Motivated by recent interest in fracton topological phases, we explore the interplay between gapped 2D ${\mathbb{Z}}_{N}$ topological phases which admit fractional excitations with restricted mobility and geometry of the lattice on which such phases are placed. We investigate the properties of the phases in a geometric context---graph theory. By placing the phases on a 2D lattice consisting of two arbitrary connected graphs, ${G}_{x}\ensuremath{\boxtimes}{G}_{y}$, we study the behavior of fractional excitations of the phases. We derive the formula of the ground-state degeneracy of the phases, which depends on invariant factors of the Laplacian.
Motivated by recent interest in fracton topological phases, we explore the interplay between gapped 2D ZN topological phases which admit fractional excitations with restricted mobility and geometry of the lattice on which such phases are placed. We investigate the properties of the phases in a geometric context---graph theory. By placing the phases on a 2D lattice consisting of two arbitrary connected graphs, Gx\ensuremath⊠Gy, we study the behavior of fractional excitations of the phases. We derive the formula of the ground-state degeneracy of the phases, which depends on invariant factors of the Laplacian.
Anisotropic higher rank ZN topological phases on graphs
We study unusual gapped topological phases where they admit \mathbb{Z}_N ℤ N fractional excitations in the same manner as topologically ordered phases, yet their ground state degeneracy depends on the local geometry of the system. Placing such phases on 2D lattice, composed of an arbitrary connected graph and 1D line, we find that the fusion rules of quasiparticle excitations are described by the Laplacian of the graph and that the number of superselection sectors is related to the kernel of the Laplacian. Based on this analysis, we further show that the ground state degeneracy is given by \bigl[N\times \prod_{i}\text{gcd}(N, p_i)\bigr]^2 [ N × ∏ i gcd ( N , p i ) ] 2 , where p_i p i ’s are invariant factors of the Laplacian that are greater than one and gcd stands for the greatest common divisor. We also discuss braiding statistics between quasiparticle excitations.
Three-dimensional (3D) gapped topological phases with fractional excitations are divided into two subclasses: one has topological order with point-like and loop-like excitations fully mobile in the 3D space, and the other has fracton order with point-like excitations constrained in lower-dimensional subspaces. These exotic phases are often studied by exactly solvable Hamiltonians made of commuting projectors, which, however, are not capable of describing those exhibiting surface states with gapless chiral dispersion. Here we introduce a systematic way, based on cellular construction recently proposed for 3D topological phases, to construct another type of exactly solvable models in terms of coupled quantum wires with given inputs of cellular structure, two-dimensional Abelian topological order, and their gapped interfaces. We show that our models can describe both 3D topological and fracton orders (and even their hybrid) and study their universal properties such as quasiparticle statistics and topological ground-state degeneracy. We also apply this construction to two-dimensional coupled-wire models with ordinary topological orders and translation-symmetry-enriched topological orders. Our results pave the way for effective quantum field theory descriptions or microscopic model realizations of fracton orders with chiral gapless surface states.
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 $12{L}^{2}\ensuremath{-}12L+4$ 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.
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.
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.