We introduce "fractalization", a procedure by which spin models are extended to higher-dimensional "fractal" spin models. This allows us to interpret type-II fracton phases, fractal symmetry-protected topological phases, and more, in terms of well understood lower-dimensional spin models. Fractalization is also useful for deriving new spin models and quantum codes from known ones. We construct higher dimensional generalizations of fracton models that host extended fractal excitations. Finally, by applying fractalization to a 2D subsystem code, we produce a family of locally generated 3D subsystem codes that are conjectured to saturate a quantum information storage tradeoff bound.
This report offers a comprehensive academic synthesis of the information-theoretic paradigm in fundamental physics, in which spacetime geometry, vacuum structure, and gravitational dynamics arise from the entanglement and error-correcting properties of quantum many-body systems. Beginning with the Toric Code, Color Codes, and string-net condensates, the review explains how gauge fields, fermionic statistics, and virtual particle structure emerge from stabilizer constraints and topological order. It then explores Fracton phases and subsystem symmetries as geometric substrates capable of reproducing the constraints and potentials of linearized gravity. The analysis extends into holographic quantum error correction—including MERA, the HaPPY code, and random tensor networks—to show how bulk locality, black hole entropy, and AdS geometry emerge from entanglement structure. The Hyperbolic Fracton Model is examined as a synthesis that unifies fractonic mobility constraints with holographic scaling and area-law entropy. The report supports a unified “code-theoretic” framework in which physical laws and spacetime arise from the structure of an underlying quantum code.
Haah's cubic code is the prototypical type-II fracton topological order. It instantiates the no stringlike operator property that underlies the favorable scaling of its code distance and logical energy barrier. Previously, the cubic code was only explored in translation-invariant systems on infinite and periodic lattices. In these settings, the code distance scales superlinearly with the linear system size, while the number of logical qubits within the degenerate ground space exhibits a complicated functional dependence that undergoes large fluctuations within a linear envelope. Here, we extend the cubic code to systems with open boundary conditions and crystal lattice defects. We characterize the condensation of topological excitations in the vicinity of these boundaries and defects, finding that their inclusion can introduce local stringlike operators and enhance the mobility of otherwise fractonic excitations. Despite this, we use these boundaries and defects to define new encodings where the number of logical qubits scales linearly without fluctuations, and the code distance scales superlinearly, with the linear system size. These include a subsystem encoding with open boundary conditions and a subspace encoding using lattice defects.
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.
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.
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.