Topological quantum phase transitions of topological Euler insulators

For two-dimensional topological band models, the Euler number is a topological invariant to characterize the non-trivial interband topology for adjacent two bands with real eigenstates in the presence of the combined parity and time-reversal symmetry, which is different from the Chern number to characterize the single-band topology of complex eigenstates in the absence of time reversal symmetry. In this work, we study a three-orbital square lattice model that hosts topological Euler insulators, clearly manifested by the bulk band structures, the Wilson loop spectra and the non-trivial Euler numbers. And we observe a bulk-edge correspondence of topological Euler insulators in stark contrast to Chern insulators. We also present various topological phase diagrams based on the Euler numbers and related calculations for different sets of model parameters. Such results enrich our understanding of intriguing Euler class topological phases and may also pave the way for material realizations.