Homotopy continuation method for solving Dyson equation fully self-consistently: Theory and application to NdNiO2

The solution of the Dyson equation for the small-gap systems can be plagued by large non-converging iterations. In addition to the convergence issues, due to a high non-linearity, the Dyson equation may have multiple solutions. We apply the homotopy continuation approach to control the behavior of iterations. We used the homotopy continuation to locate multiple fully self-consistent GW solutions for the NdNiO2 solid and to establish the corresponding Hartree-Fock limits. Some of the solutions found are qualitatively new and help to understand the nature of electron correlation in this material. We show that there are multiple low-energy charge-transfer solutions leading to the formation of charge-density waves. Our results qualitatively agree with the experimental conductivity measurements. To rationalize the structure of solutions, we compare the k-point occupations and generalize the concept of natural difference orbitals for correlated periodic solids.