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

Abstract

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.

arXiv comment19 pages, 3 figures. Updates in v2: References added. v3: Published version

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• Confinement-Deconfinement Crossover in the Lattice $\mathbb{C}{P}^{N-1}$ Model
• Lattice $\mathbb{C}{P}^{N-1}$ model with ${\mathbb{Z}}_N$ twisted boundary condition: bions, adiabatic continuity and pseudo-entropy
• Large-$N$ ${\mathbb{CP}}^{N-1}$ sigma model on a finite interval: general Dirichlet boundary conditions
• Nonlinear dynamical Casimir effect at weak nonstationarity
• Ground state modulations in the $\mathbb{C}{P}^{N-1}$ model
• Nambu-Jona Lasinio and Nonlinear Sigma Models in Condensed Matter Systems
• Confining solitons in the Higgs phase of $\mathbb{C}{P}^{N-1}$ model: Self-consistent exact solutions in large-$N$ limit
• Large-$N$ ${\mathbb{CP}}^{N-1}$ sigma model on a Euclidean torus: uniqueness and stability of the vacuum
• Remarks on the large-$N$ $\mathbb{C}{P}^{N-1}$ model
• Physics-informed neural networks for solving functional renormalization group on a lattice
• One-loop effective action of the $\mathbb{C}{P}^{N-1}$ model at large $\mu \beta$
• Lessons from $O(N)$ models in one dimension