Chern-Simons System Research Articles

Global Classical Solutions of the 1.5D Relativistic Vlasov–Maxwell–Chern–Simons System

We investigate the kinetic model of the relativistic Vlasov–Maxwell–Chern–Simons system, which originates from gauge theory. This system can be seen as an electromagnetic fields (i.e., Maxwell–Chern–Simons fields) perturbation for the classical Vlasov equation. By virtue of a nondecreasing function and an iteration method, the uniqueness and existence of the global solutions for the 1.5D case are obtained.

Bifurcation of fixed points in a O(N)-symmetric ( 2+1 )-dimensional gauged Φ6 theory with a Chern-Simons term

We examine the phase structure of an Abelian Chern-Simons system with relativistic charged matter fields with a sixth-order potential. Using a large $N$ technique, we compute the quantum effective potential and the renormalization group function of the coupling to the next-to-leading order of the $1/N$ expansion in terms of the Chern-Simons coefficient. The model has a phase which exhibits spontaneous breaking of scale symmetry accompanied by a massless dilaton which is a Goldstone mode. We show that the beta function of the sextic coupling exhibits, at the next order in the $1/N$ expansion, nontrivial running that we analyze explicitly in terms of the Chern-Simons coefficient. Viewed as a dynamical system, the renormalization group (RG) flow exhibits a topological normal form of a generic one-dimensional system having a fold bifurcation. We demonstrate that the corresponding IR and UV fixed points, each describing a conformal phase of the theory, approach each other until they merge, giving rise to a scaling behavior similar to Berezinskii-Kosterlitz-Thouless phase transitions. Our study identifies a window in the parameter space of the Chern-Simons coefficient where the renormalization group flow has a stable infrared fixed point and where scale invariance is recovered. We also find that the Chern-Simons interaction modifies the scaling dimension of the operator crossing marginality at the bifurcation points.

A new type of non-topological bubbling solutions to a competitive Chern-Simons model

We study a non-Abelian Chern-Simons system in $\mathbb{R}^2$, including the simple Lie algebras $A_2$ and $B_2$. In a previous work, we proved the existence of radial non-topological solutions with prescribed asymptotic behaviors via the degree theory. We also constructed a sequence of bubbling solutions with only one component blowing up partially at infinity. In this paper, we construct a sequence of radial non-topological bubbling solutions of another type via the shooting argument. One component of these bubbling solutions locally converge to a non-topological solution of the Chern-Simons-Higgs scalar equation, but both components blow up partially in different regions at infinity at the same time. This generalizes a recent work by Choe, Kim and the second author, where the $SU(3)$ case (i.e. $A_2$) was studied. Our result is new even for the $SU(3)$ case and also confirms the difference between the $SU(3)$ case and the $B_2$ case.

Self-dual radial non-topological solutions to a competitive Chern–Simons model

We study a non-Abelian Chern–Simons system of rank 2:(Δu1Δu2)+K(eu1eu2)−K(eu100eu2)K(eu1eu2)=(4πN1δ04πN2δ0)in R2, where N1,N2∈N∪{0}, δ0 is the Dirac measure at 0, and K=(aij) is a 2×2 matrix satisfying a11,a22>0, a12,a21<0 and det⁡K>0, including the Cartan matrix B2. The existence of non-topological solutions has remained a long-standing open problem. Here by applying the degree theory, we prove the existence of radial non-topological solutions (u1,u2) satisfying the prescribed asymptotic condition uk(x)=−2αkln⁡|x|+O(1) as |x|→∞ for some αk>1. We also construct bubbling solutions to show that the range of (α1,α2) is optimal in some sense. This generalizes a recent work by Choe, Kim and the second author, where the SU(3) case (i.e. K is the Cartan matrix A2) was investigated.

Doubly periodic vortices for a Chern–Simons model

In this paper we establish the existence of vortex solutions for a relativistic self-dual U(1)×U(1) Chern–Simons system over a doubly periodic domain. We use a constrained minimization approach to obtain the existence result, the system admits a solution when the Chern–Simons coupling parameter κ>0 is sufficiently small; while no solution exists when κ>0 is not sufficiently small. And we also get the quantized integral.

Bubbling solutions for a skew-symmetric Chern–Simons system in a torus

We establish the existence of bubbling solutions for the following skew-symmetric Chern–Simons system{Δu1+1ε2eu2(1−eu1)=4π∑i=1N1δpi1Δu2+1ε2eu1(1−eu2)=4π∑i=1N2δpi2 over a parallelogram Ω with doubly periodic boundary condition, where ε>0 is a coupling parameter, and δp denotes the Dirac measure concentrated at p. We obtain that if (N1−1)(N2−1)>1, there exists an ε0>0 such that, for any ε∈(0,ε0), the above system admits a solution (u1,ε,u2,ε) satisfying u1,ε and u2,ε blow up simultaneously at the point p⁎, and1ε2euj,ε(1−eui,ε)→4πNiδp⁎,1≤i,j≤2,i≠j as ε→0, where the location of the point p⁎ defined by (1.12) satisfies the condition (1.13).

Optical properties of Chern-Simons systems

Chern-Simons (CS) systems interacting with electromagnetic radiation are described by a term f FɅF added to the Maxwell action. In (3+1)D, this CS term is a boundary term affecting the system behaviour in its borders. We study the consequences of the above in the properties of electromagnetic radiation, in particular, by considering the interplay between magneto-electric properties and topology. Apart from a modified Kerr polarization rotation, compared to that found for the particular case of topological insulators, we also found two Brewster angles, for s and p polarization of reflected radiation, respectively. Energy distribution between reflected and transmitted radiation is also studied in terms of the magneto-electric properties and topological condition of the system.

Topological multivortex solutions for the Chern–Simons system with two Higgs particles

ABSTRACTIn this paper, we prove the uniqueness of topological multivortex solutions to the self-dual abelian Chern–Simons model if either the Chern–Simons coupling parameter is sufficiently small or sufficiently large. In addition, we also establish the sharp region of the flux for nontopological solutions with a single vortex point.

Yang–Mills–Chern–Simons system in the presence of a Gribov horizon with fundamental Higgs matter

In this work we study the behaviour of Yang–Mills–Chern–Simons theory coupled to a Higgs field in the fundamental representation by taking into account the effects of the presence of the Gribov horizon. By analyzing the infrared structure of the gauge field propagator, both confined and de-confined regions can be detected. The confined region corresponds to the appearance of complex poles in the propagators, while the de-confined one to the presence of real poles. One can move from one region to another by changing the parameters of the theory.

On non-topological solutions of the $\mathbf{G}_2$ Chern–Simons system

On non-topological solutions of the $\mathbf{G}_2$ Chern–Simons system

On Non-topological Solutions of the 𝐴₂ and 𝐵₂ Chern-Simons System

On Non-topological Solutions of the 𝐴₂ and 𝐵₂ Chern-Simons System

Existence of non-topological solutions for a skew-symmetric Chern-Simons system

We investigate the existence of non-topological solutions $(u_1,u_2)$ satisfying $$u_{i}(x)=-2\beta_i\ln|x|+O(1),\quad\text{as }|x|\rightarrow +\infty,$$ such that $\beta_i>1$ and $$(\beta_1-1)(\beta_2-1)>(N_1+1)(N_2+1),$$ for a skew-symmetric Chern-Simons system. By the bubbling analysis and the Leray-Schauder degree theory, we get the existence results except for a finite set of curves: $$\frac{N_1}{\beta_1+N_1}+\frac{N_2}{\beta_2+N_2}=\frac{k-1}{k},k=2,\cdots,\max(N_1,N_2).$$ This generalizes a previous work by Choe-Kim-Lin \cite{ChoeKimLin2011}.

Mixed Type Solutions of the $${SU(3)}$$ S U ( 3 ) Models on a Torus

In this paper, we study mixed-type solutions of \({SU(3)}\) Chern–Simons system (see (1.4) below) on a two dimensional flat torus. Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), among other things, Nolasco and Tarantello obtained solutions of (1.4) as minimizers of several functionals closely related to (1.4), and showed that if \({N_1+N_2=1}\), then one of those minimizers turns out to be a mixed-type solution, that is, one component tends to \({\ln\frac{1}{2}}\) pointwise a.e. and the other component converges to a solution of a mean field equation. We call these kinds of solutions mixed-type (I) solutions. In this paper, we prove two main results: (i) the asymptotic analysis of mixed-type (I) solutions with arbitrary configuration of vortex points, and (ii) the existence of mixed-type (I) solutions under a non-degenerate condition. This non-degenerate condition also ensures some uniqueness result. In particular, our results imply that when \({N_1+N_2=1}\), there are only two mixed-type (I) solutions of (1.4).

Supersymmetric 3D model for gravity with SU(2) gauge symmetry, mass generation and effective cosmological constant

A Chern–Simons system in 2+1 dimensions invariant under local Lorentz rotations, SU(2) gauge transformations, and local supersymmetry (SUSY) transformations is proposed. The field content is that of (2+1)-gravity plus an SU(2) gauge field, a spin-1/2 fermion charged with respect to SU(2) and a trivial free abelian gauge field. A peculiarity of the model is the absence of gravitini, although it includes gravity and SUSY. Likewise, no gauginos are present. All the parameters involved in the system are either protected by gauge invariance or emerge as integration constants. An effective mass and effective cosmological constant emerge by spontaneous breaking of local scaling invariance. The vacuum sector is defined by configurations with locally flat Lorentz and SU(2) connections sporting nontrivial global charges. Three-dimensional Lorentz-flat geometries are spacetimes of locally constant negative—or zero—Riemann curvature, which include Minkowski space, AdS3, BTZ black holes, and point particles. These solutions admit different numbers of globally defined, covariantly constant spinors and are therefore good candidates for stable ground states. The fermionic sector in this system could describe the dynamics of electrons in graphene in the long wavelength limit near the Dirac points, with the spin degree of freedom of the electrons represented by the SU(2) label. If this is the case, the SU(2) gauge field would produce a spin–spin interaction giving rise to strong correlation of electron pairs.

Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system

In this paper, we consider a skew-symmetric Chern-Simons system problem with a coupling parameter. Our main goal is that, when the coupling parameter is small, the topological type solutions to this system problem are uniquely determined by the location of their vortex points. This result follows by the bubbling analysis and the non-degeneracy of linearized equations.

Uniqueness of non-topological solutions for the Chern-Simons system with two Higgs particles

We study the non-topological radial solutions of the Abelian Chern-Simons equation with two Higgs particles. We establish the non-degeneracy property of linearized equation and the uniqueness property for the corresponding non-topological radial solutions.

Classification of the entire radial self-dual solutions to non-Abelian Chern–Simons systems

We consider the entire radial solutions to the non-Abelian Chern–Simons systems of rank 2(ΔuΔv)=−K(euev)+K(eu00ev)K(euev)+(4πN1δ04πN2δ0)(0.1)in R2, where Ni⩾0, i=1,2 and K=(aij) is a 2×2 matrix satisfying a11, a22>0, a12, a21<0 and a11a22−a12a21>0. This system is motivated by the relativistic non-Abelian Chern–Simons model, Lozano–Marqués–Moreno–Schaposnik model of bosonic sector of N=2 supersymmetric Chern–Simons–Higgs theory, and Gudnason model of N=2 supersymmetric Yang–Mills–Chern–Simons–Higgs theory. Understanding the structure of entire radial solutions is one of fundamental issues for the system of nonlinear equations. We prove that any entire radial solutions of (0.1) must be one of topological, non-topological and mixed type solutions, and completely classify the asymptotic behaviors at infinity of these solutions. As an application of this classification, we prove that the two components u and v have intersection at most finite times.

Limit solutions of the Chern–Simons equation

Given a bounded domain Ω in R2, we investigate the scalar Chern–Simons equation −Δu+eu(eu−1)=μin Ω, in cases where there is no solution for a given nonnegative finite measure μ. Approximating μ by a sequence (μn)n∈N of nonnegative L1 functions or finite measures for which this equation has a solution, we show that the sequence of solutions (un)n∈N of the Dirichlet problem converges to the solution with largest possible datum μ#≤μ and we derive an explicit formula of μ# in terms of μ. The counterpart for the Chern-Simons system with datum (μ,ν) behaves differently and the conclusion depends on how much the measures μ and ν charge singletons.

Instantonic dyons of Yang–Mills–Chern–Simons models in d = 2n + 1 dimensions, n > 2

We investigate finite energy solutions of Yang–Mills–Chern–Simons systems in odd spacetime dimensions, d = 2n + 1, with n > 2. This can be carried out systematically for all n, but the cases n = 3, 4 corresponding to seven- and nine-dimensional spacetimes are treated concretely. These are static and spherically symmetric configurations, defined in a flat Minkowski background. The value of the electric charge is fixed by the Chern–Simons coupling constant.

Einstein–Yang-Mills–Chern-Simons solutions inD=2n+1dimensions

We investigate finite energy solutions of the Einstein--Yang-Mills--Chern-Simons system in odd spacetime dimensions, $D=2n+1$, with $n&gt;1$. Our configurations are static and spherically symmetric, approaching at infinity a Minkowski spacetime background. In contrast with the Abelian case, the contribution of the Chern-Simons term is nontrivial already in the static, spherically symmetric limit. Both globally regular, particlelike solutions and black holes are constructed numerically for several values of $D$. These solutions carry a nonzero electric charge and have finite mass. For globally regular solutions, the value of the electric charge is fixed by the Chern-Simons coupling constant. The black holes can be thought of as nonlinear superpositions of Reissner-Nordstr\"om and non-Abelian configurations. A systematic discussion of the solutions is given for $D=5$, in which case the Reissner-Nordstr\"om black hole becomes unstable and develops non-Abelian hair. We show that some of these non-Abelian configurations are stable under linear, spherically symmetric perturbations. A detailed discussion of an exact $D=5$ solution describing extremal black holes and solitons is also provided.