Higher-order Topology of Axion Insulator EuIn$_2$As$_2$

Abstract: Based on first-principles calculations and symmetry analysis, we propose that EuIn$_2$As$_2$ is a long awaited axion insulator with antiferromagnetic (AFM) long range order. Characterized by the parity-based invariant $\mathbb Z_4=2$, the topological magneto-electric effect is quantized with $\theta=\pi$ in the bulk, with a band gap as large as 0.1 eV. When the staggered magnetic moment of the AFM phase is along $a/b$ axis, it's also a TCI phase. Gapless surface states emerge on (100), (010) and (001) surfaces, protected by mirror symmetries (nonzero mirror Chern numbers). When the magnetic moment is along $c$ axis, the (100) and (001) surfaces are gapped. As a consequence of a high-order topological insulator with $\mathbb Z_4=2$, the one-dimensional (1D) chiral state can exist on the hinge between those gapped surfaces. We have calculated both the topological surface states and hinge state in different phases of the system, respectively, which can be detected by ARPES or STM experiments.

Higher-order Topology in Axion Insulator EuIn(_2)As(_2)

The paper "Higher-order Topology of Axion Insulator EuIn(_2)As(_2)" explores the topological properties of the EuIn(_2)As(_2) compound through a combination of first-principles calculations and symmetry analysis. The authors propose that EuIn(_2)As(_2) can be characterized as an axion insulator (AI), demonstrating a quantized topological magneto-electric effect with a bulk band gap of approximately 0.1 eV. This work contributes significant insights into the field of condensed matter physics by advancing our understanding of higher-order topological insulators (HOTIs) and the behavior of axion fields without time reversal symmetry.

Identification of EuIn(_2)As(_2) as an Axion Insulator

EuIn(_2)As(_2) is identified as an axion insulator based on the parity-based invariant (\mathbb{Z}_4 = 2). This invariant implies a quantized topological magneto-electric (TME) effect, indexed by an axion angle (\theta = \pi). The theoretical exploration of EuIn(_2)As(_2) delves into its magnetic properties, revealing antiferromagnetic (AFM) order and the possibility of an axion insulating state with inherent inversion symmetry ((\mathcal{I})) but broken time reversal symmetry ((\mathcal{T})).

Detailed Phase Analysis

In the absence of spin-orbit coupling (SOC), EuIn(_2)As(_2) shows a nodal-line semimetal behavior, with nodal lines arising due to preserved symmetries in individual spin channels. When SOC is incorporated, the band structure becomes fully gapped, highlighting the transition into an insulating state.

The authors distinguish two antiferromagnetic phases: the afmb phase with moment orientation along the a/b-axis, and the afmc phase with moment orientation along the c-axis. These configurations demonstrate topological characteristics under varied symmetry constraints affecting the surface and hinge states.

Topological Surface and Hinge States

For the afmb phase, the material is confirmed to act as a topological crystalline insulator (TCI), with nonzero mirror Chern numbers guaranteeing the presence of gapless surface states due to mirror symmetry ((M_y) and (M_z)). Specifically, gapless Dirac cones appear on surfaces preserving these symmetries, providing potential realizations of protected surface states observable via angle-resolved photoemission spectroscopy (ARPES).

In contrast, the afmc phase, lacking surface state protection, is identified as a higher-order topological insulator (HOTI). The absence of gapless surface states leads to localization of 1D chiral hinge states, which could be detected using scanning tunneling microscopy (STM). These hinge states arise due to intertwined bulk and surface topology, presenting a new class of boundary phenomena.

Implications and Future Directions

The findings on EuIn(_2)As(_2) elucidate the rich interplay between magnetic ordering and topological phases in correlated electron systems. Practically, this compound emerges as an ideal platform to explore axion insulators and HOTIs, with particular relevance to the study of quantum magneto-electric phenomena. The theoretical framework provided could pave the way for further experimental investigations into axion fields within magnetic materials.

The conceptual robustness of EuIn(_2)As(_2) showcases the potential of stoichiometric materials as fertile grounds for investigating HOT phases—prompting inquiries into similar compounds and their electromagnetic responses. Future developments might focus on realizing such states under controlled environments, thereby broadening the horizon of material applications in quantum technologies.