Basics of Topological Solutons

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8th Aug 2017 Mathematics Reference this

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Research into topological solitons began in the 1960s, when the fully nonlinear form of the classical field equations, were being thoroughly explored by mathematicians and theoretical physicists. Topological solitons were first examined when the solutions to these equations were interpreted “as candidates for particles of the theory” [1]. The particles that were observed from the results were different from the usual elementary particles. Topological solitons appeared to behave like normal particles in the sense that they were found to be “localised and have finite energy” [4]. However, the solitons topological structure distinguished them from the other particles. Topological solitons carry a topological charge (also known as the winding number), which results in these particlelike objects being stable. The topological charge is usually denoted by a single integer, N; it is a conserved quantity, i.e. it is constant unless a collision occurs, and it is equal to the total number of particles, which means as |N| increases, the energy also increases. The conservation of the topological charge is due to the topological structure of the target space in which the soliton is defined. The most basic example of soliton has topological charge, N = 1, which is a stable solution, due to the fact a single soliton is unable to decay. 3 If the solution to a nonlinear classical field equation has the properties of being particle-like, stable, have finite mass; and the energy density is localised to a finite region of space, with a smooth structure; then this solution is a topological soliton. In addition to solitons existing with topological charge, N, there also exist antisolitons with -N. In the event of a collision between a soliton and an antisoliton, it is possible for them to “annihilate” each other or “be pair-produced” [1]. It is also possible for multi-soliton states to exist. Any field composition where N > 1, is known as a multi-soliton state. Likewise, multi-solitons also carry a topological charge which again means they are stable. Multi-state solitons either “decay into N well separated charge 1 solitons” or “they can relax to a classical bound state of N solitons” [1]. The energy and “length scale” [1] (a particular length which is determined to one order of magnitude.) the constant in the Lagrangian and field equations which represents the strength of the interaction between the particle and the field, also known as the “coupling constant”. The energy of a topological soliton is equal to its rest mass in a Lorentz invariant theory. [5] [6] Lorentz invariant: A quantity that does not change due to a transformation relating the space-time coordinates of one frame of reference to another in special relativity; a quantity that is independent of the inertial frame. In contrast to the topological soliton, the elementary particles mass is proportional to Planck’s constant, ~. In the limit ~ → 0, the elementary particles mass goes to zero where as the topological solitons mass is finite. The “quantization” of the “wave-like fields which satisfy the linearized field equations” [1] determines the elementary particle states, where the interactions between the particles are determined by the nonlinear terms

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A fundamental discovery in supporting the research of topological solitons is that, given the coupling constants take special values, then the field equations can be reduced from second order to first order partial differential equations.[1] In general, the resulting first order equations are known as Bogomolny equations. These equations do not involve any time derivatives, and their solutions are either “static soliton or multi-soliton configurations”. [1] In these given field theories, if the field satisfies the Bogomolny equation then “the energy is bounded below by a numerical multiple of the modulus of the topological charge, N”, so the solutions of a Bogomolny equation with a certain 4 charge will all have the same energy value. [1] The solutions of the Bogomolny equations are “automatically stable” [1] because the fields “minimize the energy” [1]. As well as this they naturally satisfy the Euler-Lagrange equations of motion, which implies the static solutions are a “stationary point of the energy”. [1] Kinks are solutions to the first-order Bogomolny equation which we shall see in the following chapter

Figure 2.2 shows a model of an infinite pendulum strip, with the angle φ being the angle to the downward vertical [3]. The energy (with all constraints set to 1) is E = Z ∞ −∞  1 2 φ 02 + 1 − cos φ  dx (2.1) where φ 0 = dφ dx . For the energy density to be finite this requires φ → 2Ï€n− as x → −∞ and φ → 2Ï€n+ as x → ∞, where n± ∈ Z. To find the number of “twists”, N, this is simply N = n+ − n− = φ (∞) − φ (−∞) 2Ï€ = 1 2Ï€ Z ∞ −∞ φ 0 dx ∈ Z

This is the equation for the topological charge or the winding number. If we set n− = 0 and n+ = 1 then N = 1, this gives the lowest possible energy for a topological soliton. This is called a kink, and it is the term we use for the one spatial dimension soliton with a single scalar field. The name ‘kink’ is due to the shape of the scalar field when plotted as a function of x [1]. Knowing that a kink gives the minimum of the energy, it is possible to “apply the calculus of variations to derive a differential equation φ(x) and then solve it”[3] to give the shape of the kink. Given a differentiable function on the real line, f(x), it is possible to find the minimum of f(x) by finding the solutions of f 0 (x) = 0, i.e. by finding the stationary points of f(x) [3]. It is achievable to derive this differential equation, f(x), by making a small change to x, i.e. x → x + δx, and from this “calculate the change in the value of the function to leading order in the variaton δx” [3]. δf(x) = f(x + δx) − f(x) = f(x) + δxf0 (x) + … − f(x) = f 0 (x)δx + … If f 0 (x) < 0 then we can make δf(x) < 0 by taking δx > 0. If f 0 (x) > 0 then we can make δf(x) < 0 by taking δx < 0. So the only way that f(x) can be a minimum is if f 0 (x) = 0. The energy for one "twist" is, E = Z ∞ −∞  1 2 φ 02 + 1 − cos φ  dx Returning back to kinks; in order for the function, φ(x), to minimise the energy, E; the function φ(x), needs to be changed by a small function δφ(x). δE = Z ∞ −∞ 1 2 (φ 0 + δφ0 ) 2 + 1 − cos (φ + δφ) − 1 2 φ 02 − 1 + cos φ  dx (2.3) Then taking the Taylor expansion of the linear terms and cancelling, the equation becomes, δE = Z ∞ −∞ {φ 0 δφ0 + sin φδφ} dx (2.4) Then integrating by parts gives, δE = [φ 0 δφ] ∞ −∞ + Z ∞ −∞ {−φ 00δφ + sin φδφ} dx

The term [φ 0 δφ] ∞ −∞ equates to zero on the boundary because it must satisfy δφ(±∞) = 0 as we cannot change the boundary conditions, so δE = Z ∞ −∞ {(−φ 00 + sin φ)δφ} dx (2.6) This equation can be minimised minimised further to the second order nonlinear differential equation, φ 00 = sin φ (2.7) The solution of this differential equation with the boundary conditions, φ(−∞) = 0 and φ(∞) = 2Ï€ is the kink. Therefore the kink solution is, φ(x) = 4 tan−1 e x−a (2.8) where a is an arbitrary constant. When x = a, this is the position of the kink (φ(a) = Ï€). It is clear to see φ = 0 is also a solution to the differential equation, however, it does not satisfy the boundary conditions. It is possible to find “a lower bound on the kink energy without solving a differential equation” [3]. First of all we need to rewrite the energy equation (2.1), using the double angle formula the equation becomes, E = 1 2 Z ∞ −∞  φ 02 + 4 sin2  φ 2  dx (2.9) By completing the square the equation becomes, E = 1 2 Z ∞ −∞  φ 0 − 2 sin  φ 2 2 + 4φ 0 sin  φ 2 dx (2.10) Therefore the energy satisfies the inequality,

E > 2 Z ∞ −∞ φ 0 sin  φ 2  dx = 2 Z ∞ −∞ sin  φ 2  dφ dxdx = 2 Z 2π 0 sin  φ 2  dφ = −4  cos  φ 2 2π 0 = 8 (2.11) In order to obtain the solution which is exactly 8, the term φ 0 − 2 sin φ 2 2 would have to be exactly 0. Therefore the lower bound on the kink energy is calculated by the solution to the equation, φ 0 = 2 sin  φ 2  (2.12) This is a first order Bogomolny equation. Taking this Bogomolny equation and differentiating with respect to φ 0 gives, φ 00 = cos  φ 2  φ 0 = cos  φ 2  2 sin  φ 2  = sin φ (2.13) This shows that a solution of the Bogomolny equation (2.12) gives the output of the kink solution (2.7). To calculate the energy density ε, equation (2.1), we need to use the fact that the Bogomolny equation shows that ε = φ 02 . From equation (2.8) we have, tan φ 4  = e x−a , therefore 1 4 φ 0 sec2  φ 4 = e x−a

This equation gives, φ 0 = 4 e x−a 1 + tan2 φ 4  = 4e x−a 1 + e 2(x−a) = 2 cosh (x − a) = 2 (x − a) (2.15) Therefore it can be seen that the energy density is given by ε = 42 (x − a) From this we get the solution of a lump with a maximal value of 4 when x = a. This maximal value is the position of the kink. The position of the kink is also the position of the pendulum strip when it is exactly upside down, this is due to the fact φ(a) = Ï€ [3]. Using this interpretation for the energy density, it can be verified that the energy is equal to the lower bound E = Z ∞ −∞ εdx = 4 Z ∞ −∞ 2 (x − a) dx = 4 [tanh (x − a)]∞ −∞ = 8 (2.16) For N > 1 i.e. more than one kink, E > 8|N|. In order to obtain the lower bound of N > 1 kinks, the kinks must be infinitely apart to create N infinitely separated kinks. This means there must be a repulsive force between kinks. We shall now look at applying Derrick’s theorem [3] to kinks to show that it does not rule out the existence of topological solitons. Derrick’s Theorem: If the energy E has no stationary points with respect to spatial rescaling then it has no solutions with 0 < E < ∞ (hence no topological solitons).

Derrick’s theorem can only be applied to an infinite domain. Firstly, the energy terms need to be split according to the powers of the derivative, E = E2 + E0 = Z ∞ −∞ 1 2 φ 02 dx + Z ∞ −∞ (1 − cos φ) dx (2.17) Now consider the spatial rescaling x 7→ x λ = X, so that φ (x) 7→ φ (X), with dx = λdX, d dx = 1 λ d dX . Under this rescaling the energy becomes E (λ), E(λ) = Z ∞ −∞ 1 2 ( 1 λ dφ dX ) 2λdX + Z ∞ −∞ (1 − cos φ) λdX = 1 λ E2 + λE0 (2.18) It is now important to see whether E(λ) has a stationary point with respect to λ, dE (λ) dλ = − 1 λ 2 E2 + E0 = 0 (2.19) if λ = qE2 E0 , where λ equals the size of the soliton. From this we can see a stationary point exists, so by Derrick’s theorem we cannot rule out the possibility of a topological soliton solution existing. We already know this is the case due to already finding the kink solution earlier. If it is found that φ(x) is a solution then “the stationary point corresponds to no rescaling” [3], so λ = 1, meaning E2 = E0. This is known as a virial relation.

In order to extend the kink example to higher spatial dimensions, we will rewrite it using different variables. If we let φ = (φ1, φ2) be a two-component unit vector, where φ · φ = |φ| 2 = 1. By writing φ = (sin φ, cos φ), the energy from (2.1) can be rewritten as E = Z ∞ −∞ ( 1 2     dφ dx     2 − H · φ + |H| ) dx (2.20) where H = (0, 1). [3] In this new formulation φ represents the “direction of the local magnetization (restricted to the plane) in a ferromagnetic medium” [3] and H represents the constant background magnetic field which is also restricted to lie within the same plane as φ. There is only one point in which the systems ground state is equal to zero in terms of φ, which is φ = H |H| = (0, 1). Any structure with finite energy has to approach this zero energy ground state at spatial infinity, therefore the boundary conditions are φ → (0, 1) as x → ±∞. As φ takes the same value at x = −∞ and x = +∞, then these points can be identified so the target space, which is the real line R, topologically becomes a circle, S 1 of infinite radius. Therefore we have the mapping φ : S 1 7→ S 1 between circles, because φ is a two-component vector so it also lies on a circle of unit radius. [3] The mapping between circles has a topological charge (winding number), N, which counts the number of times φ “winds around the unit circle as x varies over the whole real line”. [3] The topological charge is equal to the equation defined earlier in (2.2), but using the new variables it is given by the expression N = 1 2Ï€ Z ∞ −∞  dφ1 dx φ2 − dφ2 dx φ1  dx (2.21) If we consider a restricted ferromagnetic system in which there is the absence of a background magnetic field (H = 0); it is still possible for a topological soliton to exist if there is an “easy axis anisotropy”. [3] Magnetic anisotropy is the directional dependence of a material’s magnetic property, and the “easy axis” is a energetically favorable direction if spontaneous magnetization occurs.[7] The energy for this system is E = Z ∞ −∞ ( 1 2     dφ dx     2 + A 1 − (φ · k) 2  ) dx (2.22) where A > 0 is the anisotropy constant and k is the unit vector which specifies the easy axis. [3] For this type of system there are two zero energy ground states, φ = ±k. The kink in this system, also called a domain wall, interpolates between the two zero energy ground states and has boundary conditions φ → k as x → −∞ and φ → −k 15 as x → +∞. Therefore the domain wall does not have a full twist of a kink and only has a half-twist. It is possible to map this system to our original kink example by a change of variables. If we set k = (0, 1) for convenience, and choose A = 1 2 . Setting φ = sin φ 2  , cos φ 2 , then the energy equation becomes E = 1 4 Z ∞ −∞  1 2 φ 02 + 1 − cos φ  dx (2.23) which is equal to the energy equation (2.1) but with a normalization factor of 1 4 . The domain wall boundaries are φ → (0, ±1) as x ∓ ∞ are exactly the kink boundary conditions φ (−∞) = 0 and φ (∞) = 2Ï€. [1]

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This chapter will focus on topological solitons in (2+1) spatial dimensions. It would be incorrect to use the term soliton for these solutions due to their lack of stability, instead they are often referred to as lumps. The solutions for these lumps are “given explicitly by rational maps between Riemann spheres”. [1] For this chapter we shall be looking at one of the simplest Lorentz invariant sigma models in (2+1) spatial dimensions which renders static topological soliton solutions; the O(3) sigma model in the plane. [1] “A sigma model is a nonlinear scalar field theory, where the field takes values in a target space which is a curved Riemannian manifold, usually with large symmetry”. [1] For the O(3) sigma model the target space is the unit 2-sphere, S 2 . This model uses three real scalar fields, Φ = (φ1, φ2, φ3), which are functions of the space-time coordinates (t, x, y) in (2+1) spatial dimensions. [2] The O(3) model is defined by the Lagrangian density L = 1 4 (∂µΦ) · (∂ µΦ) with the constraint Φ · Φ = 1. For this equation the indices represent the space-time coordinates and take the values 0, 1, 2, and ∂µ is partial differentiation with respect to Xµ . [2] From (3.1), the Euler-Lagrange equation can be derived, which is ∂µ∂ µΦ + (∂µΦ · ∂ µΦ) Φ = 0 (3.2) Due to the dot product in ∂µΦ · ∂ µΦ, this shows that the “Euclidean metric of R 3 is being used, and this becomes the standard metric on the target space S 2 when the constraint Φ · Φ = 1 is being imposed”. [1] For the sigma model we are exploring, the O(3) represents the global symmetry in the target space corresponding to the rotations: Φ 7→ MΦ Where M ∈ O(3) is a constant matrix. [1] The “sigma” in the models name represents the fields (φ1, φ2, σ), where φ1 and φ2 are “locally unconstrained” [1] and σ = p 1 − φ 2 1 − φ 2 2 is dependent on φ1 and φ2. The energy for the O(3) sigma model is E = 1 4 Z ∂iΦ · ∂iΦd 2x (3.3) where i = 1, 2 runs over the spatial indices. In order for the energy to be finite, Φ has to tend to a constant vector at spatial infinity, so without loss of generality we are able to set the boundary condition Φ → (0, 0, 1) as x 2 + y 2 → ∞. Topologically we have R 2 ∪ {∞}, which is interpreted as a sphere S 2 via the stereographic projection. (The sphere itself has finite radius.) Therefore we are considering mapping between spheres Φ : S 2 7→ S 2 . Just like in our kink example, mapping between spheres means there exists a topological charge, which can be found using N = 1 4Ï€ Z Φ · (∂1Φ Ã- ∂2Φ) d 2x (3.4) The topological charge represents the number of lumps in the field configuration [1], since generally there are N well-separated, localized areas where the energy density is concentrated and each area has one unit of charge. However, as the lumps approach each other this is no longer the case. In order to apply Derrick’s theorem to the energy (3.3), we would need to consider the scaling x 7→ x λ = X and y 7→ y λ = Y which would give E (λ) = E. The energy is independent of λ, therefore any value of λ is a stationary point since the energy does not change from spatial rescaling.

If we integrate the inequality (∂iΦ ± εijΦ Ã- ∂jΦ) · (∂iΦ ± εikΦ Ã- ∂kΦ) ≥ 0 (3.5) over the plane and use the equations (3.3) and (3.4) for the energy density and the topological charge respectively [1], then we get the Bogomolny bound E ≥ 2Ï€ |N| (3.6) This Bogomolny bound is the lower bound of the energy “in terms of lumps”. [1] If the field is a solution to one of the first-order Bogomolny equations ∂iΦ ± εijΦ Ã- ∂jΦ = 0 (3.7) then the energy is equal to the Bogomolny bound. In order to analyse the Bogomolny equations it is best to make the following changes of variables. For the first change in variable let X = (X1, X2, X3) denote the Cartesian coordinates in R 3 and take X = Φ to be a point on the unit sphere, (X2 1 , X2 2 , X2 3 ) = 1. Let L be the line going through X = (0, 0, −1) and Φ and set W = X1 + iX2 to be the complex coordinate of the point where L intersects the plane at X3 = 0. We then get W = (φ1 + iφ2) (1 + φ3) (3.8) where φ1 =  W + W 1 + |W| 2  , φ2 = i  W − W 1 + |W| 2  , φ3 = 1 − |W| 2 1 + |W| 2 ! (3.9) As Φ tends to the point (0, 0, −1) then L only intersects X3 = 0 at ∞, therefore the point (0, 0, −1) maps to the point W = ∞. This method of assigning each point on the sphere to a point in C ∪ {∞} is called stereographic projection as seen in Figure 3.1.[3] The next change in variable comes from using a complex coordinate in the (x, y) plane by letting z = x + iy. Using this formation it is possible to rewrite the Lagrangian density, from (3.1) L = 1 4 (∂µφ1) 2 + (∂µφ2) 2 + (∂µφ3) 2  . Firstly we need to partially differentiate φ1, φ2, φ3, giving ∂µφ1 = ∂µW + ∂µW 1 + |W| 2 − (∂µW) W + W ∂µW  1 + |W| 2 2 W + W  (3.10) ∂µφ2 = i ” ∂µW − ∂µW 1 + |W| 2 − (∂µW) W + W ∂µW  1 + |W| 2 2 W − W

Finally, from simplifying (3.37) we get the equation for the topological charge in the new formulation to be N = 1 4Ï€ Z 4 1 + |W| 2 2 ∂zW ∂zW − ∂zW ∂zW  d 2x = 1 Ï€ Z |∂zW| 2 − |∂zW| 2  1 + |W| 2 2 d 2x (3.38) In this formulation it is clear to see E ≥ 2Ï€ |N|, with equality if and only if Bogomolny equation is satisfied ∂W ∂z = 0 (3.39) This equation shows that W is a holomorphic function of z only. [4] Due to the “requirement that the total energy is finite, together with the boundary condition” [4] W → 0 as |z| → ∞, this means that N is finite. [3] The simplest solution for the Bogomolny equation would be W = λ z , where λ is a real and positive constant. Applying this to the equation (3.9) yields the solution for the N = 1 solution Φ =  2 λ 2 + x 2 + y 2 , −2 λ 2 + x 2 + y 2 , x 2 + y 2 − λ 2 λ 2 + x 2 + y 2 (3.40) If we change the negative sign in the second component to a positive sign then we get the solution of the anti-Bogomolny equation (3.7) (with the minus sign), which also has E = 2Ï€ but has N = −1. This soliton is located at thee origin because W(0) = ∞. [3] The N = 1 general solution has 4 real parameters and is given by the Bogomolny solution W = λeiθ z − a (3.41) where λ is the size of the soliton, θ is the constant angle of rotation in the (φ1, φ2) plane and a ∈ C is the position of the soliton in the complex plane, z = x + iy.

The O(3) sigma model can be modified to stabilise a lump, and the simplest way in doing this is by introducing extra terms into the Lagrangian which break the “conformal invariance of the static energy”. [1] These new terms must scale as negative and positive powers of a “spatial dilation factor”. [1] An example of this is the Baby Skyrme model which is given by the Lagrangian L = 1 4 ∂µΦ · ∂ µΦ − 1 8 (∂µΦ Ã- ∂νΦ) · (∂ µΦ Ã- ∂ νΦ) − m2 2 (1 − φ3) (3.42) where the constraint Φ · Φ = 1 is implied. As we can see the first term in this Lagrangian is simply that of the O(3) sigma model. The second term in (3.42), is known as the Skyrme term and the final term in this Lagrangian is the mass term.

The complete understanding of topological solitons is unknown and there are very limited experimental tests of many of the theories of topological solitons and their mathematical results. However, there is evidence of topological solitons existing in some physical systems, for example in one-dimensional systems they exist in optical fibres and narrow water channels. [1]

Topological solitons can be applied to a range of different areas including particle physics, condensed matter physics, nuclear physics and cosmology. They also can be applied within technology, which involves using topological solitons in the design for the next generation of data storage devices. [3] In August 2016, a 7 million pound research programme, being led by Durham University, was announced into looking at how magnetic skyrmions can be used in creating efficient ways to store data. [10] Magnetic skyrmions are a theoretical particle in three spatial dimensions which have been “observed experimentally in condensed matter systems”. [11] This type of soliton was first predicted by scientists back in 1962, but was first observed experimentally in 2009. [10] In certain types of magnetic material it is possible for these magnetic skyrmions to be “created,manipulated and controlled”[10], and because of their size and structure it is possible for them to be tightly packed together. The structure inside the skyrmions [10] Due to this and the force which “locks” the magnetic field into the skyrmion arrangement, any magnetic information which is “encoded by skyrmions is very robust”. [10] It is thought that it will be possible to move these magnetic skyrmions with a lot less energy than the “ferromagnetic domain” being used in current data storage devices of smartphones and computers. Therefore, these magnetic skyrmions could revolutionise data storage devices, as the devices could be created on a smaller scale and use a lot less energy, meaning they would be more cost effective and would generate less heat.

This project has given an insight into the very basics of topological solutons by analysing the energy and topological charge equations for kinks in one spatial dimension and lumps in (2+1) spatial dimensions. From the energy equation for a kink, we could derive the solution of a kink and find the lower energy bound. From the lump model, we successfully changed the variables for the energy, topological charge and the Lagrange equation for a lump to be able to analyse the Bogomolny equation. From this change of variables of the Lagrange equation we successfully solved the Euler-Lagrange equations of motion for the lump model. This research project has been captivating and has given me an insight into how the complex mathematics we learn is applied to real world situations. I first became interested in this topic after attending the London Mathematical Society’s summer 33 school in 2016, where I had the privilege of attending a few lectures given by Dr Paul Sutcliffe, one of the author’s of the book on Topological Solitons. It was in these few lectures where I first learnt about topological solitons and some of their applications, and this inspired me for my research project as I wanted to study the topic further. Although this project has been thoroughly enjoyable, it came with challenging aspects, due to its complex mathematics in such a specialised subject. As a result of this topic being so specific, I was very limited in the resources I had for my research, my main resource being the book on topological solitons by Dr Paul Sutcliffe and Dr Nicholas Manton. I have gained a lot of new skills from this research project and it has given me an opportunity to apply my current mathematical knowledge. There is an endless amount of research that can be continued within this subject. I, for example, would have liked to do some further research into the (2+1) spatial dimension model of the Baby Skyrmion and, like the lump example, solve the EulerLagrange equations motion. As well as this, I would have liked to input the equations of motion I solved for the lump model in Maple, so it was possible to simulate two lumps colliding and from this graph the energy density. It would have been really interesting to research further into topological solitons in three spatial dimensions, specifically Skyrmions, to learn further about their technological applications. However, the mathematics used for this model is very challenging and specialised, and goes beyond my understanding and knowledge.

Research into topological solitons began in the 1960s, when the fully nonlinear form of the classical field equations, were being thoroughly explored by mathematicians and theoretical physicists. Topological solitons were first examined when the solutions to these equations were interpreted “as candidates for particles of the theory” [1]. The particles that were observed from the results were different from the usual elementary particles. Topological solitons appeared to behave like normal particles in the sense that they were found to be “localised and have finite energy” [4]. However, the solitons topological structure distinguished them from the other particles. Topological solitons carry a topological charge (also known as the winding number), which results in these particlelike objects being stable. The topological charge is usually denoted by a single integer, N; it is a conserved quantity, i.e. it is constant unless a collision occurs, and it is equal to the total number of particles, which means as |N| increases, the energy also increases. The conservation of the topological charge is due to the topological structure of the target space in which the soliton is defined. The most basic example of soliton has topological charge, N = 1, which is a stable solution, due to the fact a single soliton is unable to decay. 3 If the solution to a nonlinear classical field equation has the properties of being particle-like, stable, have finite mass; and the energy density is localised to a finite region of space, with a smooth structure; then this solution is a topological soliton. In addition to solitons existing with topological charge, N, there also exist antisolitons with -N. In the event of a collision between a soliton and an antisoliton, it is possible for them to “annihilate” each other or “be pair-produced” [1]. It is also possible for multi-soliton states to exist. Any field composition where N > 1, is known as a multi-soliton state. Likewise, multi-solitons also carry a topological charge which again means they are stable. Multi-state solitons either “decay into N well separated charge 1 solitons” or “they can relax to a classical bound state of N solitons” [1]. The energy and “length scale” [1] (a particular length which is determined to one order of magnitude.) the constant in the Lagrangian and field equations which represents the strength of the interaction between the particle and the field, also known as the “coupling constant”. The energy of a topological soliton is equal to its rest mass in a Lorentz invariant theory. [5] [6] Lorentz invariant: A quantity that does not change due to a transformation relating the space-time coordinates of one frame of reference to another in special relativity; a quantity that is independent of the inertial frame. In contrast to the topological soliton, the elementary particles mass is proportional to Planck’s constant, ~. In the limit ~ → 0, the elementary particles mass goes to zero where as the topological solitons mass is finite. The “quantization” of the “wave-like fields which satisfy the linearized field equations” [1] determines the elementary particle states, where the interactions between the particles are determined by the nonlinear terms

A fundamental discovery in supporting the research of topological solitons is that, given the coupling constants take special values, then the field equations can be reduced from second order to first order partial differential equations.[1] In general, the resulting first order equations are known as Bogomolny equations. These equations do not involve any time derivatives, and their solutions are either “static soliton or multi-soliton configurations”. [1] In these given field theories, if the field satisfies the Bogomolny equation then “the energy is bounded below by a numerical multiple of the modulus of the topological charge, N”, so the solutions of a Bogomolny equation with a certain 4 charge will all have the same energy value. [1] The solutions of the Bogomolny equations are “automatically stable” [1] because the fields “minimize the energy” [1]. As well as this they naturally satisfy the Euler-Lagrange equations of motion, which implies the static solutions are a “stationary point of the energy”. [1] Kinks are solutions to the first-order Bogomolny equation which we shall see in the following chapter

Figure 2.2 shows a model of an infinite pendulum strip, with the angle φ being the angle to the downward vertical [3]. The energy (with all constraints set to 1) is E = Z ∞ −∞  1 2 φ 02 + 1 − cos φ  dx (2.1) where φ 0 = dφ dx . For the energy density to be finite this requires φ → 2Ï€n− as x → −∞ and φ → 2Ï€n+ as x → ∞, where n± ∈ Z. To find the number of “twists”, N, this is simply N = n+ − n− = φ (∞) − φ (−∞) 2Ï€ = 1 2Ï€ Z ∞ −∞ φ 0 dx ∈ Z

This is the equation for the topological charge or the winding number. If we set n− = 0 and n+ = 1 then N = 1, this gives the lowest possible energy for a topological soliton. This is called a kink, and it is the term we use for the one spatial dimension soliton with a single scalar field. The name ‘kink’ is due to the shape of the scalar field when plotted as a function of x [1]. Knowing that a kink gives the minimum of the energy, it is possible to “apply the calculus of variations to derive a differential equation φ(x) and then solve it”[3] to give the shape of the kink. Given a differentiable function on the real line, f(x), it is possible to find the minimum of f(x) by finding the solutions of f 0 (x) = 0, i.e. by finding the stationary points of f(x) [3]. It is achievable to derive this differential equation, f(x), by making a small change to x, i.e. x → x + δx, and from this “calculate the change in the value of the function to leading order in the variaton δx” [3]. δf(x) = f(x + δx) − f(x) = f(x) + δxf0 (x) + … − f(x) = f 0 (x)δx + … If f 0 (x) < 0 then we can make δf(x) < 0 by taking δx > 0. If f 0 (x) > 0 then we can make δf(x) < 0 by taking δx < 0. So the only way that f(x) can be a minimum is if f 0 (x) = 0. The energy for one "twist" is, E = Z ∞ −∞  1 2 φ 02 + 1 − cos φ  dx Returning back to kinks; in order for the function, φ(x), to minimise the energy, E; the function φ(x), needs to be changed by a small function δφ(x). δE = Z ∞ −∞ 1 2 (φ 0 + δφ0 ) 2 + 1 − cos (φ + δφ) − 1 2 φ 02 − 1 + cos φ  dx (2.3) Then taking the Taylor expansion of the linear terms and cancelling, the equation becomes, δE = Z ∞ −∞ {φ 0 δφ0 + sin φδφ} dx (2.4) Then integrating by parts gives, δE = [φ 0 δφ] ∞ −∞ + Z ∞ −∞ {−φ 00δφ + sin φδφ} dx

The term [φ 0 δφ] ∞ −∞ equates to zero on the boundary because it must satisfy δφ(±∞) = 0 as we cannot change the boundary conditions, so δE = Z ∞ −∞ {(−φ 00 + sin φ)δφ} dx (2.6) This equation can be minimised minimised further to the second order nonlinear differential equation, φ 00 = sin φ (2.7) The solution of this differential equation with the boundary conditions, φ(−∞) = 0 and φ(∞) = 2Ï€ is the kink. Therefore the kink solution is, φ(x) = 4 tan−1 e x−a (2.8) where a is an arbitrary constant. When x = a, this is the position of the kink (φ(a) = Ï€). It is clear to see φ = 0 is also a solution to the differential equation, however, it does not satisfy the boundary conditions. It is possible to find “a lower bound on the kink energy without solving a differential equation” [3]. First of all we need to rewrite the energy equation (2.1), using the double angle formula the equation becomes, E = 1 2 Z ∞ −∞  φ 02 + 4 sin2  φ 2  dx (2.9) By completing the square the equation becomes, E = 1 2 Z ∞ −∞  φ 0 − 2 sin  φ 2 2 + 4φ 0 sin  φ 2 dx (2.10) Therefore the energy satisfies the inequality,

E > 2 Z ∞ −∞ φ 0 sin  φ 2  dx = 2 Z ∞ −∞ sin  φ 2  dφ dxdx = 2 Z 2π 0 sin  φ 2  dφ = −4  cos  φ 2 2π 0 = 8 (2.11) In order to obtain the solution which is exactly 8, the term φ 0 − 2 sin φ 2 2 would have to be exactly 0. Therefore the lower bound on the kink energy is calculated by the solution to the equation, φ 0 = 2 sin  φ 2  (2.12) This is a first order Bogomolny equation. Taking this Bogomolny equation and differentiating with respect to φ 0 gives, φ 00 = cos  φ 2  φ 0 = cos  φ 2  2 sin  φ 2  = sin φ (2.13) This shows that a solution of the Bogomolny equation (2.12) gives the output of the kink solution (2.7). To calculate the energy density ε, equation (2.1), we need to use the fact that the Bogomolny equation shows that ε = φ 02 . From equation (2.8) we have, tan φ 4  = e x−a , therefore 1 4 φ 0 sec2  φ 4 = e x−a

This equation gives, φ 0 = 4 e x−a 1 + tan2 φ 4  = 4e x−a 1 + e 2(x−a) = 2 cosh (x − a) = 2 (x − a) (2.15) Therefore it can be seen that the energy density is given by ε = 42 (x − a) From this we get the solution of a lump with a maximal value of 4 when x = a. This maximal value is the position of the kink. The position of the kink is also the position of the pendulum strip when it is exactly upside down, this is due to the fact φ(a) = Ï€ [3]. Using this interpretation for the energy density, it can be verified that the energy is equal to the lower bound E = Z ∞ −∞ εdx = 4 Z ∞ −∞ 2 (x − a) dx = 4 [tanh (x − a)]∞ −∞ = 8 (2.16) For N > 1 i.e. more than one kink, E > 8|N|. In order to obtain the lower bound of N > 1 kinks, the kinks must be infinitely apart to create N infinitely separated kinks. This means there must be a repulsive force between kinks. We shall now look at applying Derrick’s theorem [3] to kinks to show that it does not rule out the existence of topological solitons. Derrick’s Theorem: If the energy E has no stationary points with respect to spatial rescaling then it has no solutions with 0 < E < ∞ (hence no topological solitons).

Derrick’s theorem can only be applied to an infinite domain. Firstly, the energy terms need to be split according to the powers of the derivative, E = E2 + E0 = Z ∞ −∞ 1 2 φ 02 dx + Z ∞ −∞ (1 − cos φ) dx (2.17) Now consider the spatial rescaling x 7→ x λ = X, so that φ (x) 7→ φ (X), with dx = λdX, d dx = 1 λ d dX . Under this rescaling the energy becomes E (λ), E(λ) = Z ∞ −∞ 1 2 ( 1 λ dφ dX ) 2λdX + Z ∞ −∞ (1 − cos φ) λdX = 1 λ E2 + λE0 (2.18) It is now important to see whether E(λ) has a stationary point with respect to λ, dE (λ) dλ = − 1 λ 2 E2 + E0 = 0 (2.19) if λ = qE2 E0 , where λ equals the size of the soliton. From this we can see a stationary point exists, so by Derrick’s theorem we cannot rule out the possibility of a topological soliton solution existing. We already know this is the case due to already finding the kink solution earlier. If it is found that φ(x) is a solution then “the stationary point corresponds to no rescaling” [3], so λ = 1, meaning E2 = E0. This is known as a virial relation.

In order to extend the kink example to higher spatial dimensions, we will rewrite it using different variables. If we let φ = (φ1, φ2) be a two-component unit vector, where φ · φ = |φ| 2 = 1. By writing φ = (sin φ, cos φ), the energy from (2.1) can be rewritten as E = Z ∞ −∞ ( 1 2     dφ dx     2 − H · φ + |H| ) dx (2.20) where H = (0, 1). [3] In this new formulation φ represents the “direction of the local magnetization (restricted to the plane) in a ferromagnetic medium” [3] and H represents the constant background magnetic field which is also restricted to lie within the same plane as φ. There is only one point in which the systems ground state is equal to zero in terms of φ, which is φ = H |H| = (0, 1). Any structure with finite energy has to approach this zero energy ground state at spatial infinity, therefore the boundary conditions are φ → (0, 1) as x → ±∞. As φ takes the same value at x = −∞ and x = +∞, then these points can be identified so the target space, which is the real line R, topologically becomes a circle, S 1 of infinite radius. Therefore we have the mapping φ : S 1 7→ S 1 between circles, because φ is a two-component vector so it also lies on a circle of unit radius. [3] The mapping between circles has a topological charge (winding number), N, which counts the number of times φ “winds around the unit circle as x varies over the whole real line”. [3] The topological charge is equal to the equation defined earlier in (2.2), but using the new variables it is given by the expression N = 1 2Ï€ Z ∞ −∞  dφ1 dx φ2 − dφ2 dx φ1  dx (2.21) If we consider a restricted ferromagnetic system in which there is the absence of a background magnetic field (H = 0); it is still possible for a topological soliton to exist if there is an “easy axis anisotropy”. [3] Magnetic anisotropy is the directional dependence of a material’s magnetic property, and the “easy axis” is a energetically favorable direction if spontaneous magnetization occurs.[7] The energy for this system is E = Z ∞ −∞ ( 1 2     dφ dx     2 + A 1 − (φ · k) 2  ) dx (2.22) where A > 0 is the anisotropy constant and k is the unit vector which specifies the easy axis. [3] For this type of system there are two zero energy ground states, φ = ±k. The kink in this system, also called a domain wall, interpolates between the two zero energy ground states and has boundary conditions φ → k as x → −∞ and φ → −k 15 as x → +∞. Therefore the domain wall does not have a full twist of a kink and only has a half-twist. It is possible to map this system to our original kink example by a change of variables. If we set k = (0, 1) for convenience, and choose A = 1 2 . Setting φ = sin φ 2  , cos φ 2 , then the energy equation becomes E = 1 4 Z ∞ −∞  1 2 φ 02 + 1 − cos φ  dx (2.23) which is equal to the energy equation (2.1) but with a normalization factor of 1 4 . The domain wall boundaries are φ → (0, ±1) as x ∓ ∞ are exactly the kink boundary conditions φ (−∞) = 0 and φ (∞) = 2Ï€. [1]

This chapter will focus on topological solitons in (2+1) spatial dimensions. It would be incorrect to use the term soliton for these solutions due to their lack of stability, instead they are often referred to as lumps. The solutions for these lumps are “given explicitly by rational maps between Riemann spheres”. [1] For this chapter we shall be looking at one of the simplest Lorentz invariant sigma models in (2+1) spatial dimensions which renders static topological soliton solutions; the O(3) sigma model in the plane. [1] “A sigma model is a nonlinear scalar field theory, where the field takes values in a target space which is a curved Riemannian manifold, usually with large symmetry”. [1] For the O(3) sigma model the target space is the unit 2-sphere, S 2 . This model uses three real scalar fields, Φ = (φ1, φ2, φ3), which are functions of the space-time coordinates (t, x, y) in (2+1) spatial dimensions. [2] The O(3) model is defined by the Lagrangian density L = 1 4 (∂µΦ) · (∂ µΦ) with the constraint Φ · Φ = 1. For this equation the indices represent the space-time coordinates and take the values 0, 1, 2, and ∂µ is partial differentiation with respect to Xµ . [2] From (3.1), the Euler-Lagrange equation can be derived, which is ∂µ∂ µΦ + (∂µΦ · ∂ µΦ) Φ = 0 (3.2) Due to the dot product in ∂µΦ · ∂ µΦ, this shows that the “Euclidean metric of R 3 is being used, and this becomes the standard metric on the target space S 2 when the constraint Φ · Φ = 1 is being imposed”. [1] For the sigma model we are exploring, the O(3) represents the global symmetry in the target space corresponding to the rotations: Φ 7→ MΦ Where M ∈ O(3) is a constant matrix. [1] The “sigma” in the models name represents the fields (φ1, φ2, σ), where φ1 and φ2 are “locally unconstrained” [1] and σ = p 1 − φ 2 1 − φ 2 2 is dependent on φ1 and φ2. The energy for the O(3) sigma model is E = 1 4 Z ∂iΦ · ∂iΦd 2x (3.3) where i = 1, 2 runs over the spatial indices. In order for the energy to be finite, Φ has to tend to a constant vector at spatial infinity, so without loss of generality we are able to set the boundary condition Φ → (0, 0, 1) as x 2 + y 2 → ∞. Topologically we have R 2 ∪ {∞}, which is interpreted as a sphere S 2 via the stereographic projection. (The sphere itself has finite radius.) Therefore we are considering mapping between spheres Φ : S 2 7→ S 2 . Just like in our kink example, mapping between spheres means there exists a topological charge, which can be found using N = 1 4Ï€ Z Φ · (∂1Φ Ã- ∂2Φ) d 2x (3.4) The topological charge represents the number of lumps in the field configuration [1], since generally there are N well-separated, localized areas where the energy density is concentrated and each area has one unit of charge. However, as the lumps approach each other this is no longer the case. In order to apply Derrick’s theorem to the energy (3.3), we would need to consider the scaling x 7→ x λ = X and y 7→ y λ = Y which would give E (λ) = E. The energy is independent of λ, therefore any value of λ is a stationary point since the energy does not change from spatial rescaling.

If we integrate the inequality (∂iΦ ± εijΦ Ã- ∂jΦ) · (∂iΦ ± εikΦ Ã- ∂kΦ) ≥ 0 (3.5) over the plane and use the equations (3.3) and (3.4) for the energy density and the topological charge respectively [1], then we get the Bogomolny bound E ≥ 2Ï€ |N| (3.6) This Bogomolny bound is the lower bound of the energy “in terms of lumps”. [1] If the field is a solution to one of the first-order Bogomolny equations ∂iΦ ± εijΦ Ã- ∂jΦ = 0 (3.7) then the energy is equal to the Bogomolny bound. In order to analyse the Bogomolny equations it is best to make the following changes of variables. For the first change in variable let X = (X1, X2, X3) denote the Cartesian coordinates in R 3 and take X = Φ to be a point on the unit sphere, (X2 1 , X2 2 , X2 3 ) = 1. Let L be the line going through X = (0, 0, −1) and Φ and set W = X1 + iX2 to be the complex coordinate of the point where L intersects the plane at X3 = 0. We then get W = (φ1 + iφ2) (1 + φ3) (3.8) where φ1 =  W + W 1 + |W| 2  , φ2 = i  W − W 1 + |W| 2  , φ3 = 1 − |W| 2 1 + |W| 2 ! (3.9) As Φ tends to the point (0, 0, −1) then L only intersects X3 = 0 at ∞, therefore the point (0, 0, −1) maps to the point W = ∞. This method of assigning each point on the sphere to a point in C ∪ {∞} is called stereographic projection as seen in Figure 3.1.[3] The next change in variable comes from using a complex coordinate in the (x, y) plane by letting z = x + iy. Using this formation it is possible to rewrite the Lagrangian density, from (3.1) L = 1 4 (∂µφ1) 2 + (∂µφ2) 2 + (∂µφ3) 2  . Firstly we need to partially differentiate φ1, φ2, φ3, giving ∂µφ1 = ∂µW + ∂µW 1 + |W| 2 − (∂µW) W + W ∂µW  1 + |W| 2 2 W + W  (3.10) ∂µφ2 = i ” ∂µW − ∂µW 1 + |W| 2 − (∂µW) W + W ∂µW  1 + |W| 2 2 W − W

Finally, from simplifying (3.37) we get the equation for the topological charge in the new formulation to be N = 1 4Ï€ Z 4 1 + |W| 2 2 ∂zW ∂zW − ∂zW ∂zW  d 2x = 1 Ï€ Z |∂zW| 2 − |∂zW| 2  1 + |W| 2 2 d 2x (3.38) In this formulation it is clear to see E ≥ 2Ï€ |N|, with equality if and only if Bogomolny equation is satisfied ∂W ∂z = 0 (3.39) This equation shows that W is a holomorphic function of z only. [4] Due to the “requirement that the total energy is finite, together with the boundary condition” [4] W → 0 as |z| → ∞, this means that N is finite. [3] The simplest solution for the Bogomolny equation would be W = λ z , where λ is a real and positive constant. Applying this to the equation (3.9) yields the solution for the N = 1 solution Φ =  2 λ 2 + x 2 + y 2 , −2 λ 2 + x 2 + y 2 , x 2 + y 2 − λ 2 λ 2 + x 2 + y 2 (3.40) If we change the negative sign in the second component to a positive sign then we get the solution of the anti-Bogomolny equation (3.7) (with the minus sign), which also has E = 2Ï€ but has N = −1. This soliton is located at thee origin because W(0) = ∞. [3] The N = 1 general solution has 4 real parameters and is given by the Bogomolny solution W = λeiθ z − a (3.41) where λ is the size of the soliton, θ is the constant angle of rotation in the (φ1, φ2) plane and a ∈ C is the position of the soliton in the complex plane, z = x + iy.

The O(3) sigma model can be modified to stabilise a lump, and the simplest way in doing this is by introducing extra terms into the Lagrangian which break the “conformal invariance of the static energy”. [1] These new terms must scale as negative and positive powers of a “spatial dilation factor”. [1] An example of this is the Baby Skyrme model which is given by the Lagrangian L = 1 4 ∂µΦ · ∂ µΦ − 1 8 (∂µΦ Ã- ∂νΦ) · (∂ µΦ Ã- ∂ νΦ) − m2 2 (1 − φ3) (3.42) where the constraint Φ · Φ = 1 is implied. As we can see the first term in this Lagrangian is simply that of the O(3) sigma model. The second term in (3.42), is known as the Skyrme term and the final term in this Lagrangian is the mass term.

The complete understanding of topological solitons is unknown and there are very limited experimental tests of many of the theories of topological solitons and their mathematical results. However, there is evidence of topological solitons existing in some physical systems, for example in one-dimensional systems they exist in optical fibres and narrow water channels. [1]

Topological solitons can be applied to a range of different areas including particle physics, condensed matter physics, nuclear physics and cosmology. They also can be applied within technology, which involves using topological solitons in the design for the next generation of data storage devices. [3] In August 2016, a 7 million pound research programme, being led by Durham University, was announced into looking at how magnetic skyrmions can be used in creating efficient ways to store data. [10] Magnetic skyrmions are a theoretical particle in three spatial dimensions which have been “observed experimentally in condensed matter systems”. [11] This type of soliton was first predicted by scientists back in 1962, but was first observed experimentally in 2009. [10] In certain types of magnetic material it is possible for these magnetic skyrmions to be “created,manipulated and controlled”[10], and because of their size and structure it is possible for them to be tightly packed together. The structure inside the skyrmions [10] Due to this and the force which “locks” the magnetic field into the skyrmion arrangement, any magnetic information which is “encoded by skyrmions is very robust”. [10] It is thought that it will be possible to move these magnetic skyrmions with a lot less energy than the “ferromagnetic domain” being used in current data storage devices of smartphones and computers. Therefore, these magnetic skyrmions could revolutionise data storage devices, as the devices could be created on a smaller scale and use a lot less energy, meaning they would be more cost effective and would generate less heat.

This project has given an insight into the very basics of topological solutons by analysing the energy and topological charge equations for kinks in one spatial dimension and lumps in (2+1) spatial dimensions. From the energy equation for a kink, we could derive the solution of a kink and find the lower energy bound. From the lump model, we successfully changed the variables for the energy, topological charge and the Lagrange equation for a lump to be able to analyse the Bogomolny equation. From this change of variables of the Lagrange equation we successfully solved the Euler-Lagrange equations of motion for the lump model. This research project has been captivating and has given me an insight into how the complex mathematics we learn is applied to real world situations. I first became interested in this topic after attending the London Mathematical Society’s summer 33 school in 2016, where I had the privilege of attending a few lectures given by Dr Paul Sutcliffe, one of the author’s of the book on Topological Solitons. It was in these few lectures where I first learnt about topological solitons and some of their applications, and this inspired me for my research project as I wanted to study the topic further. Although this project has been thoroughly enjoyable, it came with challenging aspects, due to its complex mathematics in such a specialised subject. As a result of this topic being so specific, I was very limited in the resources I had for my research, my main resource being the book on topological solitons by Dr Paul Sutcliffe and Dr Nicholas Manton. I have gained a lot of new skills from this research project and it has given me an opportunity to apply my current mathematical knowledge. There is an endless amount of research that can be continued within this subject. I, for example, would have liked to do some further research into the (2+1) spatial dimension model of the Baby Skyrmion and, like the lump example, solve the EulerLagrange equations motion. As well as this, I would have liked to input the equations of motion I solved for the lump model in Maple, so it was possible to simulate two lumps colliding and from this graph the energy density. It would have been really interesting to research further into topological solitons in three spatial dimensions, specifically Skyrmions, to learn further about their technological applications. However, the mathematics used for this model is very challenging and specialised, and goes beyond my understanding and knowledge.

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