Hedgehog Ansatz Research Articles

Ordered Patterns of (3+1)-Dimensional Hadronic Gauged Solitons in the Low-Energy Limit of Quantum Chromodynamics at a Finite Baryon Density, Their Magnetic Fields and Novel BPS Bounds

In this paper, we will review two analytical approaches to the construction of non-homogeneous Baryonic condensates in the low-energy limit of QCD in (3+1) dimensions. In both cases, the minimal coupling with the Maxwell U(1) gauge field can be taken explicitly into account. The first approach (which is related to the generalization of the usual spherical hedgehog ansatz to situations without spherical symmetry at a finite Baryon density) allows for the construction of ordered arrays of Baryonic tubes and layers. When the minimal coupling of the Pions to the U(1) Maxwell gauge field is taken into account, one can show that the electromagnetic field generated by these inhomogeneous Baryonic condensates is of a force-free type (in which the electric and magnetic components have the same size). Thus, it is natural to wonder whether it is also possible to analytically describe magnetized hadronic condensates (namely, Hadronic distributions generating only a magnetic field). The idea of the second approach is to construct a novel BPS bound in the low-energy limit of QCD using the theory of the Hamilton–Jacobi equation. Such an approach allows us to derive a new topological bound which (unlike the usual one in the Skyrme model in terms of the Baryonic charge) can actually be saturated. The nicest example of this phenomenon is a BPS magnetized Baryonic layer. However, the topological charge appearing naturally in the BPS bound is a non-linear function of the Baryonic charge. Such an approach allows us to derive important physical quantities (which would be very difficult to compute with other methods), such as how much one should increase the magnetic flux in order to increase the Baryonic charge by one unit. The novel results of this work include an analysis of the extension of the Hamilton–Jacobi approach to the case in which Skyrme coupling is not negligible. We also discuss some relevant properties of the Dirac operator for quarks coupled to magnetized BPS layers.

Skyrmions at finite density

In this paper, we will describe recent advances in analytical methods to construct exact solutions of the Skyrme model (and its generalizations) representing inhomogeneous Hadronic condensates living at finite Baryon density. Such novel analytical tools are based on the idea to generalize the well-known spherical hedgehog ansatz to situations (relevant for the analysis of finite density effects) in which there is no spherical symmetry anymore. Besides the intrinsic mathematical interest to find exact solutions with nonvanishing Baryonic charge confined to a finite volume, this framework opens the possibility to compute important physical quantities which would be difficult to compute otherwise.

BPS skyrmions of generalized Skyrme model in higher dimensions

In this work we consider the higher dimensional Skyrme model, with spatial dimension d > 3, focusing on its BPS submodels and their corresponding features. To accommodate the cases with a higher topological degree, B ≥ 1, a modified generalized hedgehog ansatz is used where we assign an integer ni for each rotational plane, resulting in a topological degree that proportional to product of these integers. It is found via BPS Lagrangian method that there are only two possible BPS submodels for this spherically symmetric ansatz which shall be called as BPS Skyrme model and scale-invariant model. The properties of the higher dimensional version of both submodels are studied and it is found that the BPS Skyrmions with B ≥ 1 exist in the first submodel but there is only B = 1 BPS Skyrmion in the second submodel. We also study the higher dimensional version of self-duality conditions in terms of strain tensor eigenvalues and find that, in general, the scale-invariant model has a stronger self-duality condition than the BPS Skyrme model.

Analytic meronic black holes, gravitating solitons, and higher-spins in the EinsteinSU(N)-Yang-Mills theory

We construct meronic black holes and solitons in the Einstein $SU(N)$-Yang-Mills theory in $D=4$ and $D=5$ dimensions. These analytical solutions are found by combining the generalized hedgehog ansatz with the Euler parameterization of the $SU(N)$ group from which the Yang-Mills equations are automatically satisfied for all values of $N$ while the Einstein equations can be solved analytically. We explicitly show the role that the color number $N$ plays in the black hole thermodynamics as well as in the gravitational spin from isospin effect. Two remarkable results of our analysis are that, firstly, meronic black holes can be distinguished by colored black holes by looking at the spin from isospin effect (which is absent in the latter but present in the former). Secondly, using the theory of non-embedded ansatz for $SU(N)$ together with the spin from isospin effect, one can build fields of arbitrary high spin out of scalar fields charged under the gauge group. Hence, one can analyze interacting higher spin fields in asymptotically flat space-times without "introducing by hand" higher spin fields. Our analysis also discloses an interesting difference between the spin from isospin effect in $D=4$ and in $D=5$.

Analytic SU(N) Skyrmions at finite baryon density

We construct analytic (3+1)-dimensional Skyrmions living at finite Baryon density in the SU(N) Skyrme model that are not trivial embeddings of SU(2) into SU(N). We used Euler angles decomposition for arbitrary N and the generalized hedgehog Ansatz at finite Baryon density. The Skyrmions of high topological charge that we find represent smooth Baryonic layers whose properties can be computed explicitly. In particular, we determine the energy to Baryon charge ratio for any N showing the smoothness of the large N limit. The closeness to the BPS bound of these configurations can also be analyzed. The energy density profiles of these finite density Skyrmions have \textit{lasagna-like} shape in agreement with recent experimental findings. The shear modulus can be precisely estimated as well and our analytical result is close to recent numerical studies in the literature.

Analytic self-gravitating 4-Baryons, traversable NUT-AdS wormholes, flat space-time multi-Skyrmions at finite volume and a novel transition in the SU(3)-Skyrme model

We construct the first analytic self-gravitating Skyrmions with higher Baryon charge in four dimensions for the SU(3)-Skyrme–Einstein-Lambda theory by combining the generalized hedgehog ansatz with the approach developed by Balachandran et al. to describe the first (numerical) example of a non-embedded solution. These are genuine SU(3) analytic solutions instead of trivial embeddings of SU(2) into SU(3) and its geometry is that of a Bianchi IX Universe. The Skyrme ansatz is chosen in such a way that the Skyrme field equations are identically satisfied in the sector with Baryon charge 4. The field equations reduce to a dynamical system for the three Bianchi IX scale factors. Particular solutions are explicitly analyzed. Traversable wormholes with NUT-AdS asymptotics supported by a topologically non-trivial SU(3)-sigma soliton are also constructed. The self-gravitating solutions admit also a suitable flat limit giving rise to Skyrmions of charge 4 confined in a box of finite volume maintaining the integrability of the SU(3) Skyrme field equations. This formalism discloses a novel transition at finite Baryon density arising from the competition between embedded and non-embedded solutions in which the non-embedded solutions prevail at high density while are suppressed at low densities.

Fullerenes as solitons in chiral model of graphene

Taking into account the sp2- hybridization effect for valence electrons in carbon atoms, we introduce a unitary matrix U as an order parameter and suggest a scalar chiral model of graphene for the description of graphene-like configurations. Using the well-known hedgehog ansatz for modeling Fullerene C60, we estimate the polarizability of a single Fullerene specie in a uniform external electric field. Finally, we use this result for calculating the parameters of the van der Waals potential for the two interacting Fullerene species.

No skyrmion hair for stationary spherically symmetric reflecting stars

We investigate the existence of the skyrmion field in the background of an asymptotically flat stationary reflecting star. For this purpose, we consider the Einstein-Skyrme system for which there is a skyrmion hair in the black hole case. We discuss spherically symmetric skyrmions and employ the hedgehog ansatz for the skyrmion field. We show that, in contrast to the black hole, there is no skyrmion hair for a reflecting star.

Extensions of the generalized hedgehog ansatz for the Einstein-nonlinear \u03c3-model system: black holes with NUT, black strings and time-dependent solutions

We consider a class of ansätze for the construction of exact solutions of the Einstein-nonlinear σ-model system with an arbitrary cosmological constant in (3+1) dimensions. Exploiting a geometric interplay between the SU(2) field and Killing vectors of the spacetime reduces the matter field equations to a single scalar equation (identically satisfied in some cases) and simultaneously simplifies Einstein’s equations. This is then exemplified over various classes of spacetimes, which allows us to construct stationary black holes with a NUT parameter and uniform black strings, as well as time-dependent solutions such as Robinson-Trautman and Kundt spacetimes, Vaidya-type radiating black holes and certain Bianchi IX cosmologies. In addition to new solutions, some previously known ones are rederived in a more systematic way.

Extended thermodynamics of self-gravitating skyrmions

A modification of the hedgehog ansatz has recently lead to novel exact black hole solutions with selfgravitating SU(2) Skyrme fields. Considering a negative cosmological constant the black holes are not asymptotically anti-de Sitter (AdS) but rather asymptote to an AdS version of Barriola–Vilenkin spacetime. We examine the thermodynamics of the system interpreting the cosmological constant as a bulk pressure. We use the standard counterterm method to obtain a finite Euclidean action. For a given coupling of the matter action the system behaves as does a charged AdS black hole in the fixed charged ensemble. We find that in the limit case when the Skyrme model becomes a nonlinear sigma model the system exhibits a first order phase transition of Hawking–Page type. The universality class of these Einstein–Skyrme black holes is that of van der Waals.

Bianchi IX cosmologies in the Einstein-Skyrme system in a sector with nontrivial topological charge

The dynamics of the most general Bianchi IX cosmology with three time dependent scale factors for the Einstein-Skyrme system is analyzed. For the Skyrmion, a generalized hedgehog ansatz with unit baryon charge is introduced. The most remarkable feature of this ansatz is that, in the above topologically non-trivial sector with unit topological charge, the Skyrme field equations are identically satisfied on any Bianchi IX metric. We will show that due to this feature the complete set of coupled Einstein-Skyrme field equations can be deduced from a suitable minisuperspace Lagrangian. The latter allows to perform a systematic study of the integrability properties of the Einstein-Skyrme system for the Bianchi IX cosmology. Moreover, some analytic and algebraic solutions for the Einstein-Skyrme model are derived. Another remarkable consequence of the present formalism is that it is possible to derive the Wheeler de-Witt equation for the Bianchi IX metric in the Einstein-Skyrme cosmology in which all the effects of the Skyrmion are encoded in an effective potential of the minisuperspace Lagrangian.

Solitons in a cavity for the Einstein-SU(2) Non-linear Sigma Model and Skyrme model

In this work, taking advantage of the Generalized Hedgehog Ansatz, we construct new self-gravitating solitons in a cavity with mirror-like boundary conditions for the SU(2) Non-linear Sigma Model and Skyrme model. For spherically symmetric spacetimes, we are able to reduce the system to three independent equations that are numerically integrated. There are two branches of well-behaved solutions. The first branch is defined for arbitrary values of the Skyrme coupling and therefore also leads to a gravitating soliton in the Non-linear Sigma Model, while the second branch exists only for non-vanishing Skyrme coupling. The solutions are static and in the first branch are characterized by two integration constants that correspond to the frequency of the phase of the Skyrme field and the value of the Skyrme profile at the origin, while in the second branch the latter is the unique parameter characterizing the solutions. These parameters determine the size of the cavity, the redshift at the boundary of the cavity, the energy of the scalar field and the charge associated to a U(1) global symmetry. We also show that within this ansatz, assuming analyticity of the matter fields, there are no spherically symmetric black hole solutions.

Yang–Mills instantons in Kähler spaces with one holomorphic isometry

We consider self-dual Yang–Mills instantons in 4-dimensional Kähler spaces with one holomorphic isometry and show that they satisfy a generalization of the Bogomol'nyi equation for magnetic monopoles on certain 3-dimensional metrics. We then search for solutions of this equation in 3-dimensional metrics foliated by 2-dimensional spheres, hyperboloids or planes in the case in which the gauge group coincides with the isometry group of the metric (SO(3), SO(1,2) and ISO(2), respectively). Using a generalized hedgehog ansatz the Bogomol'nyi equations reduce to a simple differential equation in the radial variable which admits a universal solution and, in some cases, a particular one, from which one finally recovers instanton solutions in the original Kähler space. We work out completely a few explicit examples for some Kähler spaces of interest.

Magnetic excitations of carbon nanotubes in chiral model of graphene

Within the scope of the chiral model of graphene in 8-spinor realization, suggested earlier [1, 2], the combined spinor field Ψ is used as a new order parameter, which contains information on spin and quasispin excitations in graphene (8-spinor part) and also that on sp-hybridization effect of carbon valence electrons [SU(2)-matrix part]. This model admits two kinds of configurations describing, in the long-wave approximation, the graphene mono-atomic plane and the carbon nanotube respectively. The first one is given by the domain-wall (or kink-like) solution and the second one is described by the hedgehog ansatz with large topological charge (winding number n). We consider the infinite carbon nanotube in a static external magnetic field oriented along the tube and analyze the structure of the magnetic intensity on the outside. The diamagnetic behavior of the material is shown by the perturbation method which proves to be self-consistent for n ≫ 1.

Entropy in Spacetime and Topological Hair

Global topological soliton of the hedgehog ansatz is added to de Sitter spacetime in arbitrary dimensions larger than three, and then thermodynamic law is checked at the cosmological horizon. All geometric and thermodynamic quantities are varied in the presence of a long-range interacting matter distribution with negative pressure, however the entropy-area relation is satisfied in the exact form. Its geometry involves deficit solid angle but maintains a single horizon which allows unique temperature normalization, different from the case of Schwarzschild-de Sitter spacetime.

SU(N) multi-Skyrmions at finite volume

We study multi-soliton solutions of the four-dimensional SU(N) Skyrme model by combining the hedgehog ansatz for SU(N) based on the harmonic maps of $S^{2}$ into $CP^{N-1}$ and a geometrical trick which allows to analyze explicitly finite-volume effects without breaking the relevant symmetries of the ansatz. The geometric setup allows to introduce a parameter which is related to the 't Hooft coupling of a suitable large $N$ limit, in which $N\rightarrow\infty$ and the curvature of the background metric approaches zero, in such a way that their product is constant. The relevance of such a parameter to the physics of the system is pointed out. In particular, we discuss how the discrete symmetries of the configurations depend on it.

Kantowski-Sachs universes sourced by a Skyrme fluid

The Kantowski-Sachs cosmological model sourced by a Skyrme field and a cosmological constant is considered in the framework of General Relativity. Assuming a constant radial profile function for the hedgehog ansatz, the Skyrme contribution to Einstein equations is shown to be equivalent to an anisotropic fluid. Using dynamical system techniques, a qualitative analysis of the cosmological equations is presented. Physically interesting features of the model such as isotropization, bounce and recollapse are discussed.

Hedgehog ansatz and its generalization for self-gravitating Skyrmions

The hedgehog ansatz for spherically symmetric spacetimes in self-gravitating nonlinear sigma models and Skyrme models is revisited and its generalization for non-spherically symmetric spacetimes is proposed. The key idea behind our construction is that, even if the matter fields depend on the Killing coordinates in a nontrivial way, the corresponding energy-momentum tensor can still be compatible with spacetime symmetries. Our generalized hedgehog ansatz reduces the Skyrme equations to coupled differential equations for two scalar fields together with several constraint equations between them. Some particular field configurations satisfying those constraints are presented in several physically important spacetimes, including stationary and axisymmetric spacetimes. Incidentally, several new exact solutions are obtained under the standard hedgehog ansatz, one of which represents a global monopole inside a black hole with the Skyrme effect.

Generalized hedgehog ansatz and Gribov copies in regions with nontrivial topologies

In this paper the arising of Gribov copies both in Landau and Coulomb gauges in regions with non-trivial topologies but flat metric, (such as closed tubes S1XD2, or RXT2) will be analyzed. Using a novel generalization of the hedgehog ansatz beyond spherical symmetry, analytic examples of Gribov copies of the vacuum will be constructed. Using such ansatz, we will also construct the elliptic Gribov pendulum. The requirement of absence of Gribov copies of the vacuum satisfying the strong boundary conditions implies geometrical constraints on the shapes and sizes of the regions with non-trivial topologies.

IMPROVED APPROXIMATE PROFILE FUNCTION OF HEDGEHOG SKYRMION

The profile function for Skyrme model is investigated in Hedgehog ansatz. An improved analytical solution to the Hedgehog Skyrmion is obtained by using tentative function method. It is found that ensuing calculated static energy is smaller than that in Ref. 13, and the isoscalar electric mean square radius [Formula: see text] and the isoscalar magnetic mean square radius [Formula: see text] well agree with experiment results.