We investigate the presence of vortex configurations in generalized Maxwell-Chern-Simons models with nonminimal coupling, in which we introduce a function that modifies the dynamical term of the scalar field in the Lagrangian. We first follow a route already considered in previous works to develop the Bogomol'nyi procedure, and, in this context, we use the first order equations to obtain a vortex with a novel behavior at its core. We then go further and introduce a novel procedure to develop the Bogomol'nyi methodology. It supports distinct first order equations, and we then investigate another model, in which the vortex may engender inversion of the magnetic flux, an effect with no precedents in the study of vortices within the nonminimal context.
In this work we consider an Abelian O(3) sigma model coupled nonminimally with a gauge field governed by a Maxwell and Chern-Simons terms. Bogomol'nyi equations are constructed for a specific form of the potential and generic nonminimal coupling constant. Furthermore, topological and nontopological self-dual soliton solutions are obtained for a critical value of the nonminimal coupling constant. Some particular static vortex solutions (topological and nontopological) satisfying the Bogomol'nyi bound are numerically solved and presented.
In this work we study a nonlinear gauged O(3)-sigma model with both minimal and nonminimal coupling in the covariant derivative. Using an asymmetric scalar potential, the model is found to exhibit both topological and non-topological soliton solutions in the Bogomol'nyi limit.
The (2+1) dimensional gauged O(3) nonlinear sigma model with Chern-Simons term is canonically quantized. Furthermore, we study a nonminimal coupling in this model implemented by means of a Pauli-type term. It is shown that the set of constraints of the model is modified by the introduction of the Pauli coupling. Moreover, we found that the quantum commutator relations in the nominimal case is independent of the Chern-Simons coefficient, in contrast to the minimal one.
Various aspects including the construction and the symmetries of Abelian Chern-Simons vortices are reviewed. Extended version of the Lectures delivered at NIKHEF (Amsterdam), July 2006. Typos corrected, some refernces added.
We search for vortices in a generalized Abelian Chern-Simons model with a nonstandard kinetic term. We illustrate our results, plotting and comparing several features of the vortex solution of the generalized model with those of the vortex solution found in the standard Chern-Simons model.
The non-relativistic `Dirac' equation of Lévy-Leblond is used to describe a spin {\small 1/2} particle interacting with a Chern-Simons gauge field. Static, purely magnetic, self-dual spinor vortices are constructed. The solution can be `exported' to a uniform magnetic background field.
The construction and the symmetries of Chern-Simons vortices in harmonic and uniform magnetic force backgrounds found by Ezawa, Hotta and Iwazaki, and by Jackiw and Pi are generalized using the non-relativistic Kaluza-Klein-type framework presented in our previous paper. All Schrödinger-symmetric backgrounds are determined. (10 pages. PlainTex)
The vortex solutions of various classical planar field theories with (Abelian) Chern-Simons term are reviewed. Relativistic vortices, put forward by Paul and Khare, arise when the Abelian Higgs model is augmented with the Chern-Simons term. Adding a suitable sixth-order potential and turning off the Maxwell term provides us with pure Chern-Simons theory with both topological and non-topological self-dual vortices, as found by Hong-Kim-Pac, and by Jackiw-Lee-Weinberg. The non-relativistic limit of the latter leads to non-topological Jackiw-Pi vortices with a pure fourth-order potential. Explici…
A non-relativistic version of the 2+1 dimensional gauged Chern-Simons O(3) sigma model, augmented by a Maxwell term, is presented and shown to support topologically stable static self-dual vortices. Exactly like their counterparts of the ungauged model, these vortices are shown to exhibit Hall behaviour in their dynamics.
We study BPS vortex configurations in three dimensional U(N) Yang-Mills theories with Chern-Simons interaction coupled to scalar fields carrying flavor. We consider two kind of configurations: local vortices (when the number of flavors $N_f=N$), and semi-local vortices (when $N_f>N$). In both cases we carefully analyze the electric and magnetic properties and present explicit numerical solutions.
Based on the gauge potential decomposition theory and the $ϕ$-mapping theory, the topological inner structure of the Chern-Simons-Higgs vortex has been showed in detail. The evolution of CSH vortices is studied from the topological properties of the Higgs scalar field. The vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the scalar field $ϕ.$
We study the structure and properties of vortices in a recently proposed Abelian Maxwell-Chern-Simons model in $2 +1 $ dimensions. The model which is described by gauge field interacting with a complex scalar field, includes two parity and time violating terms: the Chern-Simons and the anomalous magnetic terms. Self-dual relativistic vortices are discussed in detail. We also find one dimensional soliton solutions of the domain wall type. The vortices are correctly described by the domain wall solutions in the large flux limit.
We study vortex dynamics in three-dimensional theories with Chern-Simons interactions. The dynamics is governed by motion on the moduli space M in the presence of a magnetic field. For Abelian vortices, the magnetic field is shown to be the Ricci form over M; for non-Abelian vortices, it is the first Chern character of a suitable index bundle. We derive these results by integrating out massive fermions and following the fate of their zero modes.
In this paper we study fractional as well as semi-local Chern-Simons vortices in G = U(1) x SO(2M) and G = U(1) x USp(2M) theories. The master equations are solved numerically using appropriate Ansatze for the moduli matrix field. In the fractional case the vortices are solved in the transverse plane due to the broken axial symmetry of the configurations (i.e. they are non-rotational invariant). It is shown that unless the fractional vortex-centers are all coincident (i.e. local case) the ring-like flux structure, characteristic of Chern-Simons vortices, will become bell-like fluxes - just as …
We discuss a novel method of obtaining the fractional spin of abelian and nonabelian Chern-Simons vortices. This spin is interpreted as the difference between the angular momentum obtained by modifying Schwinger's energy momentum tensor by the Gauss constraint, and the canonical (Noether) angular momentum. It is found to be a boundary term depending only on the gauge field and, hence, is independent of the matter sector to which the Chern-Simons term couples. Addition of the Maxwell term does not alter the fractional spin.
We compute the spin of both the topological and nontopological solitons of the Chern - Simons - Higgs model by using our approach based on constrained analysis. We also propose an extension of our method to the non - relativistic Chern - Simons models. The spin formula for both the relativistic and nonrelativistic theories turn out to be structurally identical. This form invariance manifests the topological origin of the Chern - Simons term responsible for inducing fractional spin. Also, some comparisons with the existing results are done.
We study the locality properties of the vortex operators in compact U(1) Maxwell-Chern-Simons and SU(N) Yang-Mills-Chern-Simons theories in 2+1 dimensions. We find that these theories do admit local vortex operators and thus in the UV regularized versions should contain stable magnetic vortices. In the continuum limit however the energy of these vortex excitations generically is logarithmically UV divergent. Nevertheless the classical analysis shows that at small values of CS coefficient $κ$ the vortices become light. It is conceivable that they in fact become massless and condense due to quan…
We present a (2+1)-dimensional gauged $O(3) σ$-model with an Abelian Chern--Simons term. It shows topologically stable, anyonic vortices as classical solutions. The fields are studied in the case of rotational symmetry and analytic approximations are found for their asymptotic behaviour. The static Euler--Lagrange equations are solved numerically, where particular attention is paid to the dependence of the vortex' properties on the coupling to the gauge field. We compute the vortex mass and charge as a function of this coupling and obtain bound states for two--vortices as well as two--vortices…
We study non-Abelian Chern-Simon BPS-saturated vortices enjoying N=2 supersymmetry in d=2+1 dimensions, with generic gauge groups of the form U(1) x G', with G' being a simple group, allowing for orientational modes in the solutions. We will keep the group as general as possible and utilizing the powerful moduli matrix formalism to provide the moduli spaces of vortices and derive the corresponding master equations. Furthermore, we study numerically the vortices applying a radial Ansatz to solve the obtained master equations and we find especially a splitting of the magnetic fields, when the co…
We have studied BPS vortices in a CPT-odd and Lorentz-violating Maxwell-Chern-Simons-Higgs (MCSH) electrodynamics attained from the dimensional reduction of the Carroll-Field-Jackiw-Higgs model. The Lorentz-violating parameter induces a pronounced behavior at origin (for the magnetic/electric fields and energy density) which is absent in the MCSH vortices. For some combination of the Lorentz-violating coefficients there always exist a sufficiently large winding number $n_{0}$ such that for all $% |n|\geq |n_{0}|$ the magnetic field flips its signal, yielding two well defined regions with oppos…
We study BPS vortices in the mass-deformed non-relativistic ${\C N}=6$ $U(N)_k\times U(N)_{-k}$ Chern-Simons-matter theory. We focus on the massive deformation that preserves the maximal ${\C N}=6$ supersymmetry, and consider a non-relativistic limit that carry 14 supercharges. In this non-relativistic field theory we find Jackiw-Pi type exact vortex solutions combined with $S^3$ fuzzy sphere geometry. We analyse their properties and show that they preserve one dynamical, one conformal and five kinematical supersymmetries among the full super Schrödinger symmetry.
We consider the bosonic sector of a N=2 supersymmetric Chern-Simons-Higgs theory in 2+1 dimensions. The gauge group is U(1)xU(N) and has N_f flavors of fundamental matter fields. The model supports non-Abelian (axially symmetric) vortices when N_f\geq N, which have internal (orientational) moduli. When N_f > N, the solutions acquire additional collective coordinates parameterizing their transverse size. We solve the BPS equations numerically and obtain local (N_f = N) and semi-local (N_f > N) string solutions.
The purpose of this Letter is to present a computation of the interaction energy of gauged O(3) Chern-Simons vortices which are infinitely separated. The results will show the behaviour of the interaction energy as a function of the constant coupling the potential, which measures the relative strength of the matter self-coupling and the electromagnetic coupling. We find that vortices attract each other for $λ> 1 $ and repel when $λ< 1 $. When $λ=1 $ there is a topological lower bound on the energy. It is possible to saturate the bound if the fields satisfy a set of first order partial differen…
The two-dimensional self-dual Chern-Simons equations are equivalent to the conditions for static, zero-energy vortex-like solutions of the (2+1) dimensional gauged nonlinear Schrödinger equation with Chern-Simons matter-gauge coupling. The finite charge vacuum states in the Chern-Simons theory are shown to be gauge equivalent to the finite action solutions to the two-dimensional chiral model (or harmonic map) equations. The Uhlenbeck-Wood classification of such harmonic maps into the unitary groups thereby leads to a complete classification of the vacuum states of the Chern-Simons model. This …
Self-dual vortex solutions are studied in detail in the generalized abelian Higgs model with independent Chern-Simons interaction. For special choices of couplings, it reduces to a Maxwell-Higgs model with two scalar fields, a Chern-Simons-Higgs model with two scalar fields, or other new models. We investigate the properties of the static solutions and perform detailed numerical analyses. For the Chern-Simons-Higgs model with two scalar fields in an asymmetric phase, we prove the existence of multisoliton solutions which can be viewed as hybrids of Chern-Simons vortices and $CP^1$ lumps. We al…
The non-relativistic Maxwell-Chern-Simons model recently introduced by Manton is shown to admit self-dual vortex solutions with non-zero electric field. The interrelated ``geometric'' and ``hidden'' symmetries are explained. The theory is also extended to (non-relativistic) spinors. A relativistic, self-dual model, whose non-relativistic limit is the Manton model is also presented. The relation to previous work is discussed.
We find self-dual vortex solutions in a Maxwell-Chern-Simons model with anomalous magnetic moment. From a recently developed N=2-supersymmetric extension, we obtain the proper Bogomol'nyi equations together with a Higgs potential allowing both topological and non-topological phases in the theory.
We investigate the vortex-type BPS equations in the ABJM theory without and with mass-deformation. We systematically classify the BPS equations in terms of the number of supersymmetry and the R-symmetries of the undeformed and mass-deformed ABJM theories. For the undeformed case, we analyze the ${\cal N}=2$ BPS equations for U(2)$\times$U(2) gauge symmetry and obtain a coupled differential equation which can be reduced to either Liouville- or Sinh-Gordon-type vortex equations according to the choice of scalar functions. For the mass-deformed case with U($N$)$\times$U($N$) gauge symmetry, we ob…
We study the effect of a Chern-Simons term on the electrically charged and spinning solitons of several $U(1)$ gauged models in $2+1$ dimensions. These are vortices of complex scalar field theories, both with and without symmetry breaking dynamics, and the $O(3)$ Skyrme model. In all cases the gauge decoupling limits are also considered. It is well known that the effect of the Chern-Simons dynamics is to endow vortices with electric charge $Q_e$ and spin $J$, but our main aim here is to reveal a new feature: that the mass-energy $E$ of the electrically charged vortex can be lower than that of …
Making use of $ϕ$-mapping topological current method, we discuss the self-dual vortices in the Abelian Chern-Simons model with two complex scalar fields. For each scalar field, an exact nontrivial equation with a topological term which is missing in many references is derived analytically. The general angular momentum is obtained. The magnetic flux which relates the two scalar fields is calculated. Furthermore, we investigate the vortex evolution processes, and find that because of the present of the vortex molecule, these evolution processes is more complicated than the vortex evolution proce…
In (2+1) dimensions, the Maxwell term $-(1/4) F_{αβ}F^{αβ}$ can be replaced by the Chern-Simons three-form $(κ/4)ε^{αβγ}A_αF_{βγ}$, yielding a novel type of `electromagnetism'. This has been proposed for studying the Quantum Hall Effect as well as High-Temperature Superconductivity. The gauge field can be coupled to a scalar field either relativistically or non-relativistically. In both cases, one admits finite-energy, vortex solutions.
We investigate probe limit vortex solutions of a charged scalar field in Einstein-Maxwell theory in 3+1 dimensions, for an asymptotically AdS Schwarzschild black hole metric with the addition of an axionic coupling to the Maxwell field. We show that the inclusion of such a term, together with a suitable potential for the axion field, can induce an effective Chern-Simons term on the 2+1 dimensional boundary. We obtain numerical solutions of the equations of motion and find Maxwell-Chern-Simons like magnetic vortex configurations, where the magnetic field profile varies with the size of the effe…
We consider a large-N Chern-Simons theory for the attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian, collective-field approach based on the 1/N expansion. We show that the dynamics of density excitations around the ground-state semiclassical configuration is governed by the Calogero or by the Sutherland Hamiltonian, depending on the symmetry of the underlying static-soliton configuration. The relationship between the Chern-Simons coupling constant $ł$ and the Calogero-Sutherland statistical parameter $ł_c$ signalizes some sort of statistical transmutation accompanying the dimensio…
We study ${\cal N}=2$ Chern-Simons-matter theories with gauge group $U_{k_1}(1)\times U_{k_2}(1)$. We find that, when $k_1+k_2=0$, the partition function computed by localization dramatically simplifies and collapses to a single term. We show that the same condition prevents the theory from having supersymmetric vortex configurations. The theories include mass-deformed ABJM theory with $U(1)_{k}\times U_{-k}(1)$ gauge group as a particular case. Similar features are shared by a class of CS-matter theories with gauge group $U_{k_1}(1)\times \cdots \times U_{k_N}(1)$.
The self-duality equations of Chern-Simons Higgs theory in a background curved spacetime are studied by making use of the U(1) gauge potential decomposition theory and $ϕ$-mapping method. The special form of the gauge potential decomposition is obtained directly from the first of the self-duality equations. Using this decomposition, a rigorous proof of magnetic flux quantization in background curved spacetime is given and the unit magnetic flux in curved spacetime is also found . Furthermore, the precise self-dual vortex equation with topological term is obtained, in which the topological term…
We consider a generalization of the abelian Higgs model with a Chern-Simons term by modifying two terms of the usual Lagrangian. We multiply a dielectric function with the Maxwell kinetic energy term and incorporate nonminimal interaction by considering generalized covariant derivative. We show that for a particular choice of the dielectric function this model admits both topological as well as nontopological charged vortices satisfying Bogomol'nyi bound for which the magnetic flux, charge and angular momentum are not quantized. However the energy for the topolgical vortices is quantized and i…
Non-Abelian vortices for a supersymmetric {\cal N}=2 Chern-Simons-Higgs theory are explicitly constructed. We introduce N Higgs fields in the fundamental representation of the U(N) gauge group in order to have a color-flavor SU(N) group remaining unbroken in the asymmetric phase. Bogomol'nyi-like first order equations are found and rotationally symmetric solutions are proposed. These solutions are shown to be truly non-Abelian by parameterizing them in terms of orientational collective coordinates. The low energy effective action for the orientational moduli results to be the one-dimensional s…
We examine vortices in Abelian Chern-Simons theory coupled to a relativistic scalar field with a chemical potential for particle number or U(1) charge. The Gauss constraint requires chemical potential for the local symmetry to be accompanied by a constant background charge density/magnetic field. Focussing attention on power law scalar potentials $\sim|Φ|^{2s}$ which do not support vortex configurations in vacuum but do so at finite chemical potential, we numerically study classical vortex solutions for large winding number |n| >> 1. The solutions depending on a single dimensionless parameter …
We study a non-Abelian Chern-Simons gauge theory in $ 2+ 1$ dimensions with the inclusion of an anomalous magnetic interaction. For a particular relation between the Chern-Simons (CS) mass and the anomalous magnetic coupling the equations for the gauge fields reduce from second- to first order differential equations of the pure CS type. We derive the Bogomol'nyi-type or self-dual equations for a $\bphi^2$ scalar potential, when the scalar and topological masses are equal. The corresponding vortex solutions carry magnetic flux that is not quantized due to the non-toplogical nature of the solito…
Starting from gravity as a Chern-Simons action for the AdS algebra in five dimensions, it is possible to deform the theory through an expansion of the Lie algebra that leads to a system consisting of the Einstein-Hilbert action plus nonminimally coupled matter. The deformed system is gauge invariant under the Poincare group enlarged by an Abelian ideal. Although the resulting action naively looks like General Relativity plus corrections due to matter sources, it is shown that the nonminimal couplings produce a radical departure from GR. Indeed, the dynamics is not continuously connected to the…
We consider vortex dynamics in self-dual Chern-Simons Higgs systems. We show that the naive Aharanov-Bohm phase is the inverse of the statistical phase expected from the vortex spin, and that the self-dual configurations of vortices are degenerate in energy but not in angular momentum. We also use the path integral formalism to derive the dual formulation of Chern-Simons Higgs systems in which vortices appear as charged particles. We argue that besides the electromagnetic interaction, there is an additional interaction between vortices, the so-called Magnus force, and that these forces can be …
The order-disorder duality structure is exploited in order to obtain a quantum description of anyons and vortices in: a) the Maxwell theory; b) the Abelian Higgs Model; c) the Maxwell-Chern-Simons theory; d) the Maxwell-Chern-Simons-Higgs theory. A careful construction of a charge bearing order operator($σ$) and a magnetic flux bearing disorder operator (vortex operator) ($μ$) is performed, paying attention to the necessary requirements for locality. An anyon operator is obtained as the product $φ=σμ$. A detailed and comprehensive study of the euclidean correlation functions of $σ$, $μ$ and $φ…
We study the various quantum aspects of the $N=2$ supersymmetric Maxwell Chern-Simons vortex systems. The fermion zero modes around the vortices will give rise the degenerate states of vortices. We analyze the angular momentum of these zero modes and apply the result to get the supermultiplet structures of the vortex. The leading quantum correction to the mass of the vortex coming from the mode fluctuations is also calculated using various methods depending on the value of the coefficient of the Chern-Simons term $κ$ to be zero, infinite and finite, separately. The mass correction is shown to …
We apply the adiabatic approximation to investigate the low energy dynamics of vortices in the parity invariant double self-dual Higgs model with only mutual Chern-Simons interaction. When distances between solitons are large they are particles subject to the mutual interaction. The dual formulation of the model is derived to explain the sign of the statistical interaction. When vortices of different types pass one through another they behave like charged particles in magnetic field. They can form a bound state due to the mutual magnetic trapping. Vortices of the same type exhibit no statistic…
Existence of Maxwell-Chern-Simons-Higgs (MCSH) vortices in a Hermitian line bundle $Ł$ over a general compact Riemann surface $Σ$ is proved by a continuation method. The solutions are proved to be smooth both spatially and as functions of the Chern-Simons deformation parameter $κ$, and exist for all $|κ|<κ_*$, where $κ_*$ depends, in principle, on the geometry of $Σ$, the degree $n$ of $Ł$, which may be interpreted as the vortex number, and the vortex positions. A simple upper bound on $κ_*$, depending only on $n$ and the volume of $Σ$, is found. Further, it is proved that a positive {\em lowe…
Non Abelian vortices of a SU(2) Chern-Simons--Higgs theory in 2+1 dimensions are constructed numerically. They represent natural counterparts of the U(1) solutions considered by Hong, Kim and Pac, and, by Jackiw and Weinberg. The Abelian embeddings are identified, for all values of the Higgs selfinteraction strength $ν$, resulting in both attractive and repulsive phases. A detailed analysis of the properties of the solutions reveals the existence ofa number of unexpected features. For a certain range of the parameter $ν$, it is shown that the non Abelian vortices have lower energy than their t…
We consider a generalization of the abelian Higgs model with a Chern-Simons term by modifying two terms of the usual Lagrangian. We multiply a dielectric function with the Maxwell kinetic energy term and incorporate nonminimal interaction by considering generalized covariant derivative. We show that for a particular choice of the dielectric function this model admits topological vortices satisfying Bogomol'nyi bound for which the magnetic flux is not quantized even though the energy is quantized. Furthermore, the vortex solution in each topological sector is infinitely degenerate.
We have analyzed Maxwell-Chern-Simons-Higgs BPS vortices in a Lorentz-violating CPT-odd context. The Lorentz violation induces profiles with a conical behavior at the origin. For some combination of the coefficients for Lorentz violation there always exists a sufficiently large winding number for which the magnetic field flips its sign.
I derive a general formalism for finding kinetic terms of the effective Lagrangian for slowly moving Chern-Simons vortices. Deformations of fields linear in velocities are taken into account. From the equations they must satisfy I extract the kinetic term in the limit of coincident vortices. For vortices passing one over the other there is locally the right-angle scattering. The method is based on analysis of field equations instead of action functional so it may be useful also for nonvariational equations in nonrelativistic models of Condensed Matter Physics.
We compare the vortex-like solutions of two different theories in (2+1) dimensions. In the first a nonrelativistic field self-interacts through a Chern-Simons gauge connection. It is $P$ and $T$ violating. The second is the standard Maxwell scalar electrodynamics. We show that for specific values of some parameters the same vortex-configurations provide solutions for both theories.
In this work we deal with vortices in Maxwell-Higgs or Chern-Simons-Higgs models that engender long range tails. We find first order differential equations that support minimum energy solutions which solve the equations of motion. In the Maxwell scenario, we work with generalised magnetic permeabilities that lead to vortices described by solutions, magnetic field and energy density with power-law tails that extend farther than the standard exponential ones. We also find a manner to obtain a Chern-Simons model with the same scalar and magnetic field profiles of the Maxwell case. By doing so, we…
In this article we consider $SU(2)$ Chern-Simons/Higgs theory coupled to gravity in three-dimensions. It is shown that for a cylindrically symmetric vortex both the Einstein equations and the field equations can be reduced to a set of first-order Bogomol'nyi equations provided that we choose a specific eighth-order potential.
We consider a generalization of abelian Chern-Simons-Higgs model by introducing a nonstandard kinetic term. In particular we show that the Bogomolnyi equations of the abelian Higgs theory may be obtained, being its solutions Nielsen-Olesen vortices with electric charge. In addition we study the self-duality equations for a generalized non-relativistic Maxwell-Chern-Simons model.
We study various three-dimensional supersymmetric Maxwell Chern-Simons solitons by using type IIB brane configurations. We give a systematic classification of soliton spectra such as topological BPS vortices and nontopological vortices in $\mathcal{N}=2,3$ supersymmetric Maxwell Chern-Simons system via the branes of type IIB string theory. We identify the brane configurations with the soliton spectra of the field theory and obtain a nice agreement with field theory aspects. We also discuss possible brane constructions for BPS domain wall solutions.
This work introduces a procedure to obtain vortex configurations described by first order equations in generalized Maxwell-Chern-Simons models without the inclusion of a neutral field. The results show that the novel methodology is capable of inducing important modification in the vortex core, leading to vortex configurations with unconventional features.
We have studied the existence of topological self-dual configurations in a nonminimal CPT-odd and Lorentz-violating (LV) Maxwell-Higgs model, where the LV interaction is introduced by modifying the minimal covariant derivative. The Bogomol'nyi-Prasad-Sommerfield formalism has been implemented, revealing that the scalar self-interaction implying self-dual equations contains a derivative coupling. The CPT-odd self-dual equations describe electrically neutral configurations with finite total energy proportional to the total magnetic flux, which differ from the charged solutions of other CPT-odd a…
