We present a general approach to the bulk-boundary correspondence of noninvertible topological phases, including both topological and fracton orders. This is achieved by a novel bulk construction protocol where solvable (d+1) ( d + 1 ) -dimensional bulk models with noninvertible topology are constructed from the so-called generalized Ising (GI) models in d d dimensions. The GI models can then terminate on the boundaries of the bulk models. The construction generates abundant examples, including not only prototype ones such as \mathbb{Z}_2 ℤ 2 toric code models in any dimensions no less than two, and the X-cube fracton model, but also more diverse ones such as the \mathbb{Z}_2× \mathbb{Z}_2 ℤ 2 × ℤ 2 topological order, the 4d \mathbb{Z}_2 ℤ 2 topological order with pure-loop excitations, etc. The boundary of the solvable model is potentially anomalous and corresponds to precisely only sectors of the GI model that host certain total symmetry charges and/or satisfy certain boundary conditions. We derive a concrete condition for such bulk-boundary correspondence. The condition is violated only when the bulk model is either trivial or fracton ordered. A generalized notion of Kramers-Wannier duality plays an important role in the construction. Also, utilizing the duality, we find an example where a single anomalous theory can be realized on the boundaries of two distinct bulk fracton models, a phenomenon not expected in the case of topological orders. More generally, topological orders may also be generated starting with lattice models beyond the GI models, such as those with symmetry protected topological orders, through a variant bulk construction, which we provide in an appendix.
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 investigate the interplay between self-duality and spatially modulated symmetry of generalized
N-state clock models, which include the transverse-field Ising model and ordinary
N-state clock models as special cases. The spatially modulated symmetry of the model becomes trivial when the model's parameters satisfy a specific number-theoretic relation. We find that the duality is noninvertible when the spatially modulated symmetry remains nontrivial, and we show that this noninvertibility is resolved by introducing a generalized
ZN toric code, which manifests ultraviolet/infrared mixing, as the bulk topological order. In this framework, the boundary duality transformation corresponds to the boundary action of a bulk symmetry transformation, with the end point of the bulk symmetry defect realizing the boundary duality defect. Our results illuminate not only a holographic perspective on dualities but also a relationship between spatially modulated symmetry and ultraviolet/infrared mixing in one higher dimension.
The bulk-boundary correspondence is a hallmark feature of topological phases of matter. Nonetheless, our understanding of the correspondence remains incomplete for phases with intrinsic topological order, and is nearly entirely lacking for more exotic phases, such as fractons. Intriguingly, for the former, recent work suggests that bulk topological order manifests in a non-local structure in the boundary Hilbert space; however, a concrete understanding of how and where this perspective applies remains limited. Here, we provide an explicit and general framework for understanding the bulk-boundary correspondence in Pauli topological stabilizer codes. We show -- for any boundary termination of any two-dimensional topological stabilizer code -- that the boundary Hilbert space cannot be realized via local degrees of freedom, in a manner precisely determined by the anyon data of the bulk topological order. We provide a simple method to compute this "obstruction" using a well-known mapping to polynomials over finite fields. Leveraging this mapping, we generalize our framework to fracton models in three-dimensions, including both the X-Cube model and Haah's code. An important consequence of our results is that the boundaries of topological phases can exhibit emergent symmetries that are impossible to otherwise achieve without an unrealistic degree of fine tuning. For instance, we show how linear and fractal subsystem symmetries naturally arise at the boundaries of fracton phases.
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.
We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary \mathbb{Z}_2 ℤ 2 topological order with fermionic particle and fermionic loop excitations that have mutual \pi π statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary \mathbb{Z}_2 ℤ 2 symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.
We review a many-body formulation of gravitational anomalies as obstructions to realizing the complete low-energy data by a local {\it bosonic} lattice quantum system of the same dimension. In this formulation, an anomalous theory can be realized at the boundary of a gapped {\it bosonic} lattice bulk, whose topological order gives a one-to-one classification of the anomaly. An invertible bulk reproduces conventional anomaly inflow; a bulk with non-trivial topological excitations yields a non-invertible generalization. To describe the holographic theory of anomalies, we explain the distinction between uniquely defined topological order and many versions of topological quantum field theory and discuss the boundary-to-bulk map. As an application of the holographic theory of anomalies, we describe morphisms between local quantum theories (such as quantum field theories). These morphisms are built from a bulk domain wall and an invertible boundary junction, which give rise to a sandwich picture. Chiral boson/fermion theory, Laughlin edges, and boundaries of
Z2 higher gauge theory with bosonic or fermionic point-like excitations illustrate the different kinds of information about non-invertible anomalies contained in transport coefficients, superselection sectors, and condensation data. In particular, a fermionic theory can be viewed as a theory with a particular non-invertible gravitational anomaly. Those examples also reveal that {\it symmetry
∼ gravitational anomaly
∼ topological order in one higher dimension}.