In this work, we develop a coupled layer construction of fracton topological orders in
d=3 spatial dimensions. These topological phases have subextensive 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 demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.
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.
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,
ZN generalization of this fracton phase, which we introduce. We describe the rich phase structure of our layered
ZN system. By invoking a lattice duality, we demonstrate that when
N≥5, there is an intermediate phase that appears between the decoupled, layered system and the fracton topologically-ordered state, which opens the possibility of a continuous transition into the fracton topological phase. We conclude by presenting a solvable model, that interpolates between the fracton phase and a confined phase in which the phase transition is first-order.
We describe topologically ordered and fracton-ordered quantum systems on geometries which do not have an underlying manifold structure. Using tree graphs such as the
k-coordinated Bethe lattice
B(k) and a hypertree called the
(k,n)-hyper-Bethe lattice
HB(k,n) consisting of
k-coordinated hyperlinks (links defined by
n sites), we construct multidimensional arboreal arenas such as $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})$ using a generalized notion of a graph Cartesian product $\ensuremath{\square}$. We study various quantum systems such as the
Z2 gauge theory, generalized quantum Ising models (GQIMs), the fractonic X-cube model, and related X-cube gauge theory defined on these arboreal arenas, finding several fascinating results. Even the simplest
Z2 gauge theory on a two-dimensional arboreal arena is found to be fractonic---an isolated monopole excitation is rendered fully immobile on an arboreal arena. The X-cube model on a generic three-dimensional arboreal arena is found to be fully fractonic in the magnetic sector, i.e., all multipoles of magnetic excitations are rendered immobile on the arboreal arena. We obtain variational ground state phase diagrams of the gauge theories (both
Z2 and X-cube gauge theories) which are shown to have deconfined and confined phases. These phases are usually separated by a first-order transition, while continuous transitions are obtained in some cases. Further, we find an intriguing class of dualities in arboreal arenas, as illustrated by the
Z2 gauge theory defined on $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})$ being dual to a GQIM defined on $HB(2,{k}_{1})\ensuremath{\square}HB(2,{k}_{2})$. Finally, we discuss different classes of topological and fracton orders that appear on arboreal arenas. We find three distinct classes of arboreal toric code orders on two-dimensional arboreal arenas, those that occur on $\text{B}(2)\ensuremath{\square}\text{B}(2), \text{B}(k)\ensuremath{\square}\text{B}(2),k>2$, and $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2}), {k}_{1},{k}_{2}>2$. Likewise, four classes of X-cube fracton orders are found on three-dimensional arboreal arenas which correspond to those on $\text{B}(2)\ensuremath{\square}\text{B}(2)\ensuremath{\square}\text{B}(2), \text{B}(k)\ensuremath{\square}\text{B}(2)\ensuremath{\square}\text{B}(2),k>2, \text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})\ensuremath{\square}\text{B}(2),{k}_{1},{k}_{2}>2$, and $\text{B}({k}_{1})\ensuremath{\square}\text{B}({k}_{2})\ensuremath{\square}\text{B}({k}_{3}),{k}_{1},{k}_{2},{k}_{3}>2$.
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 that lack a TQFT description. 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 entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.
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.