Nielsen-Olesen Vortex Research Articles

Multi-messenger constraints on Abelian-Higgs cosmic string networks

Nielsen-Olesen vortices in the Abelian-Higgs (AH) model are the simplest realisations of cosmic strings in a gauge field theory. Large-scale numerical solutions show that the dominant decay channel of a network of AH strings produced from random initial conditions is classical field radiation. However, they also show that with special initial conditions, loops of string can be created for which classical field radiation is suppressed, and which behave like Nambu-Goto (NG) strings with a dominant decay channel into gravitational radiation. This indicates that cosmic strings are generically sources of both high-energy particles and gravitational waves. Here we adopt a simple parametrisation of the AH string network allowing for both particle and gravitational wave production. With a reference to a specific model for NG-like loop distribution, this sets the basis for a “multi-messenger” investigation of this model. We find that, in order to explain the NANOGrav detection of a possible gravitational wave background, while satisfying the constraint on NG-like loop production from simulations and bounds from the cosmic microwave background, the tension of the AH string in Planck units Gμ and the fraction of the NG-like loops f NG should satisfy Gμ f NG 2.6 ≳ 3.2 × 10-13 at 95% confidence. On the other hand, for such string tensions, constraints from the diffuse gamma-ray background (DGRB) indicate that more than 97% of the total network energy should be converted to dark matter (DM) or dark radiation. We also consider joint constraints on the annihilation cross-section, the mass, and the relic abundance of DM produced by decays of strings. For example, for a DM mass of 500 GeV, the observed relic abundance can be explained by decaying AH strings that also account for the NANOGrav signal.

Erasure of strings and vortices

The interaction of defects can lead to a phenomenon of erasure. During this process, a lower-dimensional object gets absorbed and dissolved by a higher-dimensional one. The phenomenon is very general and has a wide range of implications, both cosmological and fundamental. In particular, all types of strings, such as cosmic strings, QCD flux tubes, or fundamental strings, get erased when encountering a defect, either solitonic or a D-brane that deconfines their fluxes. This leads to a novel mechanism of cosmic string breakup, accompanied by gravitational and electromagnetic radiations. The arguments based on loss of coherence and the entropy count suggest that the erasure probability is very close to one, and strings never make it through the deconfining layer. We confirm this by a numerical simulation of the system, which effectively captures the essence of the phenomenon: a $2+1$-dimensional problem of interaction between a Nielsen-Olesen vortex of a $U(1)$ Higgs model and a domain wall, inside which the $U(1)$ gauge group is un-Higgsed and the magnetic flux is deconfined. In accordance with the entropy argument, in our simulation, the vortex never makes it across the wall.

Nonminimally coupled gravitating vortex: Phase transition at critical couplingξcinAdS3

We consider the Nielsen-Olesen vortex non-minimally coupled to Einstein gravity with cosmological constant $\Lambda$. A non-minimal coupling term $\xi\,R\,|\phi|^2$ is natural to add to the vortex as it preserves gauge-invariance (here $R$ is the Ricci scalar and $\xi$ a dimensionless coupling constant). This term plays a dual role: it contributes to the potential of the scalar field and to the Einstein-Hilbert term for gravity. As a consequence, the vacuum expectation value (VEV) of the scalar field and the cosmological constant in the AdS$_3$ background depend on $\xi$. This leads to a novel feature: there is a critical coupling $\xi_c$ where the VEV is zero for $\xi\ge \xi_c$ but becomes non-zero when $\xi$ crosses below $\xi_c$ and the gauge symmetry is spontaneously broken. Moreover, we show that the VEV near the critical coupling has a power law behaviour proportional to $|\xi-\xi_c|^{1/2}$. Therefore $\xi_c$ can be viewed as the analog of the critical temperature $T_c$ in Ginzburg-Landau (GL) mean-field theory where a second-order phase transition occurs below $T_c$ and the order parameter has a similar power law behaviour $|T-T_c|^{1/2}$ near $T_c$. The critical coupling exists only in an AdS$_3$ background; it does not exist in asymptotically flat spacetime (topologically a cone) where the VEV remains at a fixed non-zero value independent of $\xi$. However, the deficit angle of the asymptotic conical spacetime depends on $\xi$ and is no longer determined solely by the mass; remarkably, a higher mass does not necessarily yield a higher deficit angle. The equations of motion are more complicated with the non-minimal coupling term present. However, via a convenient substitution one can reduce the number of equations and solve them numerically to obtain exact vortex solutions.

Quantum corrections to soliton energies

We review recent progress in the computation of leading quantum corrections to the energies of classical solitons with topological structure, including multi-soliton models in one space dimension and string configurations in three space dimensions. Taking advantage of analytic continuation techniques to efficiently organize the calculations, we show how quantum corrections affect the stability of solitons in the Shifman–Voloshin model, stabilize charged electroweak strings coupled to a heavy fermion doublet, and bind Nielsen–Olesen vortices at the classical transition between type I and type II superconductors.

Generalized Maxwell–Higgs vortices in models with enhanced symmetry

Topological vortices in relativistic gauge theories in flat three-dimensional spacetime are investigated. We consider the symmetry mathrm {U(1)}times cdots times mathrm {U(1)}, and for each mathrm {U(1)} subgroup, a complex scalar field transforming under its action is introduced, as well as generalized permeabilities through which the subsystems are coupled. We investigate in detail the features of static, finite energy solutions within this class of generalized Maxwell–Higgs models, and study the effect of the winding numbers in the magnetic properties of each subsystem. A BPS bound and the related first order equations are introduced for a large class of models. Finally, we present some specific models and solve their equations of motion to find solutions engendering many distinct features in relation to each other and to the standard Nielsen–Olesen vortex.

Induced Vacuum Current and Magnetic Flux in Quantum Scalar Matter in the Background of a Vortex Defect with the Neumann Boundary Condition

A topological defect in the form of the Abrikosov–Nielsen–Olesen vortex in the space of an arbitrary dimension is considered as a gauge-flux-carrying tube that is impenetrable for quantum matter. The charged scalar matter field is quantized in the vortex background with the perfectly rigid (Neumann) boundary condition imposed at the side surface of the vortex. We show that a current circulating around the vortex is induced in the vacuum, if the Compton wavelength of the matter field exceeds the transverse size of the vortex considerably. The vacuum current is periodic in the value of the gauge flux of the vortex, providing a quantum-field-theoretical manifestation of the Aharonov–Bohm effect. The vacuum current leads to the appearance of an induced vacuum magnetic flux that, for some values of the tube thickness, exceeds the vacuum magnetic flux induced by a singular vortex filament. The results are compared to those obtained earlier in the case of the perfectly reflecting (Dirichlet) boundary condition imposed at the side surface of the vortex. It is shown that the absolute value of the induced vacuum current and the induced vacuum magnetic flux in the case of the Neumann boundary condition is greater than in the case of the Dirichlet boundary condition.

Black hole in Nielsen-Olesen vortex

In this article, we calculate the classical vortex solution of a spontaneously broken gauge theory interacting with gravity in (2+1)-dimension. We also calculate the conditions for the formation of a (2+1)-dimensional black hole due to magnetic vortex (a Nielsen-Olesen vortex). The semiclassical Hawking temperature for this black hole is calculated, where we see that the temperature of a BTZ black hole increases or decreases without changing the size of the horizon if we insert the magnetic vortex fields in the black hole. Finally, the first law of black hole thermodynamics is described for this particular solution, which shows that the additional work terms from the scalar and gauge fields compensate the change in the temperature relative to its usual value for the BTZ solution.

Polarization of the Vacuum of the Quantized Spinor Field by a Topological Defect in the Two-Dimensional Space

The two-dimensional space with a topological defect is a transverse section of the three-dimensional space with an Abrikosov–Nielsen–Olesen vortex, i.e. a gauge-flux-carrying tube which is impenetrable for quantum matter. Charged spinor matter field is quantized in this section with the most general mathematically admissible boundary condition at the edge of the defect. We show that a current and a magnetic field are induced in the vacuum. The dependence of results on the boundary conditions is studied, and we find that the requirement of finiteness of the total induced vacuum magnetic flux removes an ambiguity in the choice of boundary conditions. The differences between the cases of massive and massless spinor matters are discussed.

K -strings with exact Casimir law and Abelian-like profiles

We explore vortex solutions for a class of dual $SU(N)$ Yang-Mills models with $N^2-1$ Higgs fields in the adjoint representation. Initially, we show that there is a collective behavior that can be expressed in terms of a small $N$-independent number of field profiles. Then, we find a region in parameter space where the nontrivial profiles coincide with those of the Nielsen-Olesen vortex, and the energy scales exactly with the quadratic Casimir. Out of this region, we solve the ansatz equations numerically and find very small deviations from the Casimir law. The coexistence of Abelian-like string profiles and non-Abelian scaling features is welcome, as these properties have been approximately observed in pure YM lattice simulations.

Meissner effect from Landau problem

The Landau problem for inhomogeneous magnetic fields is examined in a very general context and several interesting analogies with the Nielsen–Olesen vortices are established. Firstly, we show that the Landau problem with non-homogeneous magnetic fields exhibits Meissner effect that is unstable unless two-body interactions are added and vortices emerge. Using the scaling freedom, we can write the Schrödinger equation in terms of the scales ratio [Formula: see text] where the last identification is realized simply by using the Ginzburg–Landau theory. We find our equations are valid in the superconducting regime, and it is not possible for the Cooper pairs amplitude to reach to a constant, nonzero value, and therefore the theory is unstable. The supersymmetric quantum mechanics version, by completeness, is also considered.

Vortices with scalar condensates in two-component Ginzburg–Landau systems

In a class of two-component Ginzburg-Landau models (TCGL) with a U(1)$\times$U(1) symmetric potential, vortices with a condensate at their core may have significantly lower energies than the Abrikosov-Nielsen-Olesen (ANO) ones. On the example of liquid metallic hydrogen (LMH) above the critical temperature for protons we show that the ANO vortices become unstable against core-condensation, while condensate-core (CC) vortices are stable. For LMH the ratio of the masses of the two types of condensates, $M=m_2/m_1$ is large, and then as a consequence the energy per flux quantum of the vortices, $E_n/n$ becomes a non-monotonous function of the number of flux quanta, $n$. This leads to yet another manifestation of neither type 1 nor type 2, (type 1.5) superconductivity: superconducting and normal domains coexist while various "giant" vortices form. We note that LMH provides a particularly clean example of type 1.5 state as the interband coupling between electronic and protonic Cooper-pairs is forbidden.

Note on generalization of Jackiw-Pi vortices

We analyze two Abelian Higgs systems with nonstandard kinetic terms. First, we consider a model involving the Maxwell term. For a particular choice of the nonstandard kinetics, we are able to obtain generalized Jackiw-Pi vortices. We, also, analyze a second model which is a generalization of the Jackiw-Pi model. In this case we show that the system support the Nielsen-Olesen vortices as solutions.

Induced vacuum current and magnetic field in the background of a vortex

A topological defect in the form of the Abrikosov–Nielsen–Olesen vortex is considered as a gauge-flux-carrying tube that is impenetrable for quantum matter. Charged scalar matter field is quantized in the vortex background with the perfectly reflecting (Dirichlet) boundary condition imposed at the side surface of the vortex. We show that a current circulating around the vortex and a magnetic field directed along the vortex are induced in the vacuum, if the Compton wavelength of the matter field exceeds considerably the transverse size of the vortex. The vacuum current and magnetic field are periodic in the value of the gauge flux of the vortex, providing a quantum-field-theoretical manifestation of the Aharonov–Bohm effect. The total flux of the induced vacuum magnetic field attains notable finite values even for the Compton wavelength of the matter field exceeding the transverse size of the vortex by just three orders of magnitude.

The gravity of dark vortices: effective field theory for branes and strings carrying localized flux

A Nielsen-Olesen vortex usually sits in an environment that expels the flux that is confined to the vortex, so flux is not present both inside and outside. We construct vortices for which this is not true, where the flux carried by the vortex also permeates the `bulk' far from the vortex. The idea is to mix the vortex's internal gauge flux with an external flux using off-diagonal kinetic mixing. Such `dark' vortices could play a phenomenological role in models with both cosmic strings and a dark gauge sector. When coupled to gravity they also provide explicit ultra-violet completions for codimension-two brane-localized flux, which arises in extra-dimensional models when the same flux that stabilizes extra-dimensional size is also localized on space-filling branes situated around the extra dimensions. We derive simple formulae for observables such as defect angle, tension, localized flux and on-vortex curvature when coupled to gravity, and show how all of these are insensitive to much of the microscopic details of the solutions, and are instead largely dictated by low-energy quantities. We derive the required effective description in terms of a world-sheet brane action, and derive the matching conditions for its couplings. We consider the case where the dimensions transverse to the bulk compactify, and determine how the on- and off-vortex curvatures and other bulk features depend on the vortex properties. We find that the brane-localized flux does not gravitate, but just renormalizes the tension in a magnetic-field independent way. The existence of an explicit UV completion puts the effective description of these models on a more precise footing, verifying that brane-localized flux can be consistent with sensible UV physics and resolving some apparent paradoxes that can arise with a naive (but commonly used) delta-function treatment of the brane's localization within the bulk.

The Casimir Effect at Finite Temperature in a Six-Dimensional Vortex Scenario

The Casimir effect for parallel plates satisfying the Dirichlet boundary condition in the context of effective QED coming from a six-dimensional Nielsen-Olesen vortex solution of the Abelian Higgs model with fermions coupled to gravity is studied at finite temperature. We find that the sign of the Casimir energy remains negative under the thermal influence. It is also shown that the Casimir force between plates will be weaker in the higher-temperature surroundings while keeps attractive. This Casimir effect involving the thermal influence is still inconsistent with the known experiments. We find that the thermal correction can not compensate or even reduce the modification from this kind of vortex model to make the Casimir force to be in less conflict with the measurements.

Topological defects in the abelian Higgs model

We give a rigorous description of the dynamics of the Nielsen-Olesen vortex line. In particular, given a worldsheet of a string, we construct initial data such that the corresponding solution of the abelian Higgs model will concentrate near the evolution of the string. Moreover, the constructed solution stays close to the Nielsen-Olesen vortex solution.

Note on vortices from Lorentz-violating models

We consider two self-dual abelian Higgs systems obtained from Lorentz breaking symmetry models by dimensional reduction. For the first model, we show that the self-dual equations are identical to those of Nielsen-Olesen vortices. Also, we show that our vortices have electric charge. In the second case we show that self-dual Chern-Simons-Higgs vortices without electric charge are possible.

Stabilizing the semilocal string with a dilatonic coupling

We demonstrate that the stability of the semilocal vortex can be significantly improved by the presence of a dilatonic coupling of the form e^\frac{q | \Phi |^2}{\eta^2} F_{\mu \nu}F^{\mu \nu} with q>0 where \eta is the scale of symmetry breaking that gives rise to the vortex. For q=0 we obtain the usual embedded (semilocal) Nielsen-Olesen vortex. We find the stability region of the parameter \beta = (\frac{m_\Phi}{m_A})^2 (m_\Phi and m_A are the masses of the scalar and gauge fields respectively). We show that the stability region of \beta is 0<\beta<\beta_{max}(q) where \beta_{max}(q=0)=1 (as expected) and \beta_{max}(q) is an increasing function of q. This result may have significant implications for the stability of the electroweak vortex in the presence of a dilatonic coupling (dilatonic electroweak vortex).

Self-dual Maxwell–Chern–Simons solitons from a Lorentz-violating model

Self-dual abelian Higgs system, involving both the Maxwell and Chern–Simons terms are obtained from Carroll–Field–Jackiw theory by dimensional reduction. Bogomolʼnyi-type equations are studied from theoretical and numerical point of view. In particular, we show that the solutions of these equations are Nielsen–Olesen vortices with electric charge.

Lagrangian alternative to QCD string

The spectrum of radially excited hadrons provides much information about the confinement forces in QCD. The confinement is realized most naturally in terms of the QCD string whose quantization gives rise to the radially excited modes. We propose an alternative framework for the description of the excited spectrum. Namely we put forward some effective field models in which the hadrons acquire masses due to interaction with a scalar field modeling the non-perturbative gluon vacuum. The effective potential for this field is periodic with an infinite number of non-equivalent vacua. The radially excited hadrons emerge as elementary excitations over different vacua. We construct explicit examples for such effective theories in the meson sector. An interesting byproduct of the considered models is the existence of a classical field configuration in each vacuum. Depending on the model, it can represent a domain wall, Nielsen-Olesen vortex or 't Hooft-Polyakov monopole. In the pure scalar sector, it is shown that the first quantum correction leads to splitting of the radially excited modes into two nearly degenerate states. Such a phenomenon has some phenomenological support. The presented approach may be also viewed as an alternative framework to the bottom-up AdS/QCD models where the radial excitations appear on the classical level as well.