The 't Hooft-Polyakov monopole in the geometric theory of defects
aa r X i v : . [ phy s i c s . g e n - ph ] J un The ’t Hooft–Polyakov monopole in the geometrictheory of defects
M. O. Katanaev ∗ Steklov mathematical institute,119991, Moscow, ul. Gubkina, 8
July 28, 2020
Abstract
The ’t Hooft–Polyakov monopole solution in Yang–Mills theory is given newphysical interpretation in the geometric theory of defects. It describes solids withcontinuous distribution of dislocations and disclinations. The corresponding densi-ties of Burgers and Frank vectors are computed. It means that the ’t Hooft–Polyakovmonopole can be seen, probably, in solids.
Important properties of real crystals such as plasticity, melting, growth, etc., are mainlydefined by defects of the crystalline structure which are called dislocations. Moreover,many bodies posses a spin structure. For example, ferromagnets are also characterized bythe distribution of magnetic moments described by the unit vector field. This unit vectorfield may also have defects (singularities) which are called disclinations. Description ofdislocations and disclinations in elastic media is a very active field of research for morethen one century because of its importance for applications (see, e.g., [1, 2]).Real solids posses usually a crystalline structure and are often described by modelsbased on this crystalline structure especially at the quantum level. At the same time,many properties of solids can be also described by the elasticity theory in the continuousapproximation. Discrete and continuous approaches complement each other, and are bothneeded for our understanding of nature.In this paper, we consider only continuous approximation. In this approximation solidswithout dislocations are described by the displacement vector field within the ordinaryelasticity theory. The spin structure of solids without disclinations is described by the unitvector field ( n -field) satisfying appropriate field equations. In the presence of dislocationsand disclinations there as a problem: what variable are to be used? For example, realsolids posses many defects, and if we want to use continuous approximation for defect ∗ E-mail: [email protected] n -filed do not exist because they aresingular at each point. The geometric theory of defects is aimed to resolve this problem.The idea of geometric theory of defects is simple. In the continuous approximation,a crystal with a spin structure is considered as elastic media (manifold) with a givenmetric and affine connection with torsion (the Riemann–Cartan geometry). As usual,elastic deformations of media and distribution of the unit vector field are described bythe displacement and rotational angle vector fields. The absence of defects means thatdisplacement and unit vector fields are smooth. If they are not continuous then wesay that the media has defects. In general, there are two types of defects: dislocationswhich are defects of elastic media itself (discontinuity of the displacement vector field)and disclinations corresponding to discontinuities of the unit vector field. If defects areabsent, then geometry is trivial: curvature and torsion are zero. In the presence of defects,geometry becomes nontrivial. Dislocations give rise to torsion and disclinations result innontrivial curvature. The physical meaning of torsion and curvature are surface densitiesof Burgers [3, 4] and Frank [5] vectors, respectively, [6, 7]. The geometric theory of defectsallows one to describe single defects as well as their continuous distribution. For singledefects, torsion and curvature are zero everywhere except some points, lines or surfaceswhere defects are located and where they have singularities. In the case of continuousdistribution of dislocations and disclinations, torsion and curvature become nontrivial onthe whole media, and instead of the displacement and angular rotation field we use tetradand SO (3)-connection as the independent variables. The advantage is that these variablesexist even in the absence of the displacement and unite vector fields.The history of geometric theory of defects goes back to 1950s [8–11] when dislocationswere related to torsion for the first time. The review and earlier references can be foundin the book [12].In the geometric approach to the theory of defects [6, 7, 13], we discuss the model whichis different from others in two respects. Firstly, we do not have the displacement and unitvector fields as independent variables because, in general, they are not continuous. Instead,the triad field and SO (3)-connection are considered as the only independent variables. Ifdefects are absent, then the triad and SO (3)-connection reduce to partial derivatives of thedisplacement and rotational angle vector fields (pure gauge because torsion and curvaturevanish). In this case, the latter can be reconstructed. Secondly, the set of equilibriumequations is different. We proposed the purely geometric set which coincides with that ofEuclidean three dimensional gravity with torsion. The nonlinear elasticity equations andprincipal chiral SO (3)-model for the unit vector field enter the model through the elasticand Lorentz gauge conditions [14, 15, 7] which allow us to reconstruct the displacementand unit vector fields in the absence of defects in full agreement with classical models.When a new model is proposed then one has to show how to obtain previous resultswithin new approach. A number of dislocations were described in the geometric theoryof defects and shown to be in agreement with the elasticity theory [7], which correspondsto linear approximation. Therefore the geometric theory of defects does not contradictexperimental data in the domain where elasticity theory is valid. At the same time, thegeometric theory of defects have also different predictions, for example, for the deformationtensor near the core of wedge dislocation. As far as we know, there is no experimentalconfirmation or refutation of geometric theory of defects. So, the model is still undertheoretical development.In this paper, we consider the possibility of physical interpretation of the ’t Hooft–2olyakov monopole solution [16, 17] in the geometric theory of defects. The famous ’tHooft–Polyakov solution in the SU (2) gauge theory interacting with the triplet of scalarfields attracted much interest in physics and mathematics (for review, see, for example,[18, 19]). The solution is static and spherically symmetric. Therefore, it reduces tominimization of three-dimensional Euclidean energy expression which can be regardedas the free energy expression in solid state physics. We consider the SU (2)-connectioncomponents as the SO (3)-connection because their Lie algebras coincide, the triplet ofscalar fields being the source of defects. Moreover, we assume that the SO (3) group actsnot in the isotopic space but in the tangent space to space manifold R . The metric of thespace remains Euclidean. So the ’t Hooft–Polyakov monopole corresponds to Euclideanvielbein and nontrivial SO (3)-connection which give rise to nontrivial Riemann–Cartangeometry of space.So, the ’t Hooft–Polyakov monopole solution has natural interpretation in solid statephysics describing elastic media with continuous distribution of disclinations and disloca-tions. We compute the corresponding densities of Frank and Burgers vectors. In this section we give short review of the geometric theory of defects and introduce basicgeometric notions: triad field and SO (3)-connection. More details can be found in [7].We consider a three dimensional continuous media described by a topologically trivialRiemann–Cartan manifold. We use triad field e µi and SO (3)-connection ω µij = − ω µji ,where Greek letters µ = 1 , , i, j = 1 , , g µν := e µi e ν j δ ij = δ µν is an ordinary flat Euclidean metric, but connection is nontrivial andmay have singularities on some points, lines, or surfaces.The simplest and most widespread examples of linear dislocations are shown in Fig. 1(see, e.g., [1, 2]). They are produced as follows. We cut the medium along the half-plane x = 0, x >
0, move the upper part of the medium located over the cut x > x > b towards the dislocation axis x , and glue the cutting surfaces. The vector b is called the Burgers vector. In a general case, the Burgers vector may not be constant onthe cut. For the edge dislocation, it varies from zero to some constant value b as it movesfrom the dislocation axis. After the gluing, the media comes to the equilibrium statecalled the edge dislocation, see Fig. 1 a . If the Burgers vector is parallel to the dislocationline, it is called the screw dislocation (Fig. 1 b ).From the topological standpoint, the medium containing several dislocations or eventhe infinite number of them is still the Euclidean space R . In contrast to the case ofelastic deformations, the displacement vector in the presence of dislocations is no longera smooth function because of the presence of cutting surfaces where it jumps.The main idea of the geometric approach amounts to the following. To describesingle dislocations in the framework of elasticity theory, we must solve equations for thedisplacement vector with some boundary conditions on the cuts. This is possible for smallnumber of dislocations. But, with an increasing number of dislocations, the boundaryconditions become so complicated that the solution of the problem becomes unrealistic.Besides, one and the same dislocation can be created by different cuts which leads toan ambiguity in the displacement vector field. Another shortcoming of this approach is3 x x x b a b x b x x Figure 1: Straight linear dislocations. ( a ) The edge dislocation. The Burgers vector b isperpendicular to the dislocation line. ( b ) The screw dislocation. The Burgers vector b isparallel to the dislocation line.that it cannot be applied to the description of a continuous distribution of dislocationsbecause the displacement vector field does not exist in this case at all because it musthave discontinuities at every point. In the geometric approach, we consider the triad fieldinstead of the displacement vector field which is introduced as follows.Let a point of the medium has Cartesian coordinates y i in the ground equilibriumstate. After elastic deformation, this point has the coordinates y i x i ( y ) = y i + u i ( x ) , (1)where u i ( x ) is the displacement vector field. We consider its components as functions offinal point position x .In a general dislocation-present case, we do not have a preferred Cartesian coordinatesystem in the equilibrium because there is no symmetry. Therefore, we consider arbitraryglobal coordinates x µ , µ = 1 , ,
3, in R . We use Greek letters for coordinates allow-ing arbitrary coordinate changes. Then the Burgers vector for linear dislocation can beexpressed as the integral of the displacement vector I C dx µ ∂ µ u i ( x ) = − I C dx µ ∂ µ y i ( x ) = − b i , (2)where C is a closed contour surrounding the dislocation axis. This integral is invari-ant under arbitrary coordinate transformations x µ x µ ′ ( x ) and covariant under global SO (3)-rotations of y i . Here, components of the displacement vector field u i ( x ) are consid-ered with respect to the orthonormal basis in the tangent space, u = u i e i . If components ofthe displacement vector field are considered with respect to the coordinate basis u = u µ ∂ µ ,the invariance of the integral (2) under general coordinate changes is violated.In the geometric approach, we introduce new independent variable – the triad – insteadof partial derivatives ∂ µ u i : e µi ( x ) := ( ∂ µ y i , outside the cut, lim ∂ µ y i , on the cut. (3)4he triad is a smooth function on the cut by construction. We note that if the vielbein wassimply defined as partial derivatives ∂ µ y i , then it would have the δ -function singularityon the cut because functions y i ( x ) have a jump. The Burgers vector can be expressedthrough the integral over a surface S having contour C as the boundary: I C dx µ e µi = Z Z S dx µ ∧ dx ν ( ∂ µ e νi − ∂ ν e µi ) = b i , (4)where dx µ ∧ dx ν is the surface element. As a consequence of the definition of the vielbein in(3), the integrand is equal to zero everywhere except at the dislocation axis. For the edgedislocation with constant Burgers vector, the integrand has a δ -function singularity at theorigin. The criterion for the presence of a dislocation is a violation of the integrabilityconditions for the system of equations ∂ µ y i = e µi : ∂ µ e νi − ∂ ν e µi = 0 . (5)If dislocations are absent, then the functions y i ( x ) exist and define transformation to aCartesian coordinates frame.In the geometric theory of defects, the field e µi is identified with the triad. Next, wecompare the integrand in (4) with the expression for the torsion in Cartan variables T µν i := ∂ µ e νi − ∂ ν e µj − e µj ω νji + e νj ω µji . (6)They differ only by terms containing the SO (3)-connection ω µji . This is the ground for theintroduction of the following postulate. In the geometric theory of defects, the Burgersvector corresponding to a surface S is defined by the integral of the torsion tensor: b i := Z Z S dx µ ∧ dx ν T µν i . This definition is invariant with respect to general coordinate transformations of x µ andcovariant with respect to global rotations. Thus, the torsion tensor has straightforwardphysical interpretation: it is equal to the surface density of the Burgers vector.If the curvature tensor for the SO (3)-connection R µν ij := ∂ µ ω νij − ∂ ν ω µij − ω µik ω νkj + ω νik ω µkj , (7)is zero, then the connection is locally trivial, and there exists such SO (3) rotation that ω µij = 0. In this case, we return to expression (4).Next we give physical interpretation of the SO (3)-connection entering the expressionfor torsion (6). To this end we consider more general solids possessing spin structure,for example, ferromagnets or liquid crystals. The spin structure is the unit vector field n i ( x ) ( n i n i = 1). It can be described as follows. We fix some direction in the medium n i . Then the field n i ( x ) at a point x can be uniquely defined by the angular rotationfield θ ij ( x ) = − θ ji ( x ) = ε ijk θ k , where ε ijk is the totally antisymmetric tensor and θ k isa covector directed along the rotation axis, its length being the rotation angle. Here andin what follows, Latin tangent indices are raised and lowered with the help of the flatEuclidean metric δ ij . So, n i = n j S ji ( θ ) , (8)5 x x a b x x C Figure 2: Distribution of unit vector field in the x := x , y := x plane for straight lineardisclinations parallel to the x axis, for | Θ | = 2 π ( a ) and | Θ | = 4 π ( b ).where S j i ∈ SO (3) is the rotation matrix corresponding to θ ij and parameterized as S ij = ( e ( θε ) ) ij = cos θ δ ji + ( θε ) ij θ sin θ + θ i θ j θ (1 − cos θ ) ∈ SO (3) , (9)where ( θε ) ij := θ k ε kij and θ := √ θ i θ i . If the unit vector field is continuous then there areno disclinations. Disclinations arise when the angular rotation field has discontinuities.The simplest examples of linear disclinations are shown in Fig. 2, where the discontinuityof the angular rotation field occurs on a half-plane cut from the x axis to infinity, andthe vector field n lies in the perpendicular plane ( x , x ).A linear disclination is characterized by the Frank vectorΘ i := 12 ε ijk Θ jk , (10)where Θ ij := I C dx µ ∂ µ θ ij , (11)and the integral is taken along closed contour C surrounding the disclination axis. Thelength of the Frank vector is equal to the total angle of rotation of the field n i as itgoes around the disclination. For linear disclinations it must be a multiple of 2 π . Inthe presence of disclinations, the rotational angle field θ ij ( x ) is no longer continuous, andwe must make some cuts for a given distribution of disclinations and impose appropriateboundary conditions in order to define θ ij ( x ). In geometric theory of defects, instead ofthe rotational angle field, we introduce the SO (3)-connection ω µij := ( ∂ µ ω ij , outside the cut, lim ∂ µ ω ij , on the cut. (12)in the way similar to the introduction of the triad field. Sure, we assume that the limitson both sides of the cut exist and are equal. So the SO (3)-connection is less singular thenthe rotational angle field by definition. 6hen the Frank vector for a surface S is given by the integral of curvatureΩ ij := Z Z S dx µ ∧ dx ν R µν ij . (13)If we have straight linear disclination with rotational symmetry, and vector n rotatesin the perpendicular plane, then the SO (3) group reduces to abelian SO (2) group, thenonlinear terms in the curvature (7) disappear, and we return to the previous expression(11) due to the Stokes theorem.The previous discussion refers to an isolated disclinations. If there is a continuousdistribution of disclinations the curvature differs from zero everywhere, and the rotationalangle field θ ij does not exist. Disclinations are said to be absent if and only if the curvatureof SO (3)-connection vanishes, R µνij = 0. In this manner, the geometric theory of defectsdescribes single defects as well as their continuous distribution, in which the phenomenaof disclinations is replaced by the notion of curvature. Let us consider three-dimensional Euclidean space R with Cartesian coordinates x µ andEuclidean metric δ µν , µ, ν = 1 , , SU (2)-gauge fields A µi , i = 1 , ,
3, interacting with the triplet of scalar fields ϕ i in the adjoint representationminimize the three-dimensional energy [18, 19] E := Z d x (cid:18) F µνi F µνi + 12 ∇ µ ϕ i ∇ µ ϕ i + 14 λ (cid:0) ϕ − a (cid:1) (cid:19) , (14)where indices are raised and lowered by Euclidean metrics δ µν and δ ij , F µν i := ∂ µ A νi − ∂ ν A µi + A µj A ν k ε jki , ∇ µ ϕ i := ∂ µ ϕ i + A µj ϕ k ε jki . (15)– are the curvature tensor components for SU (2)-connection and the covariant derivativeof scalar fields; λ > , a > ε ijk is the totally antisymmetrictensor, ε := 1, and ϕ := ϕ i ϕ i .The spherically symmetric ansatz is A µi = ε µij x j ( K − r , ϕ i = x i Hr , (16)where K ( r ) and H ( r ) are some dimensionless functions on radius r := √ x .The Euler–Lagrange equations for functional (14) in the spherically symmetric casereduce to r K ′′ = K (cid:0) K + H − (cid:1) ,r H ′′ =2 HK + λ (cid:0) H − a r (cid:1) H. (17)At present we know only one exact analytic solution to this system of equations for λ = 0 K = ar sh ( ar ) , H = ar tanh( ar ) − , (18)7hich is called the Bogomol’nyi–Prasad–Sommerfield solution [20, 21]. It is easily checkedthat this solution has finite energy.The Lie algebra su (2) is isomorphic to so (3), and we can consider energy (14) as thethree-dimensional Euclidean functional for SO (3)-connection interacting with the tripletof scalar fields ϕ i in the fundamental representation. We assume, that this is the expressionfor the free energy describing static distribution of disclinations and dislocations in elasticmedia with defects, the triplet of scalar fields being the source of defects.The Euclidean metric means that elastic stresses are absent in media. The Cartanvariables for monopole solutions are e µi = δ iµ , ω µij = A µk ε kij = ( δ jµ x i − δ iµ x j ) K − r , (19)where we use the spherically symmetric SO (3)-connection (16). The curvature and torsionare expressed through Cartan variables as usual by Eqs.(6), (7). In the considered case,simple calculations yield the following expressions for curvature and torsion: R µν k := 12 R µν ij ε ijk = F µν k = ε µνk K ′ r − ε µνj x j x k r (cid:18) K ′ − K − r (cid:19) , (20) T µν k = (cid:0) δ kµ x ν − δ kν x µ (cid:1) K − r . (21)In the geometric theory of defects, curvature (20) and torsion (21) have physical mean-ing of surface densities of Frank and Burgers vectors, respectively. That is they are equalto k -th components of respective vectors on surface element dx µ ∧ dx ν . If s µ is normal tothe surface element, then there are the following densities of Frank and Burgers vectors: f µi := 12 ε µνρ R νρi = 13 r δ iµ (cid:18) K ′ + K − r (cid:19) − r (cid:18) ˆ x µ ˆ x i − δ iµ (cid:19) (cid:18) K ′ − K − r (cid:19) , (22) b µi := 12 ε µνρ T νρi = ε µij ˆ x j K − r , (23)where ˆ x µ := x µ /r and tensor f µi is decomposed into irreducible components.For the Bogomol’nyi–Prasad–Sommerfield solution functions K ( r ) and H ( r ) are givenin Eq. (18). They have the following asymptotics K (cid:12)(cid:12) r → ≈ − ( ar ) − ( ar ) , K (cid:12)(cid:12) r →∞ ≈ ar e − ar → ,H (cid:12)(cid:12) r → ≈ ar ) − ar ) , H (cid:12)(cid:12) r →∞ ≈ ar − → ∞ . (24)8he corresponding asymptotics of Frank and Burgers vector densities are f µi (cid:12)(cid:12) r → ≈ − δ iµ (cid:18) a + 790 a r (cid:19) + 245 x µ x i a → − δ iµ a ,b µi (cid:12)(cid:12) r → ≈ − ε µij x j (cid:18) a + a r (cid:19) → − ε µij x j a ,ϕ i (cid:12)(cid:12) r → ≈ x i (cid:18) a − a r (cid:19) → x i a ,f µi (cid:12)(cid:12) r →∞ ≈ − x µ x i r → ,b µi (cid:12)(cid:12) r →∞ ≈ − ε µij x j r → ,ϕ i (cid:12)(cid:12) r →∞ ≈ x i r (cid:18) a − r (cid:19) → x i r a. (25)It implies, in particular, that the total energy (14) is finite. The geometric theory of defects is aimed for description of dislocations and disclinations inthe continuous approximation. It is well suited for description of single defects as well astheir continuous approximation. In the present paper, we consider media with Euclideanmetric but nontrivial SO (3)-connection. The ’t Hooft–Polyakov monopole solution is thestatic spherically symmetric solution of SU (2) Yang–Mills theory. The isomorphism of su (2) and so (3) Lie algebras implies that the ’t Hooft–Polyakov monopole may have newphysical interpretation in solid state physics. In contrast to the original model, the SO (3)group acts now not in the isotopic space but in the tangent space, giving rise to nontrivialtorsion and curvature. These geometrical notions have physical interpretation as surfacedensities of Burgers and Frank vectors, respectively, in the geometric theory of defects.These are explicitly computed for the Bogomol’nyi–Prasad–Sommerfield solution. We arenot aware what kind of media is to be chosen for experimental observations and whatkind of experiment can be taken to confirm or disprove the geometric theory of defectsbut the mere existence of such possibility seems to be interesting. References [1] L. D. Landau and E. M. Lifshits.
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