Fracton phases exhibit striking behavior which appears to render them beyond the standard topological quantum field theory (TQFT) paradigm for classifying gapped quantum matter. Here, we explore fracton phases from the perspective of defect TQFTs and show that topological defect networks---networks of topological defects embedded in stratified 3+1D TQFTs---provide a unified framework for describing various types of gapped fracton phases. In this picture, the sub-dimensional excitations characteristic of fractonic matter are a consequence of mobility restrictions imposed by the defect network. We conjecture that all gapped phases, including fracton phases, admit a topological defect network description and support this claim by explicitly providing such a construction for many well-known fracton models, including the X-Cube and Haah's B code. To highlight the generality of our framework, we also provide a defect network construction of a novel fracton phase hosting non-Abelian fractons. As a byproduct of this construction, we obtain a generalized membrane-net description for fractonic ground states as well as an argument that our conjecture implies no type-II topological fracton phases exist in 2+1D gapped systems. Our work also sheds light on new techniques for constructing higher order gapped boundaries of 3+1D TQFTs.
Fracton phase of matter host fractionalized topological quasiparticles with restricted mobility. A wide variety of fracton models with abelian excitations had been proposed and extensively studied while the candidates for non-Abelian fracton phases are less explored. In this work, we investigate the effect of twisted defect in abelian fracton models. The twisted defect is launched by introducing a branch cut line hosting anyon condensate. In particular, these twisted defects, which alter different types of quasiparticles, carry projective non-Abelian zero modes. En route, such defects can be e…
A topological defect network (TDN) is formed by a network of topological defects embedded within a topological quantum field theory (TQFT). TDNs were introduced recently for the purpose of describing fracton topological phases of matter using the framework of defect TQFT. Their effectiveness has been demonstrated through numerous examples, yet a systematic construction was lacking. Here we solve this problem by formulating a method to construct TDNs for a wide range of lattice Hamiltonians. Our method takes a lattice Hamiltonian as input, applies an ungauging procedure, then creates a refined …
We propose a set of constraints on the ground-state wavefunctions of fracton phases, which provide a possible generalization of the string-net equations used to characterize topological orders in two spatial dimensions. Our constraint equations arise by exploiting a duality between certain fracton orders and quantum phases with "subsystem" symmetries, which are defined as global symmetries on lower-dimensional manifolds, and then studying the distinct ways in which the defects of a subsystem symmetry group can be consistently condensed to produce a gapped, symmetric state. We numerically solve…
In three dimensions, gapped phases can support "fractonic" quasiparticle excitations, which are either completely immobile or can only move within a low-dimensional submanifold, a peculiar topological phenomenon going beyond the conventional framework of topological quantum field theory. In this work we explore fractonic topological phases using three-dimensional coupled wire constructions, which have proven to be a successful tool to realize and characterize topological phases in two dimensions. We find that both gapped and gapless phases with fractonic excitations can emerge from the models.…
In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which …
We provide a new perspective on fracton topological phases, a class of three-dimensional topologically ordered phases with unconventional fractionalized excitations that are either completely immobile or only mobile along particular lines or planes. We demonstrate that a wide range of these fracton phases can be constructed by strongly coupling mutually intersecting spin chains and explain via a concrete example how such a coupled-spin-chain construction illuminates the generic properties of a fracton phase. In particular, we describe a systematic translation from each coupled-spin-chain const…
We describe a construction of topological orders from coupled lower dimensional symmetry-protected topological orders, which is closely related to gauging a subsystem symmetry. Our construction yields both conventional topological orders and exotic fracton topological orders of type-I and type-II. In particular, we find a coupled spin chain construction of Haah's cubic code, and a coupled layer construction of Yoshida's fractal spin liquids.
As an unconventional realization of topological orders with an exotic interplay of topology and geometry, fracton (topological) orders feature subextensive topological ground state degeneracy and subdimensional excitations that are movable only within a certain subspace. It has been known in the exactly solvable three-dimensional X-cube model that universally represents the type-I fracton orders, that mobility constraints on subdimensional excitations originate from the absence of spatially deformable string-like operators. To unveil the interplay of topology and geometry, in this paper, we st…
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) point-like topological excitations with restricted mobility, with non-trivial 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 t…
Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter, but the existence of a TQFT description for these phases remains an open question. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground state wave functions. We explicitly compute the en…
We study gapped boundaries of Abelian type-I fracton systems in three spatial dimensions. Using the X-cube model as our motivating example, we give a conjecture, with partial proof, of the conditions for a boundary to be gapped. In order to state our conjecture, we use a precise definition of fracton braiding and show that bulk braiding of fractons has several features that make it \textit{insufficient} to classify gapped boundaries. Most notable among these is that bulk braiding is sensitive to geometry and is "nonreciprocal," that is, braiding an excitation $a$ around $b$ need not yield the …
Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In a previous work, we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-p…
We investigate the entanglement-renormalization group flows of translation-invariant topological stabilizer models in three dimensions. Fracton models are observed to bifurcate under entanglement renormalization, generically returning at least one copy of the original model. Based on this behavior we formulate the notion of bifurcated equivalence for fracton phases, generalizing foliated fracton equivalence. The notion of quotient superselection sectors is also generalized accordingly. We calculate bifurcating entanglement-renormalization group flows for a wide range of examples and, based on …
Fracton models, a collection of exotic gapped lattice Hamiltonians recently discovered in three spatial dimensions, contain some 'topological' features: they support fractional bulk excitations (dubbed fractons), and a ground state degeneracy that is robust to local perturbations. However, because previous fracton models have only been defined and analyzed on a cubic lattice with periodic boundary conditions, it is unclear to what extent a notion of topology is applicable. In this paper, we demonstrate that the X-cube model, a prototypical type-I fracton model, can be defined on general three-…
The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy $S_{topo}$ for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy $S_{nonlocal}$ (a generalization of $S_{topo}$). The lower bound applies to Abelian stabilizer models including conventional topological ord…
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 however very challenging to identify. In this work, we propose to characterize quantum phase transition of the type-I FTOs induced by…
Employing the fracton-elastic duality, we develop a low-energy effective theory of a zero-temperature vortex crystal in a two-dimensional bosonic superfluid which naturally incorporates crystalline topological defects. We extract static interactions between these defects and investigate several continuous quantum transitions triggered by the Higgs condensation of vortex vacancies/interstitials and dislocations. We propose that the quantum melting of the vortex crystal towards the hexatic or smectic phase may occur via a pair of continuous transitions separated by an intermediate vortex superso…
We present a realization of fracton-elasticity duality purely formulated in terms of ordinary gauge fields, encompassing standard elasticity and incommensurate crystals as those describing twisted bilayer graphene, quasicrystals or more general moiré lattices. Our construction comprises a description of all types of two-dimensional defects: disclinations, dislocations, discompressions and point-like defects, and takes into account body forces and impurities. The original form of the duality in terms of tensor gauge fields is recovered after partial gauge fixing. We identify the coupling of eac…
Fracton topological phases have a large number of materialized symmetries that enforce a rigid structure on their excitations. Remarkably, we find that the symmetries of a quantum error-correcting code based on a fracton phase enable us to design decoding algorithms. Here we propose and implement decoding algorithms for the three-dimensional X-cube model. In our example, decoding is parallelized into a series of two-dimensional matching problems, thus significantly simplifying the most time consuming component of the decoder. We also find that the rigid structure of its point excitations enabl…
The Lieb-Schultz-Mattis (LSM) theorem implies that gapped phases of matter must satisfy non-trivial conditions on their low-energy properties when a combination of lattice translation and $U(1)$ symmetry are imposed. We describe a framework to characterize the action of symmetry on fractons and other sub-dimensional fractional excitations, and use this together with the LSM theorem to establish that X-cube fracton order can occur only at integer or half-odd-integer filling. Using explicit parton constructions, we demonstrate that "odd" versions of X-cube fracton order can occur in systems at h…
As new kinds of stabilizer code models, fracton models have been promising in realizing quantum memory or quantum hard drives. However, it has been shown that the fracton topological order of 3D fracton models occurs only at zero temperature. In this Letter, we show that higher dimensional fracton models can support a fracton topological order below a nonzero critical temperature $T_c$. Focusing on a typical 4D X-cube model, we show that there is a finite critical temperature $T_c$ by analyzing its free energy from duality. We also obtained the expectation value of the 't Hooft loops in the 4D…
Fractons are particles with restricted mobility. We give a symmetry-based derivation of effective field theories of gapless phases with fractonic topological defects, such as solids and supersolids, using a coset construction. The resulting theory is identified as the Cosserat elasticity theory, which reproduces the conventional symmetric elasticity theory at low energies. The construction can be viewed as a dynamical realization of the inverse Higgs mechanism. We incorporate topological defects such as dislocations and disclinations, which are nontrivially related by the Bianchi identities of…
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 arXiv:2102.09555 -- can also host immobile, point-like excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate betwee…
Starting from an isotropic configuration of intersecting, two-dimensional toric codes, we construct a fracton topological phase introduced in Ref. [26], which is characterized by immobile, point- like topological excitations ("fractons"), and degenerate ground-states on the torus that are locally indistinguishable. Our proposal leads to a simple description of the fracton excitations and of the ground-state as a "loop" condensate, and provides a basis for building new 3D topological orders such as a natural, $Z_{N}$ generalization of this fracton phase, which we introduce. We describe the rich…
We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, point-like topological excitations, and sub-extensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical toolset for our results. Our work paves the way…
We classify subsystem symmetry-protected topological (SSPT) phases in $3+1$D protected by planar subsystem symmetries, which are dual to abelian fracton topological orders. We distinguish between weak SSPTs, which can be constructed by stacking $2+1$D SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite abelian group. Finally, we show that fracton orders realizable via $p$-string condensation are dual to weak SSPTs, while strong SSPTs do not…
We demonstrate the existence of a fundamentally new type of excitation, fractonic lines, which are line-like excitations with the restricted mobility properties of fractons. These excitations, described using an amalgamation of higher-form gauge theories with symmetric tensor gauge theories, see direct physical realization as the topological lattice defects of ordinary three-dimensional quantum crystals. Starting with the more familiar elasticity theory, we show how it maps onto a rank-4 tensor gauge theory, with phonons corresponding to gapless gauge modes and disclination defects correspondi…
Motivated by recent studies of fractons, we demonstrate that elasticity theory of a two-dimensional quantum crystal is dual to a fracton tensor gauge theory, providing a concrete manifestation of the fracton phenomenon in an ordinary solid. The topological defects of elasticity theory map onto charges of the tensor gauge theory, with disclinations and dislocations corresponding to fractons and dipoles, respectively. The transverse and longitudinal phonons of crystals map onto the two gapless gauge modes of the gauge theory. The restricted dynamics of fractons matches with constraints on the mo…
We present a three-dimensional cubic lattice spin model, anisotropic in the $\hat{z}$ direction, that exhibits fracton topological order. The latter is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground state degeneracy: On an $L_x\times L_y\times L_z$ three-torus, it has a $2^{2L_z}$ topological degeneracy, and an additional non-topological degeneracy equal to $2^{L_xL_y-2}$. The fractons can be…
Fractons are gapped point-like excitations in $d=3$ topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor ${\rm U}(1)$ gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where cha…
Fracton phases are recent entrants to the roster of topological phases in three dimensions. They are characterized by subextensively divergent topological degeneracy and excitations that are constrained to move along lower dimensional subspaces, including the eponymous fractons that are immobile in isolation. We develop correlation function diagnostics to characterize Type I fracton phases which build on their exhibiting {\it partial deconfinement}. These are inspired by similar diagnostics from standard gauge theories and utilize a generalized gauging procedure that links fracton phases to cl…
We analyze the "higher rank" gauge theories, that capture some of the phenomenology of the Fracton order. It is shown that these theories lose gauge invariance when arbitrarily weak and smooth curvature is introduced. We propose a resolution to this problem by introducing a theory invariant under area-preserving diffeomorphisms, which reduce to the "higher rank" gauge transformations upon linearization around a flat background. The proposed theory is geometric in nature and is interpreted as a theory of chiral topological elasticity. This theory exhibits some of the Fracton phenomenology. We e…
Fracton topological order describes a remarkable phase of matter which can be characterized by fracton excitations with constrained dynamics and a ground state degeneracy that increases exponentially with the length of the system on a three-dimensional torus. However, previous models exhibiting this order require many-spin interactions which may be very difficult to realize in a real material or cold atom system. In this work, we present a more physically realistic model which has the so-called X-cube fracton topological order but only requires nearest-neighbor two-spin interactions. The model…
Fracton topological phases host fractionalized excitations that are either completely immobile or only mobile along certain lines or planes. We demonstrate how such phases can be understood in terms of two fundamentally different types of parton constructions, in which physical degrees of freedom are decomposed into clusters of "parton" degrees of freedom subject to emergent gauge constraints. First, we employ non-interacting partons subject to multiple overlapping constraints to describe a fermionic fracton model. Second, we demonstrate how interacting partons can be used to develop new model…
Fractons are topological quasiparticles with limited mobility. While there exists a variety of models hosting these excitations, typical fracton systems require rather complicated many-particle interactions. Here, we discuss fracton behavior in the more common physical setting of classical kagome spin models with frustrated two-body interactions only. We investigate systems with different types of elementary spin degrees of freedom (three-state Potts, XY, and Heisenberg spins) which all exhibit characteristic subsystem symmetries and fracton-like excitations. The mobility constraints of isolat…
Foliated fracton order is a qualitatively new kind of phase of matter. It is similar to topological order, but with the fundamental difference that a layered structure, referred to as a foliation, plays an essential role and determines the mobility restrictions of the topological excitations. In this work, we introduce a new kind of field theory to describe these phases: a foliated field theory. We also introduce a new lattice model and string-membrane-net condensation picture of these phases, which is analogous to the string-net condensation picture of topological order.
We formulate a fracton-elasticity duality for twisted moiré superlattices, taking into account that they are incommensurate crystals with dissipative phason dynamics. From a dual tensor-gauge formulation, as compared to standard crystals, we identify twice the number of conserved charges that describe topological lattice defects, namely, disclinations and a new type of defect that we dub discompressions. The key implication of these conservation laws is that both glide and climb motions of lattice dislocations are suppressed, indicating that dislocation networks may become exceptionally stable…
Fracton topological phases host fractionalized topological quasiparticles with restricted mobility, with promising applications to fault-tolerant quantum computation. While a variety of exactly solvable fracton models have been proposed, there is a need for platforms to realize them experimentally. We show that a rich set of fracton phases emerges in interacting Majorana band models whose building blocks are within experimental reach. Specifically, our Majorana constructions overcome a principal obstacle, namely the implementation of the complicated spin cluster interactions underlying fracton…
Generally, ``fracton'' topological orders are referred to as gapped phases that support \textit{point-like 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, \textit{spatially extended excitations} possess generalized fracton-like 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…
Two new recently proposed classes of topological phases, namely fractons and higher order topological insulators (HOTIs), share at least superficial similarities. The wide variety of proposals for these phases calls for a universal field theory description that captures their key characteristic physical phenomena. In this work, we construct topological multipolar response theories that capture the essential features of some classes of fractons and higher order topological insulators. Remarkably, we find that despite their distinct symmetry structure, some classes of fractons and HOTIs can be c…
We introduce a new quantity, that we term recoverable information, defined for stabilizer Hamiltonians. For such models, the recoverable information provides a measure of the topological information, as well as a physical interpretation, which is complementary to topological entanglement entropy. We discuss three different ways to calculate the recoverable information, and prove their equivalence. To demonstrate its utility, we compute recoverable information for fracton models using all three methods where appropriate. From the recoverable information, we deduce the existence of emergent $Z_2…
Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fract…
Fractons are particles that cannot move in one or more directions without paying energy proportional to their displacement. Here, we introduce the concept of symmetry enforced fractonicity, in which particles are fractons in the presence of a global symmetry, but are free to move in its absence. A simple example is dislocation defects in a two-dimensional crystal, which are restricted to move only along their Burgers vector due to particle number conservation. Utilizing a recently developed dual rank-2 tensor gauge description of elasticity, we show that accounting for their symmetry enforced …
We review what is known about fracton phases of quantum matter. Fracton phases are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions. They constitute a new class of quantum state of matter, which does not wholly fit into any of the existing paradigms, but which connects 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 sol…
