## Research Progress

## Fragile topological band in the checkerboard antiferromagnetic monolayer FeSe

author： time：2022-02-17 clicks：

Fig.1: (a) The crystal structure of cb-AFM monolayer FeSe. (b) Spin-polarized band structures without SOC. (c) Band structures with SOC. (d) The Wilson loop spectra of the isolated fragile topological flat band. (e) Spatial distribution of the corner states.(f) Edge states of the H_{S }with S_{z }(red) and H_{0 }without S_{z}(blue).

**Background**

The fragile topology, which stands between the trivial insulator and the stable topological insulator, violates the bulk-boundary correspondence in a conventional way but has attracted intensive interest for its intriguing response to magnetic field and disorder. When a flat band with fragile topology enters a superconductor phase, the fragile topology will have a nontrivial contribution to the superfluid weight and hence enhances the superconducting transition temperature. So far, most of the studies on fragile topology are based on theoretical models, and the only reported fragile topological electronic system is twisted bilayer graphene. Therefore, it is highly desirable to discover and classify more natural fragile topological materials to study such exotic properties.

**What we discover?**

In this work, by means of the first-principles calculations and magnetic topological quantum chemistry, we comprehensively study the electronic structures of checkboard antiferromagnetic (cb-AFM) monolayer FeSe (Fig.1a). We find that the low-energy physics in the cb-AFM monolayer FeSe can be well captured by a double-degenerate flat band with fragile topology just below the Fermi level. First, we calculate the electronic structures without spin-orbit coupling of cb-AFM monolayer FeSe (Fig.1b). We find that the hybridization between spin-up and spin-down bands is almost zero, and the system is a semimetal with a parabolic four-fold degenerate point the M point. When the SOC is included, the four-fold degenerate point is opened with an insulating gap, leading to an isolated fragile topological flat band just below the Fermi level (Fig.1c). The square Wilson loop shown in Fig.1d prove that the fragile topology is protected by S_{4z }symmetry. Therefore cb-AFM monolayer FeSe is the first reported S_{4z}-protected fragile topology material. In addition, we find that this fragile topology can give rise to an 2D second-order topological insulator, which can induce bound states with fractional charge e/2 at the corners of the sample (Fig.1e). This result is consistent with the previous STM experimental observations about bound states at the corner of monolayer FeSe, which further validates the theoretical results. Finally, we demonstrate that cb-AFM monolayer FeSe is very close to an antiferromagnetic topological insulator, which is characterized by a spin-conserving symmetry S_{z}. The approximately S_{z }symmetry originates from the weak hybridization between spin-up and spin-down bands (Fig.1b). When the S_{z }symmetry is restored, cb-AFM monolayer FeSe will exhibit gapless edge states (Fig.1f).

**Why is this important?**

These results explain the previous experimental observations of topological edge states and bound state in monolayer FeSe very well. More importantly, our work points out the first S_{4z}-protected fragile topological material and provides a new platform to study the intriguing properties of the magnetic fragile topology, as well as the interplay between fragile topology and superconductivity.

**Who did the research?**

Aiyun Luo^{1}, Zhida Song^{2}& Gang Xu^{1,}^{†}

1. Wuhan National High Magnetic Field Center & School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, China

2. Department of Physics, Princeton University, Princeton, NJ, 08544, USA

**Funding**

This work was supported by the National Key Research and Development Program of China (2018YFA0307000), and the National Natural Science Foundation of China (11874022).