially extended excitations with restricted mobility. We construct exactly solvable lattice models in all spatial dimensions D and divide excitations into and ones. At infrared scales, simple excitations are geometrically manifold-like, e.g., points, strings, membranes, and so on. We introduce a useful notation---a pair of integers (n,m)---to label simple excitations. n=0,1,2,⋯ represent particles, strings, membranes,⋯, respectively; m is the dimension of the submanifold where mobility of excitations is allowed. With this notation, fractons are labeled by (0,0) and the subdimensional mobility of spatially extended excitations can be discussed systematically. On the other hand, complex excitations at infrared scales are geometrically non-manifold-like, whose physical characterization (e.g., creation operators, excitation energy, fusion rules, mobility) is far more intricate than that of simple excitations. Several concrete models are analyzed in details. For example, complex excitations dubbed chairons and yuons are constructed in D=4. In D=5, except chairons, cloverions and xuons are found. Finally, we construct a family tree in which parent models and descendent models share excitation DNA. Several implications in, e.g., condensed matter physics and gravity are briefly discussed.
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.
Fracton topological order (FTO) is a new classification of correlated phases in three spatial dimensions with topological ground-state degeneracy (GSD) scaling up with system size and fractional excitations which are immobile or have restricted mobility. With the topological origin of GSD, FTO is immune to local perturbations, whereas a strong enough global external perturbation is expected to break the order. The critical point of the topological transition is either characterized by the broken GSD or the appearance of topologically distinct states with lower energy. In this work, we propose to characterize quantum phase transition of the type-I FTOs induced by external terms, when the transition can be characterized by the breaking down of GSD, and develop a theory to study analytically the critical points. In particular, for the external perturbation term creating lineon-type excitations, we predict a generic formula for the point of the quantum phase transition, characterized by the breaking down of GSD. This theory applies to a board class of FTOs, including X-cube model, and for more generic FTO models under perturbations creating two-dimensional (2D) or 3D excitations, we predict the upper and lower limits of the critical point. Our work makes a step in characterizing analytically the quantum phase transition of generic fracton orders.