The continuum limit is a widely used theoretical construct for obtaining continuum field-theories of crystalline systems. We study the formulation of a continuum limit for gapped bosonic phases and ask whether their topological properties survive passage to the continuum. Using methods in algebraic topology and category theory, we give a rigorous formulation of the continuum limit and construct a surjective global map relating crystalline topological phases across all finite point-group symmetries to continuum invertible topological phases. Consequently, we find that some lattice phases admit no continuum limit, while distinct lattice phases that do admit a continuum limit can share the same continuum image, hence implying that some lattice topological data can collapse. Conversely, we find that every continuum invertible phase admits a faithful crystalline realization. In addition to the general framework, we apply it to several examples, including studies of a 2D rotation-symmetric phase, higher-order topological phases, and the mixed spin-lattice anomaly of the deconfined quantum critical point between an antiferromagnet and valence bond solid.
The continuum limit is a widely used theoretical construct for obtaining continuum field-theories of crystalline systems. We study the formulation of a continuum limit for gapped bosonic phases and ask whether their topological properties survive passage to the continuum. Using methods in algebraic topology and category theory, we give a rigorous formulation of the continuum limit and construct a surjective global map relating crystalline topological phases across all finite point-group symmetries to continuum invertible topological phases. Consequently, we find that some lattice phases admit no continuum limit, while distinct lattice phases that do admit a continuum limit can share the same continuum image, hence implying that some lattice topological data can collapse. Conversely, we find that every continuum invertible phase admits a faithful crystalline realization. In addition to the general framework, we apply it to several examples, including studies of a 2D rotation-symmetric phase, higher-order topological phases, and the mixed spin-lattice anomaly of the deconfined quantum critical point between an antiferromagnet and valence bond solid.
Topological phases are quantum phases of matter that appear close to absolute zero temperature. In contrast to commonly known condensed phases of matter, such as a magnet or a superconductor, distinct topological phases may have the same symmetry. Instead, these phases are distinguished by the topology of the electronic wavefunction. Already the single-particle excitations described by band theory in an insulator or a metal or by Bogoliubov-de Gennes theory in a superconductor may have novel anomalous properties associated to their topology. This topological band theory is the subject of this thesis. We aim towards identifying the anomalous boundary and defect states associated to the topology of the band structure in crystalline insulators and superconductors. Furthermore, we determine criteria to easily identify the topological phases from the band structure of a given insulator or superconductor. In Chapter 2 we show that crystalline topological insulators and superconductors may host anomalous gapless or in-gap excitations on corners of a two-dimensional crystal or hinges of a three-dimensional crystal. These topological phases are called "second-order" according to the codimension "2" of their anomalous boundary states. We discuss the precise conditions for their appearance and when they can be linked to the bulk topology. Our discussion includes mirror, twofold-rotation and inversion as crystalline symmetries. In Chapter 3 we discuss the appearance of anomalous gapless or ingap states at defects in a crystalline insulator or superconductor. In particular, we link the existence of anomalous states at disclinations -- lattice defects that violate a rotation symmetry only locally -- to second-order topological crystalline phases. As a side product, we identify the possible contributions of other topological phases to the disclination anomaly. The results of this chapter allow to determine precisely how the disclination anomaly is linked to the bulk topology. In Chapter 4 we present how to construct "symmetry-based indicators" for topological superconductors that admit a description within the Bogoliubov-de Gennes framework. Symmetry-based indicators are necessary criteria for a given Bogoliubov-de Gennes Hamiltonian to realize a topological crystalline phase with anomalous boundary exitations. These criteria are formulated in terms of data from a small set of high-symmetry momenta only. Our approach guarantees to extract the maximal information on the anomalous boundary exitations that can be extracted from a point-wise evaluation of the Bogoliubov-de Gennes Hamiltonian. In the limit of weak superconducting pairing, our criteria can be formulated in terms of data from the normal-state band structure alone.
The imposition of crystalline symmetries is known to lead to a rich variety of insulating and superconducting topological phases. These include higher-order topological phases and obstructed atomic limits with and without filling anomalies. We here comprehensively classify such topological crystalline phases (TCPs) with mirror, twofold rotation, and inversion symmetries in the presence of disorder that preserves the crystalline symmetry on average. We find that the inclusion of disorder leads to a simplification of the classification in comparison to the clean case. We also find that, while clean TCPs evade a general bulk-boundary principle, disordered TCPs admit a complete bulk-boundary correspondence, according to which (bulk) topological phases are topologically equivalent if and only if they have the same anomalous boundary states and filling anomaly. We corroborate the stability of disordered TCPs by way of field-theoretic, numerical, and symmetry-based analyses in various case studies. While the boundary signatures of most disordered TCPs are similar to their clean counterparts, the addition of disorder to certain mirror-symmetric TCPs results in novel higher-order statistical topological phases, in which zero-energy hinge states have critical wave-function statistics, while remaining protected from Anderson localization.
A candidate reading of seven recent preprints from cond-mat.str-el and physics.atom-ph suggests that the classical toolkit for classifying gapped phases and their transitions—built on Landau symmetry breaking, local order parameters, and invertible group symmetries—requires systematic extension at the level of microscopic algebraic structure, not merely at the phenomenological level. The emerging picture, drawn from the corpus, is that *generalized symmetries* (non-invertible, categorical, and higher-form) impose precise algebraic constraints on what constitutes a valid order parameter, what boundary conditions are permissible, and which phase transitions can occur without any form of spontaneous symmetry breaking, even after arbitrary gauging. Concretely: (i) condensable algebras in 2+1d topological orders classify gapped boundaries, and twin algebras sharing identical anyon content but inequivalent algebra structure describe physically distinct phases with isomorphic ground-state degeneracy but inequivalent order parameters [corpus:arxiv:2605.31602]; (ii) these twin phases can undergo direct transitions that are provably beyond-Landau even after partial gauging [corpus:arxiv:2605.31601]; (iii) string order parameters in 1d systems organize into Lagrangian algebras in the Drinfel'd center of the symmetry category, making the fusion rule of the symmetry category—not just its representation theory—the operative diagnostic [corpus:arxiv:2605.27193]; (iv) non-invertible symmetry-breaking transitions can be mapped via generalized gauging to deconfined quantum critical points, and vice versa, establishing a precise duality [corpus:arxiv:2605.27672]; and (v) non-invertible symmetry enriched topological orders (NI-SETOs) can be constructed from string-net models, extending symmetry enrichment beyond the group-theoretic setting [corpus:arxiv:2605.28794]. Two additional corpus entries—a Rydberg spin-chain study [corpus:arxiv:2605.27166] and a topological lattice gauge theory paper [corpus:arxiv:2605.28688]—provide microscopic anchoring and structural context, respectively. The central falsification path is explicit: if a twin-phase transition can be shown to spontaneously break some hidden symmetry after arbitrary gauging, the framework collapses. Concrete tensor-network (DMRG/MPS) constructions of twin-phase order parameters in 1+1d would constitute a direct test. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.27166, 2605.27193, 2605.27672, 2605.28688, 2605.28794, 2605.29745, 2605.29875, 2605.30421, 2605.31185, 2605.31540, 2605.31601, 2605.31602
Topological holography is a holographic principle that describes the generalized global symmetry of a local quantum system in terms of a topological order in one higher dimension. This framework separates the topological data from the local dynamics of a theory and provides a unified description of the symmetry and duality in gapped and gapless phases of matter. In this work, we develop the topological holographic picture for (1+1) d d quantum phases, including both gapped phases as well as a wide range of quantum critical points, including phase transitions between symmetry protected topological (SPT) phases, symmetry enriched quantum critical points, deconfined quantum critical points, and intrinsically gapless SPT phases. Topological holography puts a strong constraint on the emergent symmetry and the anomaly for these critical theories. We show how the partition functions of these critical points can be obtained from dualizing (orbifolding) more familiar critical theories. The topological responses of the defect operators are also discussed in this framework. We further develop a topological holographic picture for conformal boundary states of (1+1) d d rational conformal field theories. This framework provides a simple physical picture to understand conformal boundary states and also uncovers the nature of the gapped phases corresponding to the boundary states.
Deconfined criticality and gapless topological states have recently attracted growing attention, as both phenomena go beyond the traditional Landau paradigm. However, the deep connection between these two critical states, particularly in lattice realization, remains insufficiently explored. In this Letter, we reveal that certain deconfined criticality can be regarded as an intrinsically gapless topological state without gapped counterparts in a one dimensional lattice model. Using a combination of field-theoretic arguments and large-scale numerical simulations, we establish the global phase diagram of the model, which features deconfined critical lines separating two distinct spontaneous symmetry breaking ordered phases. More importantly, we unambiguously demonstrate that the mixed anomaly inherent to deconfined criticality enforces topologically robust edge modes near the boundary, providing a general mechanism by which deconfined criticality manifests as a gapless topological state. Our findings not only offer a new perspective on deconfined criticality but also deepen our understanding of gapless topological phases of matter.
We investigate the gauging of higher-form finite Abelian symmetries and their sub-groups in quantum spin models in spatial dimensions d=2 d = 2 and 3. Doing so, we naturally uncover gauged models with dual higher-group symmetries and potential mixed ‘t Hooft anomalies. We demonstrate that the mixed anomalies manifest as the symmetry fractionalization of higher-form symmetries participating in the mixed anomaly. Gauging is realized as an isomorphism or duality between the bond algebras that generate the space of quantum spin models with the dual generalized symmetry structures. We explore the mapping of gapped phases under such gauging related dualities for 0-form and 1-form symmetries in spatial dimension d=2 d = 2 and 3. In d=2 d = 2 , these include several non-trivial dualities between short-range entangled gapped phases with 0-form symmetries and 0-form symmetry enriched Higgs and (twisted) deconfined phases of the gauged theory with possible symmetry fractionalizations. Such dualities also imply strong constraints on several unconventional, i.e., deconfined or topological transitions. In d=3 d = 3 , among others, we find, dualities between topological orders via gauging of 1-form symmetries. Hamiltonians self-dual under gauging of 1-form symmetries host emergent non-invertible symmetries, realizing higher-categorical generalizations of the Tambara-Yamagami fusion category.