Chern-textured exciton insulators with valley spiral order in moiré materials

We explore the phase diagrams of moiré materials in search of a class of intervalley-coherent correlated insulating state: the Chern texture insulator (CTI). This phase of matter, proposed in a companion paper [Kwan , .], breaks valley <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>U</a:mi><a:mo>(</a:mo><a:mn>1</a:mn><a:mo>)</a:mo></a:mrow></a:math> symmetry in a nontrivial fashion wherein the valley order parameter is forced to texture in momentum space as a consequence of band topology. Using detailed Hartree-Fock studies, we establish that the CTI emerges as an energetically competitive intermediate-coupling ground state in several moiré systems that lack a twofold rotation symmetry that forbids the single-particle topology essential to the formation of the CTI valley texture.