Distinct topological excitonic insulators characterized by quantum geometry

The intertwining of electron-hole correlation and nontrivial topology is known to give rise to exotic topological excitonic insulators. Here, we show that the involvement of quantum geometry can characterize more exotic excitonic phases exhibiting physical properties that are not influenced by their topology but by geometry. Starting from a topological band insulator and gradually reducing the band gap, many-body interaction can initially generate a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>p</a:mi> <a:mo>+</a:mo> <a:mi>i</a:mi> <a:mi>p</a:mi> </a:mrow> </a:math> -wave and then an <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>s</b:mi> </b:math> -wave excitonic insulator. Interestingly, they bear the same Chern number but exhibit completely different spin textures and magneto-optical Kerr responses, reflecting the intricate geometric distinctions in their wave functions. We also propose to enhance the correlation effect via Floquet engineering, which provides a systematic way to realize these topological excitonic insulators and their phase transitions in the nonequilibrium steady states. Our results demonstrate correlated phenomena characterized by quantum geometry, beyond the conventional topological classifications.