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.
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.
Fracton phases constitute a new class of quantum state of matter. They are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics or mobile only in certain directions. These phases do not wholly fit into any of the existing paradigms but connect to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly solvable lattice spin models. We then introduce the framework of tensor gauge theory, which provides a powerful complementary perspective and allows us to access gapless fracton phases. We discuss the basic properties of gapless fracton phases and their connections to elasticity theory and gravity. We also discuss what is known about the dynamics and thermodynamics of fractons at nonzero density before concluding with a brief survey of some open problems.
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 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.
Fracton phases are exotic quantum states of matter that are characterized by the presence of pointlike excitations with fundamentally restricted mobility. Important open questions remain about how to characterize the universal properties of distinct types of fracton order when viewed as conventional quantum phases. The authors provide substantial progress towards answering this question by deriving constraints on the ground-state wavefunctions of fracton phases, which generalize the string-net equations used to characterize two-dimensional topological order. These constraints yield new types of fracton order, and provide a starting point for characterizing their essential data.