Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators (2512.24709v1)

Abstract: We apply methods of equivariant homotopy theory, which may not previously have found due attention in condensed matter physics, to classify first the fragile/unstable topological phases of 2D crystalline Chern insulator materials, and second the possible topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) -- which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized space of these band topologies, we compute the latter in examples where this group is non-abelian, showing that any potential FQAH anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware. Mathematical details are spelled out in a supplement.

• The paper develops a classification framework using equivariant homotopy theory to capture fragile topological phases in 2D crystalline Chern insulators.
• It reveals that crystalline symmetries enforce novel invariants and momentum-space localization of anyonic excitations in fractional Chern insulators.
• The study connects modular group actions to topological order, offering insights for quantum computation via symmetry-protected defect anyons.

Fragile Topological Phases and Topological Order in 2D Crystalline Chern Insulators

Introduction

This paper introduces a comprehensive framework for classifying and characterizing fragile topological phases and topological order in two-dimensional crystalline Chern insulators, leveraging the machinery of equivariant homotopy theory. The analysis encompasses both the non-interacting case (Chern insulators protected by crystalline symmetries) and their interacting fractionalized analogs (fractional Chern insulators, FCIs), focusing on the impact of symmetry and the subtleties arising from adiabatic monodromy and general covariance. Notably, the approach captures previously unrecognized topological structures, demonstrating, for example, that anyonic excitations in FCIs with broken symmetry localize in momentum space rather than position space, and that with nontrivial crystalline symmetry, the resulting topological order is controlled by modular group actions associated with the high-symmetry points in the Brillouin torus.

Equivariant 2-Cohomotopy and Fragile Phases

The classification of topological phases of 2-band Chern insulators is recast in terms of homotopy classes of equivariant maps from the Brillouin zone (torus) to the 2-sphere, $[T^2, S^2]^G$, where $G$ is the point group symmetry of the lattice. This set is isomorphic to the equivariant 2-Cohomotopy of the Brillouin torus. Unlike stable classification schemes such as $K$-theory, equivariant 2-Cohomotopy is sensitive to the crystalline symmetry actions and captures fine structure associated with fragile (or unstable) phases.

In the absence of crystalline symmetry ($G=1$), the 2-Cohomotopy reduces to the standard Chern classification by an integer winding number (the Chern number), as is well-known from the Haldane model and related systems. However, the introduction of crystalline symmetries fundamentally alters the structure: the equivariance condition forces the map to respect fixed-point sets and the symmetry-induced stratification of the torus, producing new invariants associated with the symmetry-fixed points or "high-symmetry points" in momentum space. For example, for the $p3$ space group (the symmetry group of the triangular lattice with $C_3$ rotation), the set of topological phases becomes $3\mathbb{Z} \times [4]$, with the first factor reflecting that the Chern number is divisible by 3, and the second counting inequivalent mappings of high-symmetry points to the fixed poles on $S^2$.

Figure 1: Cell-structure associated with $p3$ crystalline symmetry highlights the three fixed points (high-symmetry points) under $C_3$ rotation and their action on the Brillouin torus.

Topological Order, Adiabatic Monodromy, and Fractional Chern Insulators

Moving to FCIs, the paper addresses the question of how adiabatic parameter transport leads to nontrivial unitary representations (monodromy) on the ground state manifold—i.e., the topological order. For usual FQH systems, this monodromy is captured by braid group representations associated with physically exchanging anyons in real space. For FCIs, the relevant parameter space involves deformations of Bloch Hamiltonians, i.e., adiabatic traversals in the space of equivariant maps from the Brillouin torus to $S^2$. The ground state Hilbert bundle forms a local system over the connected components of this (covariantized) mapping space.

Without crystalline symmetry, the fundamental group of this mapping space in a fixed Chern sector is the integer Heisenberg group $H_3(\mathbb{Z}, 2C)$ (level-2 central extension), reflecting the same non-commuting algebra observed for Wilson loop observables in toroidal FQH systems. Thus, anyonic excitations in the FCI are momentum-space solitons, in sharp correspondence with the duality between magnetic flux in position space and Berry curvature in momentum space. This dualization is mathematically precise within the cohomotopical framework advanced here.

With nontrivial crystalline symmetry, the space of available band topologies is further quotient by the action of the equivariant diffeomorphism (mapping class) group. In the $p3$ case, for example, the modular group acts as $\mathrm{Sym}(3)$, the permutation group on the three high-symmetry points of $T^2$. Consequently, topological defects (anyons) become bound to these points, and their statistics are governed by the symmetric group rather than the braid group, producing parafermionic or "defect" anyons exhibiting parastatistics; the algebraic structure of the emerging topological order is dramatically altered.

Theoretical Implications of Equivariant Covariance

The analysis makes clear that in crystalline systems, the correct parameterization of topological order is in terms of the homotopy quotient (covariantization) of the symmetric mapping space:

$$P = \mathrm{Map}(T^2, S^2)^G // \mathrm{Diff}^+(T^2)^G$$

This reflects the expectation from TQFT that diffeomorphic (or more generally, symmetry-related) configurations are physically equivalent from the perspective of the ground state sector. The monodromy group on ground states thus becomes an extension (typically a semidirect product) of the fundamental group of the symmetric mapping space with the equivariant modular group.

The practical outcome is that in the presence of symmetry, para-anyon statistics and their associated non-abelian representations (e.g., irreducible representations of $\mathrm{Sym}(3)$ for $p3$) are the primary source of topological gates in quantum systems defined on crystalline materials. The work highlights specific group actions and their quantum computational significance, such as the appearance of rotation gates generated by cyclic permutations of high-symmetry defects.

Outlook and Future Directions

This paper delivers a precise homotopical account of fragile topological phases and topological order for 2-band crystalline Chern insulators and their fractionalized relatives. The introduction of equivariant 2-Cohomotopy furnishes a complete symmetry-respecting classification of possible phases, subsuming and sharply refining K-theoretical approaches, particularly in the context of fragile phases and crystalline-protected topological invariants. The extension to topological order via monodromy in parameter space clarifies the physical meaning of anyonic excitations in these systems, emphasizing that in the crystalline context, the most robust forms of topological order are bound to defect patterns and are manifest in momentum space.

Several salient implications and directions for future work arise:

• Experimental realization and detection: The momentum-space localization of anyons, especially for symmetry-broken systems, demands novel probes for detecting topological order in FCIs, possibly via topological response functions in momentum space.
• Quantum computation: The algebraic structures (e.g., parafermion statistics associated with symmetric group actions) are candidates for topological gates. The explicit identification of these protected gates as arising from crystal symmetry actions opens avenues for hardware-oriented investigations into robust quantum gate implementation in crystalline platforms.
• Theoretical generalization: The framework extends immediately to more complicated symmetry settings, multi-band systems, and higher dimensions, and provides a template for applying equivariant (co)homotopy theory to other classes of topological and symmetry-enriched quantum phases.

Conclusion

By unifying and extending the topological classification of crystalline Chern insulators with tools from equivariant homotopy theory, this work supplies novel algebraic invariants for fragile phases, clarifies the symmetry-resolved structure of topological order, and establishes the physical and mathematical presence of defect para-anyons localized in momentum space. This synthesis has concrete ramifications for both experimental condensed matter physics and the design of topologically protected operations in quantum information science.