Robust Signatures of Fragile Topology in Nonmagnetic Materials

The gist: Fragile topology nevertheless has robust signatures, yielding bulk Dirac cones in two-dimensional materials with arbitrarily many bands, and establishing a fragile topological index that guarantees the presence of gap closing points in the band structure

Fragile Topology and Robust Signatures

Topological phases of matter are called fragile because they do not generically host protected gapless boundary states and can be trivialized by adding additional valence bands Here we show that fragile topology nevertheless has robust signatures: it yields bulk Dirac cones in two-dimensional materials with arbitrarily many bands. In systems with time-reversal and twofold rotation symmetry, we establish a fragile topological index which guarantees the presence of gap closing points in the band structure. This index provides a robust spectroscopic signature of fragile topology which can be directly accessed in experiment.

Characterization via Topological Indices

The paper establishes a relation between two fragile topological indexes, one obtained from the Wilson loop and the second obtained from the Euler loop. When these indexes do not match (i.e. their difference is nonzero), then there must exist band crossing points in the bulk. The Euler class, denoted as χ = 1/2 P i wi, characterizes an isolated band pair. This index is conserved provided the band pair remains isolated, meaning that generic (i.e. non-TRIM) Dirac cones can be created or annihilated pairwise only if they have opposite winding numbers.

Relation between Wilson and Euler Loops

The paper demonstrates a technical result showing how the Wilson loop and the Euler loop are related for real systems with two bands in the presence of C2 and T symmetries. Specifically, the relation is given by φl,± = π(1 ± El) mod 2π. This relation shows that the eigenphases of the Wilson loop matrix are related to the Euler loop along the same C2 symmetric path by φl,± = π(1 ± El) mod 2π. The difference between these two loops is always equal to ElΓ − ElBZ = wΓ + wX + wY + wM (neglecting the possibility of additional Dirac points being present on the BZ boundary).

Fragile Topological Index Q

The fragile topological index Q is defined as Q = wCWLS − 1/2 sum X TRIM wi, which should be equal to zero. If Q!= 0, then there must exist at least 2Q generic Dirac cones in the band pair. This index is independent of the total number of bands in the system and appears to take nonzero values in a wide range of realistic materials —in our first-principles calculations, five out of five crystals.

First-Principles Verification

The results are tested using first-principles calculations on five materials from the Computational 2D Materials Database. All of them host such Dirac cones, suggesting that fragile topology may have widespread consequences in real materials. For MoOBrCl, the total number of generic Dirac cones is 8, and the sum over all winding numbers of all Dirac cones (TRIM and generic) vanishes, χ = 0. This band pair is not trivial despite having a vanishing Euler class. The index Q predicts a minimum number of generic Dirac cones, which in this case is 2Q = 4.

Implications for Nonlinear Transport

The fragile-topology-enforced Dirac cones will generically be gapped under perturbations breaking rotation symmetry (e.g. a crystal field), leading to a nonzero Berry dipole. This suggests that fragile topology may be used as a route towards identifying materials with robust nonlinear transport properties. Furthermore, introducing a perturbation that breaks C2, leaving only T, will generically lift any degeneracies at bulk Dirac points, leading to non-zero Berry curvature omega(k) concentrated close to the original location of the Dirac points. This nonzero time-reversal symmetric Berry curvature generically yields a non-linear response related to its the Berry curvature dipole D.

Experimental Access

This index provides a direct signature of fragile topology that can be accessed in ARPES measurements. Beyond this, it may allow an automated search for bulk Dirac cones in ab-initio codes, without requiring energy minimization, and it is also applicable to the 2D T - and C2-invariant planes of a 3D BZ. The results for some materials show that band pairs with Q!= 0 are closer than 2 eV from the Fermi level, making them amenable to ARPES detection.

Conclusion on Robustness

The Dirac points of an isolated band pair are stable in the sense that they can be removed only by pairwise annihilation. The index Q is independent of the total number of bands in the system and appears to take nonzero values in a wide range of realistic materials —in our first-principles calculations, five out of five crystals. This index is independent of the total number of bands in the system and appears to take nonzero values in a wide range of realistic materials —in our first-principles calculations, five out of five crystals. The combination of these features makes it a potentially good candidate for ARPES experiments. This index is independent of the total number of bands in the system and appears to take nonzero values in a wide range of realistic materials —in our first-principles calculations, five out of five crystals. The combination of these features makes it a potentially good candidate for ARPES experiments.

Summary

The paper demonstrates that fragile topology enforces the presence of Dirac points in the bulk of a system, characterized by a robust spectroscopic signature known as the fragile topological index Q. This index is derived from relating two topological invariants, the Euler loop and the Wilson loop, which reveals that if Q!= 0, generic Dirac cones must exist in band pairs. The authors verify this prediction using first-principles calculations on five materials from the Computational 2D Materials Database, finding that all tested materials host such Dirac cones. This index is applicable to both experimental access via ARPES and automated searches in ab-initio simulations, suggesting its broad applicability to real crystals. The study also shows how the index distinguishes between TRIM cones and generic ones, providing a distinct signature of fragile topology compared to the Euler class.

How it works

The core mechanism involves establishing a relation between two topological indexes, one from the Wilson loop and one from the Euler loop. The paper shows that for an isolated band pair, these loops are related by φl,± = π(1 ± El) mod 2π. This relationship is crucial because it allows for the definition of a fragile topological index Q = wCWLS − 1/2 sum X TRIM wi. If Q is non-zero, it implies the existence of generic Dirac cones in the band pair.

Key Mathematical Tools

The characterization relies on defining an Euler connection a(k) = ⟨u1(k) ∇k u2(k)⟩. The Euler loop El is defined as the integral of this connection over a closed loop. The Wilson loop WL matrix is defined on a closed loop l as Wl = P exp i I I ∫ dk · A. For C2T symmetry, the eigenvalues of the WL matrix are given by eiφl,± with eigenphases φl,± = ϕl,0 ± ϕl,1 mod 2π.

Experimental and Computational Relevance

The index Q is independent of the total number of bands in the system and appears to take nonzero values in a wide range of realistic materials —in our first-principles calculations, five out of five crystals. This index provides a direct signature of fragile topology that can be accessed in ARPES measurements. It may also allow an automated search for bulk Dirac cones in ab-initio codes, without requiring energy minimization, and it is also applicable to the 2D T - and C2-invariant planes of a 3D BZ. The results for some materials show that band pairs with Q!= 0 are closer than 2 eV from the Fermi level, making them amenable to ARPES detection.

Distinction from Previous Work

The authors focus on systems obeying both T and C2 at the same time, as opposed to requiring just the combination, C2T. In spite of being a more stringent condition, this requirement is practical, since there are significantly more nonmagnetic (i.e. T-preserving) than magnetic crystals in existing databases. More importantly, however, when both T and C2 are present, it is the index Q (and not the Euler class) that enables us to formulate a distinct spectroscopic signature of fragile topology, by distinguishing between TRIM cones and generic ones.

1. Robust Signature Detection In silico: The AI system can be used to automatically search for bulk Dirac cones in ab-initio simulations by computing a fragile topological index Q using Wilson loops and Euler loops, similar to how Chern numbers are used in 3D systems.

2. Automated Material Screening: The AI can screen large databases of 2D materials, such as the Computational 2D Materials Database, to predict which materials host generic Dirac cones enforced by fragile topology by calculating Q for isolated band pairs.

3. Spectroscopic Signature Prediction: The system can predict the presence of a robust spectroscopic signature in Angle-Resolved Photoemission Spectroscopy (ARPES) experiments by calculating the fragile topological index Q for a given material's band structure.

4. Nonlinear Transport Property Identification: By using the Berry curvature dipole D, which is non-zero when C2 symmetry is broken, the AI can identify materials likely to exhibit robust nonlinear transport properties based on their predicted Dirac cone locations.

Some topological phases of matter are called fragile. They can be trivialized by adding additional valence bands and do not generically host protected gapless boundary states. Here we establish a fragile topological index with a bulk signature: it guarantees the presence of gap closing points in the bulk band structure of systems with both time-reversal and twofold rotation symmetry. This result extends the established use of the Euler class in diagnosing the presence of Dirac cones for general two-dimensional systems with a combination of time-reversal and twofold rotation symmetry, to the case of nonmagnetic materials obeying both symmetries individually. The abundance of natural nonmagnetic materials with these symmetries renders the topological invariant practically useful, despite its fragility. We demonstrate this by applying first-principles calculations to five materials and finding that all of them host Dirac points identified by the invariant, suggesting its widespread applicability. Our results thus provide a robust spectroscopic signature of fragile topology in nonmagnetic systems which can be directly accessed in experiments on practically relevant materials. Furthermore, they enable an alternate method for finding Dirac cones in ab-initio simulations, and may be used as a route towards identifying materials with nonlinear transport properties.