Abstract
We construct the first analytic examples of non-homogeneous condensates in the Georgi-Glashow model at finite density in $(2+1)$ dimensions. The non-homogeneous condensates, which live within a cylinder of finite spatial volume, possess a novel topological charge that prevents them from decaying in the trivial vacuum. Also the non-Abelian magnetic flux can be computed explicitly. These solutions exist for constant and non-constant Higgs profile and, depending on the length of the cylinder, finite density transitions occur. In the case in which the Higgs profile is not constant, the full system of coupled field equations reduce to the Lam\'e equation for the gauge field (the Higgs field being an elliptic function). For large values of this length, the energetically favored configuration is the one with a constant Higgs profile, while, for small values, it is the one with non-constant Higgs profile. The non-Abelian Chern-Simons term can also be included without spoiling the integrability properties of these configurations. Finally, we study the stability of the solutions under a particular type of perturbations.
Highlights
One of the most challenging open problems in theoretical and experimental investigations in quantum chromodynamics (QCD) is to determine the phases diagram at finite density and temperature, and especially, to shed light on the confinement mechanism
The nonhomogeneous condensates, which live within a cylinder of finite spatial volume, possess a novel topological charge that prevents them from decaying in the trivial vacuum
For large values of this length, the energetically favored configuration is the one with a constant Higgs profile, while, for small values, is the one with the nonconstant Higgs profile
Summary
One of the most challenging open problems in theoretical and experimental investigations in quantum chromodynamics (QCD) is to determine the phases diagram at finite density and temperature, and especially, to shed light on the confinement mechanism. This is a really fundamental question, as all the relevant nonperturbative configurations of Yang-Mills theory, which are important to understand the confinement mechanism, have been constructed in “infinite space” (see [56,57,58] for detailed pedagogical reviews), while the behavior of these nonperturbative configurations living at finite density is largely unknown (despite its huge interest in many applications) In this respect, one of the main issues (which is discussed in this paper) yet to be properly understood is how topologically nontrivial configurations react to nontrivial boundary conditions at finite volume. We construct the first genuine analytic examples of nonhomogeneous and topologically nontrivial condensates in the Georgi-Glashow model and in the Yang-MillsHiggs-Chern-Simons theory in (2 þ 1) dimensions. Such solutions possess a novel topological charge and have nonvanishing non-Abelian magnetic flux.
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