Self-Consistent Large-$N$ Analytical Solutions of Inhomogneous Condensates in Quantum $\mathbb{C}{P}^{N-1}$ Model

We give, for the first time, self-consistent large-$N$ analytical solutions of inhomogeneous condensates in the quantum $\mathbb{C}{P}^{N-1}$ model in the large-$N$ limit. We find a map from a set of gap equations of the ${\mathbb C}P^{N-1}$ model to those of the Gross-Neveu (GN) model (or the gap equation and the Bogoliubov-de Gennes equation), which enables us to find the self-consistent solutions. We find that the Higgs field of the ${\mathbb C}P^{N-1}$ model is given as a zero mode of solutions of the GN model, and consequently only topologically nontrivial solutions of the GN model yield nontrivial solutions of the $\mathbb{C}{P}^{N-1}$ model. A stable single soliton is constructed from an anti-kink of the GN model and has a broken (Higgs) phase inside its core,in which $\mathbb{C}{P}^{N-1}$ modes are localized,with a symmetric (confining) phase outside. We further find a stable periodic soliton lattice constructed from a real kink crystal in the GN model,while the Ablowitz-Kaup-Newell-Segur hierarchy yields multiple solitons at arbitrary separations.