Semilocal Self-Dual Chern-Simons Solitons and Toda-type Eqations

We consider a nonrelativistic Chern-Simons theory of planar matter fields interacting with the Chern-Simons gauge field in a SU(N)global×U(1)local invariant fashion. We find that this model admits static zero-energy selfdual soliton solutions. We also present a set of exact soliton solutions. The exact time-dependent solutions are also obtained, when this model is considered in the background of an external uniform magnetic field. PACS Numbers: 03.65.Ge, 11.10.Lm, 11.15. -q Typeset using REVTEX 1 Soliton solutions in Chern-Simons ( CS ) gauge theories have received considerable attention over the past few years due to their possible relevance to the planar condensed matter systems. It is known that the abelian Higgs model with a CS term admits finite energy charged vortex solutions [1]. Further, the pure CS Higgs theory admits static selfdual soliton solutions with a φ-type scalar potential [2]. Moreover, in the nonrelativistic limit of this theory [3], the charge density solves Liouville equation at the self-dual limit, all of whose solutions are well known. When this nonrelativistic model is modified by including an external magnetic field [4] or a harmonic force [5], exact time-dependent soliton solutions can be obtained. The self-dual nonrelativistic case for the nonabelian gauge group has also been considered [6], which provides a unified dynamical framework for a variety of two-dimensional nonlinear equations [7]. In this Brief Report, we consider a nonrelativistic CS theory with a gauge group as in the case of semilocal Nielsen-Olesen strings [8] or semilocal charged vortices [9,10]. In particular, we consider the Jackiw-Pi ( JP ) model [3] but with the gauge group enlarged to SU(N)global × U(1)local. We find that this model admits static zero-energy self-dual soliton solutions. Interestingly enough, we are also able to find a set of exact soliton solutions. These solitons are characterized by the magnetic flux Φ = − e κ |κ| (N +1)| n |, the charge Q = − e Φ and the angular momentum J = Q, where n is the winding number and κ and e are two dimensional constants to be discussed below. We also present exact time-dependent solutions of the model in the presence of an external uniform magnetic field. Consider the nonrelativistic Lagrangian L = iΨ ( ∂t + ieA 0 ) Ψ− 1 2m | (∂i + ieAi)Ψ | + g 2 (ΨΨ) + κ 4 ǫAμFνα (1) where Ψ is N component scalar field, i.e., Ψ = (ψ 0, ψ ∗ 1, . . . , ψ ∗ N−1) ( Here ∗ denotes the complex conjugation ). The Lagrangian (1) is invariant under a SU(N)global × U(1)local 2 transformations. For N = 1, the Lagrangian (1) essentially describes the JP model. The N = 2 case was previously discussed and some exact solutions were obtained in Ref. [11]. Note that the scalar field self-interaction may be attractive or repulsive according as g is positive or negative respectively. However, as we will see shortly, the self-interaction is always attractive for zero-energy self-dual soliton solutions as in the case of JP model. The equations of motion which follow from (1) are κ 2 ǫFαβ = eJ ν (2) i∂tΨ = − 1 2m DiDiΨ+ eA Ψ− g| Ψ |Ψ (3) where the conserved matter current J is given by, J = (ρ, J ) = [ ΨΨ, i 2m [Ψ(DΨ)− (DiΨ)Ψ] ]