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.
Understanding quantum phases and phase transitions in the presence of symmetries is a central objective of quantum many-body physics. A powerful modern paradigm for investigating this problem is topological holography, which relates symmetries in
k dimensions to "bulk" topological orders in
(k+1) dimensions. While conceptually profound, most existing bulk construction methods rely on sophisticated mathematical formalisms and can be difficult to apply to certain symmetry types. In this work, we propose a physically intuitive and versatile method, termed the layered gauging construction, to systematically generate
(k+1)-dimensional (liquid or fracton) topological orders from
k-dimensional generalized symmetries. Roughly speaking, the prescription is to stack many layers of
k-dimensional quantum systems with certain symmetries into a
(k+1)-dimensional pile, and then sequentially gauge a diagonal symmetry acting on each nearest-neighbor pair of layers. The detailed procedure depends on the specific symmetry types. We have successfully implemented the method in a number of examples in different spatial dimensions, with symmetries that are conventional, higher-form, subsystem, anomalous, nonabelian, or noninvertible. We hence conjecture the method to be very general. For example, from the subsystem symmetry of the
2d plaquette Ising model, we derive the X-cube model and also an anisotropic fracton topological order. Additionally, starting from an anomalous
Z2 symmetry in
1d, we construct a new square lattice model realizing the double semion topological order.
We formulate a construction of type-I fracton models based on gauging planar subsystem symmetries of topologically ordered two-dimensional layers that have been stacked in three ambient spatial dimensions. Via our construction, any defect of an Abelian symmetry group in a two-dimensional symmetry-enriched topological order can be promoted to a fracton. This allows us to construct fracton models supporting chiral boundaries and fractons of noninteger quantum dimension. We also find a lineon model supporting non-Abelian surface fractons on its boundary.
Coupled layer constructions are a valuable tool for capturing the universal properties of certain interacting quantum phases of matter in terms of the simpler data that characterizes the underlying layers. In the study of fracton phases, the X-Cube model in 3+1D can be realized via such a construction by starting with a stack of 2+1D Toric Codes and turning on a coupling which condenses a composite "particle-string" object. In a recent work [Phys. Rev. B 112, 125124 (2025)], we have demonstrated that in fact, the particle-string can be viewed as a symmetry defect of a topological 1-form symmetry. In this paper, we study the result of gauging this symmetry in depth. We unveil a rich gauging web relating the X-Cube model to symmetry protected topological (SPT) phases protected by a mix of subsystem and higher-form symmetries, subsystem symmetry fractionalization in the 3+1D Toric Code, and non-trivial extensions of topological symmetries by subsystem symmetries. Our work emphasizes the importance of topological symmetries in non-topological, geometric phases of matter.
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.
Topological holography (TH), or SymTFT, realizes symmetries and dualities of a quantum system as boundary data of a topological bulk in one higher dimension. We formulate fracton topological holography (FTH), extending this mechanism from liquid topological orders to fracton stabilizer codes. The construction is organized as a general four-stage framework: prepare the bulk model and compute its excitations, determine boundary data and admissible gapped top boundaries, identify the low-energy preserving operator algebra together with its symmetry, relation, and twist data, and then switch among top boundaries to compare the induced boundary descriptions. As a type-I example, we develop FTH for the X-cube model with smooth and rough top boundaries; for a minimal effective Hamiltonian, both yield transverse-field plaquette Ising models, with exchanged subsystem symmetry and twist data, and the boundary switch is implemented by a linear-depth local unitary sequential quantum circuit (SQC). As a type-II example, we formulate FTH for Haah's cubic code in the Laurent-polynomial stabilizer formalism and analyze the natural
(Z) and
(X) top boundaries, which induce two two-dimensional qubit systems related locally by exchanging generalized plaquette Ising and transverse-field terms and nonlocally by a symmetry--relation duality. These results show that FTH is a genuine extension of TH to both type-I and type-II fracton orders. FTH therefore provides a concrete framework for organizing and understanding duality, with the prospect of offering a systematic route to new dualities.