A Two-Dimensional Model. Abrikosov Vortices

Abstract

We now describe more complicated examples of theories that have topological integrals of motion. We start with the analog of the action integral (9.3) for a complex scalar field Ψ in two dimensions:

$$S = \int {l{d^3}} x = \frac{1}{2}\int {{\partial _\mu }} \bar \Psi {\partial ^\mu }\Psi {d^3}x - \frac{1}{8}\lambda {\int {({{\left| \Psi \right|}^2} - {a^2})} ^2}{d^3}x$$

(10.1)

, where μ = 0, 1, 2, x = (x ⁰, x ¹, x ² = (x ⁰, x), ∈ R ³, ∂ ₀ = ∂ ⁰ = ∂/∂x ⁰ =∂/∂t and ∂ _i = −∂ ⁱ for i = 1, 2. We can also think of Ψ as a two-component real scalar field, instead of a complex field.