An introduction to kinks in φ 4 -theory
SSciPost Physics Lecture Notes Submission
An introduction to kinks in ϕ -theory Mariya Lizunova and Jasper van Wezel Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CCUtrecht, The Netherlands Institute for Theoretical Physics Amsterdam, University of Amsterdam, Science Park904, 1098 XH Amsterdam, The Netherlands* [email protected]
Abstract
As a low-energy effective model emerging in disparate fields throughout allof physics, the ubiquitous ϕ -theory is one of the central models of moderntheoretical physics. Its topological defects, or kinks, describe stable, particle-like excitations that play a central role in processes ranging from cosmology toparticle physics and condensed matter theory. In these lecture notes, we intro-duce the description of kinks in ϕ -theory and the various physical processesthat govern their dynamics. The notes are aimed at advanced undergraduatestudents, and emphasis is placed on stimulating qualitative insight into therich phenomenology encountered in kink dynamics. The appendices containmore detailed derivations, and allow enquiring students to also obtain a quan-titative understanding. Topics covered include the topological classification ofstable solutions, kink collisions, the formation of bions, resonant scattering ofkinks, and kink-impurity interactions. Contents ϕ -theory 5 A.1 Internal excitation mode 171 a r X i v : . [ n li n . PS ] S e p ciPost Physics Lecture Notes Submission A.2 Static solutions 18A.3 Numerical method 21
References 23
Solitons were introduced into physics by J.S. Russell in 1834 [1], after he observed a solitarywave travelling for miles along the Union Canal near Edinburgh, Scotland, without alteringits shape or speed. The dynamics of this particular wave were later described usingthe Korteweg–de Vries equation [2], but the idea of solitary waves as stable, localizedconfigurations with finite energy in any medium or field [3], turned out to be much moregeneral. They are now known to occur and play an important role in almost all areas ofphysics, including particle physics [4], cosmology [5,6], (non-linear) optics [7,8], condensedmatter theory [9–11], and biophysics [12]. To understand the generic properties of solitarywaves, they can be studied in the most elementary models or field theories possible. Twofamous examples are the sine-Gordon model and the ϕ -theory.The sine-Gordon model is an integrable model [13], with infinitely many conservedquantities that allow solitary waves in its field configuration, called solitons, to passthrough one another while retaining their individual sizes and shapes [14]. It has foundmany applications including, for example, the analysis of seismic data [15], convectingnematic fluids [16], Josephson-junction arrays [17, 18], and magnetic materials [19].The ϕ -theory, on the other hand, was first introduced by Ginzburg and Landau as aphenomenological theory of second-order phase transitions [20]. Since then, it has beenidentified as a low-energy effective description of phenomena in almost any field of physics,making the detailed understanding of its fundamental properties and excitations partic-ularly relevant. The ϕ -theory can be extended to higher order [21–24], as well as morestructured fields [25–30], but the classical scalar theory already contains all essential in-gredients required to describe the emergence, dynamics, and interactions of solitary wavescalled kinks, and will be the focus of these lecture notes.The ϕ -theory is not integrable, and although it possesses stable and localized solitarywave excitations of finite energy [3], these cannot pass through one another unaffected,as they do in the sine-Gordon model [31]. Instead, collisions between kinks may result ina wide array of physical phenomena, such as the excitation of internal modes, resonantand non-resonant scattering [32–35], and the formation of bound states [35, 36]. All ofthese processes have found application throughout physics, in effective descriptions ofseemingly disparate things like molecular dynamics [37–39], the motion of domain wallsin crystals [40–42], the formation of abnormal nuclei [43–45], and the folding of proteinchains [46–48].These lecture notes aim to provide a self-contained first introduction to the descriptionof kinks in ϕ -theory and the rich collection of physical phenomena arising in their dy-namics. They are suitable for use as a short course for advanced undergraduate students.Familiarity with basic classical field theory is assumed, and some phenomenological knowl-edge of, for example, particle physics or basic condensed matter physics will be useful forappreciating the significance of presented results. Sec. 2 forms the basis of these lecturenotes, introducing the classical ϕ field theory in (1 + 1) dimensions, and explaining hownon-trivial solutions containing kinks arise. The remaining sections focus on the dynamics2 ciPost Physics Lecture Notes Submission and interactions of kinks in the ϕ -theory, with the phenomenology of kink-antikink colli-sions being introduced in Sec. 3, followed by the modelling of the resulting scattering andformation of bound states in Secs. 4 and 5, and a discussion of the effect of local disorderin Sec. 6. Exercises appear and the end of most sections, and guide the reader throughthe main results presented in the text. They are not intended to be challenging. Finally,more detailed discussions of several aspects are presented in appendix A.We hope these lecture notes provide a basis for understanding some of the ubiquitousphenomena arising throughout effective low-energy descriptions in all realms of physics.They explain why localized excitations in a continuous field behave like massive particlesthat can scatter, form bound states, and respond to impurities in the continuous medium.They allow you to appreciate the universal nature of these effects, and they prepare youfor independently investigating the detailed dynamics of solitary waves in any physicalsetting. Consider a classical, real, and scalar field ϕ = ϕ ( t, x ) in (1+1)-dimensional space-time [31,49, 50]. Its dynamics is determined by the Lagrangian density: L = 12 (cid:18) ∂ϕ∂t (cid:19) − (cid:18) ∂ϕ∂x (cid:19) − U ( ϕ ) . (2.1)The specific potential U ( ϕ ) = m ϕ yields a free massive scalar field theory, whose equa-tion of motion is described by the classical Klein-Gordon equation. More generally, thefunction U ( ϕ ) can be thought of as a self-interaction potential of the field ϕ . We canalways use the freedom to choose the zero of energy to ensure that U ( ϕ ) is a non-negativefunction of ϕ , whose minimum value is precisely zero.Using the Euler-Lagrange equation, the equation of motion for ϕ ( t, x ) is found to be: ∂ ϕ∂t − ∂ ϕ∂x + dUdϕ = 0 . (2.2)For a static, time independent solution this simplifies to d ϕ/dx = dU /dϕ . The dynamicsof an initial field configuration ϕ ( t , x ) may be studied by numerically solving the equationof motion on a discrete lattice [43,51], or employing an appropriate approximation schemesuch as the collective coordinate approximation (CCA) [22, 33, 52–55]. Both approacheswill be used in the next sections of these lecture notes.The instantaneous energy of any field configuration ϕ ( t, x ) is given by the functional: E [ ϕ ] = + ∞ (cid:90) −∞ (cid:34) (cid:18) ∂ϕ∂t (cid:19) + 12 (cid:18) ∂ϕ∂x (cid:19) + U ( ϕ ) (cid:35) dx. (2.3)Notice that in spite of the time dependence of ϕ , the energy E [ ϕ ] is a time-independent,conserved quantity. The energy of a static ground state field configuration is sometimesreferred to as the mass of that field and denoted by M . This should not be confused withthe Klein-Gordon mass parameter m in a free field theory. For the energy in Eq. (2.3)to be finite, the integral should converge. This yields the requirement that all physicalfields ϕ ( t, x ) approach a minimum of U ( ϕ ) sufficiently quickly as x approaches positive ornegative infinity. 3 ciPost Physics Lecture Notes Submission Figure 2.1: Left: Schematic representation and example of the asymptotic behavior for atopologically trivial solution. Right: Schematic representation and example of the asymp-totic behavior for a topologically non-trival solution.If there is only a single minimum, at ϕ = ϕ v , then the field configuration ϕ ( t, x ) = ϕ v will be the vacuum or ground state of the system. If there are multiple, degenerate minima ϕ (1) v , ϕ (2) v , and so on, they together form a vacuum manifold and any field configuration ϕ ( t, x ) = ϕ v is a possible ground state. It is also possible however, to find static fieldconfigurations that approach distinct minima at opposing boundaries of space ( x = ±∞ ).These types of solutions are called topological [49, 50, 56], and more generally, one maydivide all static configurations into topological sectors labelled by the set of minima theyapproach at spatial infinity. This is indicated schematically in Fig. 2.1.The energy of a static field in any topological sector may be written in a particularlyconvenient form by introducing the so-called superpotential W ( ϕ ) [49], defined by: U ( ϕ ) = 12 (cid:18) dW ( ϕ ) dϕ (cid:19) . (2.4)Notice that we can always find a smooth, continuously differentiable function W ( ϕ ) sat-isfying this equation because we assumed U ( ϕ ) to be a non-negative function of ϕ . Usingthe superpotential, the expression for the energy in Eq. (2.3) can be written for a staticfield configuration as: E [ ϕ ] = 12 + ∞ (cid:90) −∞ (cid:34)(cid:18) dϕdx (cid:19) + (cid:18) dWdϕ (cid:19) (cid:35) dx = 12 (cid:90) (cid:18) dϕdx − dWdϕ (cid:19) dx + (cid:90) dWdϕ dϕdx dx = 12 (cid:90) (cid:18) dϕdx − dWdϕ (cid:19) dx + W | ϕ ( x =+ ∞ ) − W | ϕ ( x = −∞ ) . (2.5)From the final line, it is clear that any field in a given topological sector necessarily has anenergy E ≥ E BPS , with the minimum possible energy E BPS = W | ϕ ( x =+ ∞ ) − W | ϕ ( x = −∞ ) named after Bogomolny, Prasad, and Sommerfield [57, 58]. Any field configuration withenergy equal to E BPS is said to saturate the BPS bound.Since the integral in the final line of Eq. (2.5) is over a squared function, the only way4 ciPost Physics Lecture Notes Submission
Figure 2.2: The kink (left) and antikink (right) configurations of Eq. (2.9). The charac-teristic width of the kink, l K , is indicated.to obtain a BPS saturated configuration is to have a field obeying the condition: dϕdx = dWdϕ = √ U . (2.6)If such a field configuration does exist, it will be guaranteed by the variational principleto also be a ground state for its topological sector. The Euler-Lagrange equation givenby Eq. (2.2) is therefore automatically satisfied by solutions of Eq. (2.6), even though thelatter is only a first order differential equation. ϕ -theory The simplest scalar field theory having distinct topological sectors is the so-called ϕ -theory. It is defined as a particular instance of the general model of Eq. (2.1), with theself-interaction potential equal to U ( ϕ ) = 14 (1 − ϕ ) . (2.7)Here, we write the potential in a dimensionless and mathematically convenient form. Theequation of motion with this potential becomes ∂ ϕ∂t − ∂ ϕ∂x + ϕ − ϕ = 0 . (2.8)This equation has non-topological or trivial solutions (approaching the same state at bothspatial boundaries) given by ϕ ( t, x ) = 0 and ϕ ( t, x ) = ±
1. These correspond to the fieldalways being at either a maximum or minimum of the potential. The solution ϕ = 0, sittingat a maximum, is unstable, while the solutions ϕ = ± x → −∞ , and the other minimum as x → ∞ . One such solution is: ϕ ( t, x ) = tanh (cid:18) x − al K (cid:19) , with l K = √ . (2.9)It is easy to check that this solution satisfies both the equation of motion (2.8), and theBPS condition of Eq. (2.6). It is therefore a stable, non-dissipating configuration, withthe minimum possible energy for any field connecting two distinct vacua. This topologicalsolution is often called a kink, and denoted by ϕ K . Another topological solution, calledantikink and written ϕ K = − ϕ K , connects the same two vacua, but in the oppositedirection. As shown in Fig. 2.2, the centre of the kink lies at the (arbitrary) position x = a , and it has a characteristic width l K . From Eq. (2.5) the value M K = 2 √ / ciPost Physics Lecture Notes Submission Figure 2.3: The elliptic sine configuration of Eq. (2.11), for different values of the amplitudeparameter ϕ . The left figure contains solutions with (in order of increasing wave length) ϕ = 0 . ϕ = 0 .
5, and ϕ = 0 .
9. The right figure shows the solutions with ϕ = 0 . ϕ = 0 . a , translations of the kink in space may be interpreted as zero-energy excitations.There is also a stable excitation mode of the kink at non-zero energy [50, 59], which canbe interpreted as an internal or vibrational mode (see appendix A.1). Finally, owing tothe Lorentz invariance of Eq. (2.8), the static kink solution may be boosted to yield adynamical solution in which the kink (or antikink) moves with a constant velocity ϕ ( t, x ) = ± tanh (cid:32) x − a + vt (cid:112) − v ) (cid:33) . (2.10)Here, v is the velocity of the kink measured in units of the speed of light.You may notice that there is one more static solution to the equation of motion (2.8),given by the elliptic sine: ϕ ( t, x ) = ϕ sn( bx, k ) , with k = ϕ − ϕ , b = 1 − ϕ . (2.11)Here, the amplitude ϕ is taken to lie between zero and one. In the limit ϕ →
1, theelliptic sine solution approaches the kink configuration (see Fig. 2.3). A more detailedderivation of Eq. (2.11) and its limiting form is given in appendix A.2.
Exercise 2.1 (Dimensional analysis)
The Lagrangian density defined by Eqs. (2.1)and (2.7) is written in dimensionless form, and can be used to represent the dynamicsof many types of physical fields. For example, we can consider a string or piece ofrope with displacement waves characterised by the local displacement u ( t, x ). In thiscase, the field is a physical quantity with units of length, and the physically relevantLagrangian density can be written as: L = 12 ρ (cid:18) ∂u∂t (cid:19) − τ (cid:18) ∂u∂x (cid:19) − U ( u ) . (2.12)Here, ρ is the mass density of the string, τ is the tension, and U can be interpretedas a potential energy density. The potential can be made to have any shape, forexample by exposing the string to a gravitational force and placing it on top of acurved surface. Notice that the spatial integral over the Lagrangian density L is theLagrangian, with units of energy. (a) Check that all terms in L have the correct units.To arrive at a u -theory, we can place the string on a surface whose height does6 ciPost Physics Lecture Notes Submission not change along the length of the string, but which has a double-well shape in theorthogonal direction: U ( u ) = ρgh (cid:18) − u r + u r (cid:19) . (2.13)Here, the gravitational acceleration is denoted by g , the difference in height betweenthe local maxima and minima in the potential is h , while ± r are the displacementvalues at which the string reaches a bottom of one of the wells. (b) Introduce dimensionless versions of all physical quantities and show that theLagrangian density can be written in the form of Eqs. (2.1) and (2.7).
Exercise 2.2 (The kink solution)
Obtain the kink solution of Eq. (2.9) by inte-grating the BPS equation Eq. (2.6) with the potential given by Eq. (2.7).
Exercise 2.3 (The kink mass)
The value M K = 2 √ / ϕ K can be found using the superpotential W . (a) Starting from Eq. (2.7), show that the superpotential can be written as W = ϕ/ √ − ϕ / √ (b) Since we showed in Exercise 2.2 that the kink is a BPS-saturated solution, weknow its energy is given by the BPS form E BPS = W | ϕ ( x =+ ∞ ) − W | ϕ ( x = −∞ ) .Use this to derive the mass M K = 2 √ / Field configurations with a kink, like ϕ K defined by Eq. (2.10), are exact solutions ofthe equation of motion (2.8) and are therefore stable in the sense that the kink cannotdecay or disappear over time. This changes when we consider solutions with multiplekinks [32, 43, 52]. For example, a field with both a kink moving to the right and anantikink moving to the left is described by: ϕ KK ( t, x ) = tanh x + a − v in t (cid:113) − v in ) − tanh x − a + v in t (cid:113) − v in ) − . (3.1)Here, we chose a frame of reference in which the velocities of the two kinks are preciselyopposite and equal to ± v in , while the initial positions at t = 0 equal ± a and are symmetricaround the origin. For brevity, we will from here on omit the explicit distinction betweenkink and antikink, and refer to the configuration of Eq. (3.1) simply as a field with twokinks. As long as the kinks are far apart, their overlap is negligible and Eq. (3.1) is anexact solution to the equation of motion up to corrections that are exponentially small in l K /a . In other words, the kinks are both stable and both evolve as if they were alone.When the kinks come close together, however, they start to interact. To see why thismust be the case, consider two kinks moving together at high initial speeds, as shown inFig. 3.1 (how to numerically calculate this time evolution is discussed in appendix A.3).The field configuration starts out with a vacuum solution in most of space, given by ϕ = − ciPost Physics Lecture Notes Submission Figure 3.1: The profile of the field ϕ ( t, x ) for different values of t , as a kink and antikinkcollide. While the kinks attempt to pass through one another, kinetic energy is convertedto potential energy in the field and the kinks slow down (three leftmost panels). The kinksthen come to a halt (middle panel) and reverse direction, moving away from each otheragain (two rightmost panels). Some of the initial kineteic energy is radiated away in theform of small ripples (top right panel). For low initial velocities, the outmoving kinks maynot have sufficient kinetic energy to escape to infinity, and a bion is formed instead.at the edges and ϕ = 1 between the kinks. As the kinks move closer together, the middleregion shrinks, until the kinks meet at x = 0. The kinks may then try and move past oneanother, but in doing so they create a region of ϕ = − ϕ = − ϕ -theory, the region between the kinks now harbours potential energy. As this regiongrows, the kinetic energy of the kinks must then decrease, since total energy is conserved.At some point, the kinks halt altogether, reverse their direction of travel, and start movingtowards each other again. The region between the kinks now shrinks, and potential energyis converted back into kinetic energy. This time, when the kinks pass through each other,an intermediate region of ϕ = 1 is created. Since this does not cost any energy to grow,the kinks can continue and move apart indefinitely. The entire process from beginning toend can be interpreted as a bouncing of two kinks against each other. The intuitive picture of two kinks behaving like classical particles, with well-defined po-sitions and speeds, whose only effect on each other comes from the order in which theyappear in the field, works well as long as the kinks are well-separated. During the timesthat they overlap, however, the field configuration is no longer close to a solution of theequation of motion, and the two kinks can decay. This can be seen in Fig. 3.1 as theformation of ripples around the edges of the kink, which propagate outwards as timeevolves. This decay process can be understood intuitively by considering two kinks withzero velocity that are very close together ( l K (cid:29) a ). This situation is very close to thenon-topological vacuum solution with ϕ = − x = 0 is not a solution to the equation of motion (2.8) its energy can dissipateaway to infinity, leaving behind a non-topological vacuum, without any kinks. Notice thatthis same process will also occur if the kinks are initially far apart. However, because theviolations of the equation of motion are exponentially small, it will take an exponentially8 ciPost Physics Lecture Notes Submission Figure 3.2: The behaviour of the field ϕ ( t, x ) at x = 0 as a function of time for differentvalues of the initial velocity. With v in = 0 .
15 (top left panel) the kink and antikink forma bion and keep colliding, separating, and re-approaching. At v in = 0 . v in = 0 . v in . Considering ever lower initial velocities, there must then be apoint at which the colliding kinks can no longer retain sufficient kinetic energy to escapetheir region of overlap. Below this critical initial velocity, v cr , the kinks can still cross,reverse their velocities, cross again, and move apart. However, the kinetic energy is theninsufficient to lift the field values between the kinks away from the stable value ϕ = − The phase diagram in Fig. 3.2 shows the outcomes of the kink collisions as a function oftheir initial velocities. For v in ≥ v cr , the kinks always bounce and escape to infinity. Hav-ing v in < v cr , typically results in the formation of a bion. The value of the critical velocity9 ciPost Physics Lecture Notes Submission separating these two regions can be established numerically. For an initial separation of a = 7, no collisions are observed in the field value at x = 0 up to 300 time steps after theinitial collision, suggesting the value v cr (cid:39) . v in exist that are below the criticalvelocity, but that nonetheless do not result in the formation of a bion. For such values ofthe initial velocity, the two kinks start out behaving as if they form a bion, by colliding,separating, reversing velocities, and colliding again. After a fixed number of collisions,however, the two kinks separate completely and escape to infinity, as shown in the topright panel of Fig. 3.2. The ranges of v in for which this occurs are called 2-bounce escapewindows (or reflection windows), 3-bounce windows, and so on [43].The escape process is made possible by the presence of an internal, vibrational excita-tion mode of the kinks [50, 59], whose detailed derivation is discussed in appendix A.1. Astwo kinks collide, their vibrational modes may be excited, and absorb some of the kineticenergy. If the initial velocity is such that the vibrational motion of the kink profile passesthrough its equilibrium position precisely as the kinks collide for a second (or third, orlater) time, the vibrational energy can be converted back into kinetic energy. This givesthe kinks a boost and allows them to escape to infinity. The internal excitations thuseffectively act as a storage place for kinetic energy, protected against radiative decay. Theresonance condition for efficient conversion of vibrational into kinetic energy is [43]: ω R T = 2 πn + ∆ ⇔ T = ( n + δ ) T R . (3.2)Here, ω R = 2 π/T R , and ω R and T R are the frequency and the period of the internal kinkvibrations (indicated by red dots in the top right panel of Fig. 3.2), and T is the timebetween the final and penultimate collisions of the kinks (blue dots in the top right panelof Fig. 3.2). The integer n indicates the number of internal oscillations the kink undergoesbetween collisions, and ∆ = 2 πδ is a phase shift incurred during the collision process.Eq. (3.2) thus guarantees the time between collisions is equal to an integer multiple ofthe period of internal vibrations, up to a constant phase shift. The resonance conditionhas been numerically checked to both reproduce the analytically derived value of ω R (seeappendix A.1), and to correctly predict the emergence of escape windows, both in ϕ andhigher-order theories [24, 43]. In fact, the arrangement of bounce windows for v in < v cr predicted this way, and observed in simulations, is fractal in nature [33]. Exercise 3.1 (Simulating kink dynamics)
Write a numerical code to calculatethe field configuration ϕ ( t, x ), starting from a configuration at t = 0 with only well-separated kinks and no other excitations. The initial state is defined by the positionsand velocities of the kinks. Refer to appendix A.3 to set up the code. (a) Starting from a configuration with only a single kink, show that it is stable, andpropagates without changing its shape. (b)
Consider the initial configuration with two kinks defined in Eq. (3.1), and sim-ulate one case with colliding kinks, and one in which a bion is formed. (c)
Identify some resonance windows and use Eq. (3.2) to find ω R , the frequency ofinternal kink vibrations. (d) Discuss the limitations of your code, and give a quantitative measure for howwell it performs. 10 ciPost Physics Lecture Notes Submission
Two colliding kinks in ϕ -theory behave in many aspects like colliding hard-core parti-cles with a short-range attractive interaction. If the kinetic energy of these particles issufficiently large, the particles will scatter without being affected by their short-rangedattraction. If the kinetic energy is low enough, however, the attraction dominates, anda bound state is formed. This qualitative observation can be made quantitative by con-structing a low-energy effective theory for the ϕ model, in which only the dynamics ofkinks is taken into account, and all other variations in the field are neglected. Such aneffective theory is known as the Collective Coordinate Approximation, or CCA [52].For the sake of concreteness, consider the case of two colliding kinks. In its simplestform, the CCA consists of constructing a theory in which the only degrees of freedom arethe positions of the kinks [52]. Going to a frame of reference in which the configuration issymmetric around x = 0, there is then only a single degree of freedom a ( t ) describing thepositions of the two kinks. At any point in time, the field configuration associated with agiven value of the collective coordinate a is then approximated by: ϕ KK ( t, x ) = tanh (cid:18) x + a ( t ) √ (cid:19) − tanh (cid:18) x − a ( t ) √ (cid:19) − . (4.1)The effective Lagrangian for the collective coordinate, L CCA ( a, ˙ a ), is obtained by substi-tuting the Ansatz of Eq. (4.1) into the Lagrangian density for the full ϕ -theory (Eqs. (2.1)and (2.7)) and integrating over space. Taking into account the time dependence of a ( t ) inthe calculation of ∂ϕ/∂t , this leads to a Lagrangian of the form: L CCA ( a, ˙ a ) = 12 m ( a ) ˙ a − V ( a ) . (4.2)The position-dependent mass and potential energy in this expression can be written interms of an auxiliary function as: m ( a ) = I + ( a ) ,V ( a ) = 12 I − ( a ) + 14 (cid:90) + ∞−∞ (1 − ϕ KK ) dx, with I ± ( a ) = 2 M K ± (cid:90) + ∞−∞ dx cosh (cid:16) ( x + a ) / √ (cid:17) cosh (cid:16) ( x − a ) / √ (cid:17) . (4.3)At large separation, the integral in the auxiliary functions vanishes, and we find that themass parameter m ( a ) and the effective potential V ( a ) both approach 2 M K , the mass of twoisolated static kinks. For general values of a , the integrals can be evaluated numerically,and yield the functional form for the effective potential plotted in Fig. 4.1.The effective Lagragian of Eq. (4.2) has reduced the initial relativistic field theory to anon-relativistic description of just a single point particle in an external potential. This canbe easily solved by considering the Euler-Lagrange equation for the collective coordinate: ddt ∂L CCA ∂a − ∂L CCA ∂a = 0 ⇒ m ¨ a + 12 dmda ˙ a + dVda = 0 . (4.4)The velocity-dependent term in the final line may be interpreted as a frictional forceacting on the effective particle. This arises due the change of the effective mass m ( a )11 ciPost Physics Lecture Notes Submission Figure 4.1: The effective potential V in the collective coordinate approximation, as afunction of the separation a between kinks. The dashed line at V = 2 M K denotes theasymptotic value of the potential. The green arrow on top schematically indicates aninelastic reflection of the kinks, while the red arrow tending to the bottom of the potentialwell shows the formation of a quasi-long-lived bion.with position, which is significant only when the two kinks overlap. It is thus a directmanifestation of the two-kink configuration not being a stable solution to the equation ofmotion (2.8) for the full field theory.Based on the shape of the effective potential V ( a ) shown in Fig. 4.1, we can distinguishtwo qualitatively different types of dynamics. If the initial energy of the particle describedby a ( t ) is sufficiently high, it can cross the potential well around the origin a = 0, climbup the potential barrier, which increases linearly with | a | at negative a , then turn aroundand return through the well before escaping to infinity. This describes the bouncing oftwo kinks. If the initial energy is low enough, however, the particle may loose sufficientenergy during its crossing of the potential well to become stuck. It will then oscillate backand forth across a = 0 with slowly decreasing amplitude. This describes the formation ofa quasi-long-lived bion.In many cases, numerically solving the equation of motion (4.4) for the effective modelyields good agreement with the much more involved numerical solutions for the dynamicsof the full field-theoretical model, both in ϕ -theory and its higher order generalisationslike the ϕ -model [23]. The Ansatz of Eq. (4.1) does neglect the Lorentz contraction ofmoving kinks, so that the CCA fails to reproduce relativistic effects that may appearat high velocities. Another limitation of the CCA is that it cannot describe the escapewindows in the phase diagram of Fig. 3.2 because the Ansatz does not include a degreeof freedom to describe the internal excitation modes of the kinks. Extensions of the CCAthat include resonance dynamics are possible [55]. Exercise 4.1 (Simulating effective dynamics)
Write a numerical code that usesthe collective coordinate approximation to calculate the field configuration ϕ ( t, x ),starting from a configuration at t = 0 with two well-separated kinks. (a) Consider the initial configuration with two kinks defined in Eq. (3.1), and sim-ulate one case with colliding kinks, and one in which a bion is formed. (b)
Compare your results to those in part (b) of Exercise 3.1. (c)
Adjust your initial conditions until you start seeing the limitations of the CCA.Quantify the error made in the CCA, as compared to Exercise 3.1.12 ciPost Physics Lecture Notes Submission
Figure 5.1: The field ϕ ( t, x ) obtained by gluing together half a kink, half a period of anelliptic sine, and half an antikink. Both the spatial profile at t = 0 (left panel), and theits time evolutions (shown at x = 0 in the right panel) closely approximate that of thequasi-long-lived bion. For these figures, the parameter values ϕ = 0 . δϕ = − . The qausi-long-lived bion that we described as a damped oscillator in the previous section,can also be thought of as a combination of parts of three static solutions to the equationof motion. Considering the shape of the field configuration ϕ ( t, x ) for a bion at someparticular time t , you may notice that it looks like half a kink and half an antikink, gluedtogether by half a period of the elliptic sine, as shown in Fig. 5.1. In fact, we can makethis more precise by constructing a field configuration ϕ ( t, x ) starting from the staticsolution in Eq. (2.11). Taking λ to be the period of the elliptic sine, we keep only halfa period, within the range − λ/ ≤ x ≤ λ/
4. We then attach half a kink to the left, for −∞ < x < λ/
4, and half an antikink to the right, for λ/ < x < ∞ . The positions a = − λ/ a = λ/ ϕ , and hence λ , has to be chosen such thatthe three static solutions glue together smoothly at the connection points x = ± λ/
4. Wecan then consider this configuration of glued static solutions as an Ansatz for the bionfield configuration, and calculate its time evolution. This has the advantage of the biondescription being independent of the particular process that led to its formations, be ittwo-kink collisions or otherwise [66]. Notice that in the numerical simulation of the biondynamics starting from this Ansatz for ϕ ( t, x ), the field has to be defied both at t = 0and t = ∆ t . This can be done by choosing the value of ϕ to decrease by a small amount δϕ in the second time step. The result of the ensuing time evolution is shown in Fig. 5.1(right panel), and closely resembles that found in the kink collisions described in Sec. 3.In particular, the dynamics describes a quasi-long-lived state, or bion.The strategy of gluing together static solutions to find an appropriate Ansatz or initialstate may be applied more generally. For example, dynamical kink-kink pairs and tritonsolutions (kink-antikink-kink) were described this way [67]. The latter may be used as anAnsatz for obtaining the quasi-long-lived solution called a wobbling kink [64, 68]. The dynamics of kink-antikink collisions arose from taking two exact solutions to theequation of motion in Eq. (2.2), the kink and antikink, and combining them into a singleinitial state that is not an exact solution. Similarly interesting dynamics may be obtainedby taking an exact solution to Eq. (2.2), and using it as in initial state whose dynamics is13 ciPost Physics Lecture Notes Submission determined by a slightly different equation of motion. This is the situation encounteredin the description of kinks interacting with impurities [35, 69–71].The ϕ -theory often arises as a low-energy effective description of some more micro-scopic model. Imperfections at the microscopic level may then result in local variations ofthe potential in the effective theory [72–74]. For example, in the context of waves prop-agating through a solid medium, impurities or defects embedded in the crystal structuremay cause impinging waves to scatter or form bound states. To describe such processes,we consider a modification of the potential of Eq. (2.7) by taking:14 (1 − ϕ ) −→
14 (1 − ϕ ) (1 − (cid:15)δ ( x − x )) . (6.1)Here, (cid:15) is the strength of the Dirac delta impurity located at x = x . For (cid:15) = 0, theimpurity is absent, and the potential equals that of the standard ϕ -theory. For (cid:15) < (cid:15) > ϕ crossing zero is enhanced at x = x . The potential well on the otherhand lowers the local energy cost of a kink, and will therefore attract it.To numerically simulate the dynamics of kinks encountering an impurity, we start fromthe modified equation of motion: ϕ tt − ϕ xx + ( ϕ − ϕ )(1 − (cid:15)δ ( x − x )) = 0 . (6.2)The Dirac delta function may be approximated on a discrete lattice either by a Kroneckerdelta with height equal to the inverse of a coordinate step [35,75], or by a Gaussian profileof the form [65]: δ ( x − x ) −→ σ √ π exp (cid:34) − (cid:18) x − x σ (cid:19) (cid:35) . (6.3)Here, σ is the spatial width over which the impurity affects the potential. It may eitherbe used as a free parameter in the modelling of realistic impurity potentials, or be fixedsuch that the area under the Gaussian profile equals one [65].Since the dynamics and physics of kink-impurity interactions closely resemble thoseof kink-antikink collisions, we give only a brief overview of the possible resulting states.More detailed descriptions can be found in Refs. [35, 65, 69–71, 75]. For the sake of beingspecific, we consider a single kink described by Eq. (2.10), starting from a = 6 and movingtowards an impurity located at x = 0, with fixed impurity strength | (cid:15) | = 0 .
5. We thendescribe the dynamics for various values of the initial velocity. Fig. 6.1 shows a sketch ofthe corresponding phase diagram.For an attractive impurity with (cid:15) = 0 .
5, a kink with sufficiently high initial velocitywill traverse the impurity and continue propagating on the other side. In crossing theimpurity location, it does loose some of its kinetic energy. Some of that energy may beradiated away in the form of small ripples, and some of it is absorbed in an internal,quasi-long-lived excitation mode of the impurity [35, 65].As the initial velocity of the kink is reduced, at some point a critical velocity v (cid:48) cr willbe encountered, below which the loss of kinetic energy is so large that kinks can no longertraverse the impurity. Typically, the kink is then captured by the impurity, and oscillatesaround it in a manner similar to the oscillations of kinks and antikinks in a bion. This toois a quasi-long-lived solution to the equations of motion, in the sense that the amplitudeof oscillations will slowly decrease to zero. Within a suitably defined CCA, the two typesof processes, kinks crossing an impurity and kinks being captured, can again be describedas an effective particle with variable mass subject to a resistive force [35].14 ciPost Physics Lecture Notes Submission Figure 6.1: Phase diagrams (top row) and example evolutions (bottom row) for kink-impurity collisions with varying initial velocity v in [65]. The left side involves a repulsiveimpurity of strength (cid:15) = − .
5, and the right side an attractive impurity with (cid:15) = 0 .
5. Theexample evolutions on the left show the reflection ( v in = 0 .
4) and transmission ( v in = 0 . v in = 0 . v in = 0 . v in = 0 .
5) by the attractive impurity.For specific ranges of initial velocities below the critical value, v in < v (cid:48) cr , the kink maybe scattered from an attractive impurity after a finite number of oscillations around it.Analogous to the bounce windows in two-kink collisions, this process may be understoodas a resonance between the oscillation time of the kink around the impurity, and thefrequency of internal vibration modes [35].For a repulsive impurity with (cid:15) = − .
5, kinks of sufficiently high initial velocity aretransmitted with some loss of kinetic energy. At low enough initial velocities, kinks againcannot overcome the impurity potential, and are reflected instead. The critical velocityseparating the two behaviours, however, does not have the same value as that associatedwith the attractive impurity, and there are no bounce windows.
Exercise 6.1 (Simulating kink-impurity interactions)
Write a numerical codeto calculate the field configuration ϕ ( t, x ), starting from a configuration at t = 0 witha single kink and a single impurity. (a) Consider an initial configuration with the kink at a = −
6, and a repulsiveimpurity of strength (cid:15) = − . x = 0. Simulate its dynamics for severalvalues of the initial velocity. Include at least one velocity for which the kinktraverses the impurity, and one for which it is reflected. (b) Consider an initial configuration with the kink at a = −
6, and an attractiveimpurity of strength (cid:15) = 0 . x = 0. Simulate its dynamics for several valuesof the initial velocity. Include at least one velocity for which the kink traversesthe impurity, and one for which a bound state is formed. (c) Identify some resonance windows and use Eq. (3.2) to find the frequencies ofthe involved internal excitation modes.15 ciPost Physics Lecture Notes Submission
In these lecture notes, we encountered a wide array of qualitatively different types ofdynamics involving kinks in ϕ -theory. Using straightforward numerical calculations andeffective models, we were able to describe effects ranging from the existence of topologicallydistinct static solutions to the scattering of kinks, and from the formation of quasi-long-lived states to resonant interactions.This entire range of phenomena arises in the simplest possible classical field theory,which itself is an effective low-energy description for many microscopic models in both solidstate and high-energy physics. We hope that advanced undergraduate students studyingthese lecture notes will come to appreciate both the universality of the rich phenomenologyencountered within ϕ -theory, and the fact that they can qualitatively understand anddescribe all of it. It may then serve as a first introduction to the powerful and ubiquitousidea of emergence in physics. Acknowledgements
This work was performed in the Delta Institute for TheoreticalPhysics (DITP) consortium, a program of the Netherlands Organization for Scientific Re-search (NWO), funded by the Dutch Ministry of Education, Culture and Science (OCW).16 ciPost Physics Lecture Notes Submission
A Appendices
A.1 Internal excitation mode
To describe internal excitations of any static configuration, we can consider small pertur-bations on top of the static solution [50]: ϕ ( t, x ) = ϕ s ( x ) + δϕ ( t, x ) . (A.1)Here, ϕ s is a time-independent field satisfying the equation of motion (2.2). The pertur-bation δϕ ( t, x ) may have any shape, but we assume its amplitude | δϕ | (cid:28) | ϕ s | to be smallcompared to that of the static field.Substituting this equation for the field ϕ ( t, x ) into the equation of motion (2.2), andkeeping only terms up to linear order in the perturbation δϕ ( t, x ), yields the condition: ∂ δϕ∂t − ∂ δϕ∂x − δϕ + 3 ϕ s δϕ = 0 . (A.2)This equation can be solved using a separation of variables, by substituting the Ansatz: δϕ ( t, x ) = (cid:88) i ψ i ( x ) cos( ω i t + θ i ) . (A.3)The temporal phase differences θ i between distinct excitation modes will be set to zerofrom here on without loss of generality. Inserting the Ansatz into the linearised equationof motion results in a set of Schr¨odinger-like equations, ˆ Hδψ i = E i δψ i , with:ˆ H = − d dx + 3 ϕ s − ,E i = ω i . (A.4)Thus, the original problem of solving the linearised equation of motion can be restated asthe search for all eigenvalues ω i and eigenstates δψ i ( x ) of the linear operator ˆ H . Noticehowever, that these eigenfunctions accurately describe the excitation modes only in thelimit of small amplitude, where ignoring higher order terms in Eq. (A.2) is justified.The exponential dependence of δϕ on ω i in Eq. (A.3) implies that any eigenstate of theHamiltonian with eigenvalue ω i < ϕ s . If the initialconfiguration ϕ s is a stable, static solution to the equation of motion, such negative-energyexcitation modes should not occur.For Lagrangian densities with translational symmetry, there always exists a zero-energyexcitation that corresponds to a rigid translation of the initial state [56]. To see thisexplicitly for the ϕ -theory, first notice that the Hamiltonian in Eq. (A.4) can be writtenin terms of the potential given by Eq. (2.7) as:ˆ H = − ∂ ∂x + ∂ U∂ϕ . (A.5)Then, compare this to the spatial derivative of the static equation of motion, which canbe written as: − d ϕ (cid:48) dx + d Udϕ · ϕ (cid:48) = 0 . (A.6)Here, ϕ (cid:48) = ∂ϕ/∂x is the spatial derivative of the static field ϕ ( x ). Any solution ϕ s to theequation of motion, is also a solution to Eq. (A.6). On the other hand, this equation may17 ciPost Physics Lecture Notes Submission Figure A.1: The spatial profiles of the field ϕ ( t, x ) for kinks with the internal excitationsof Eq. (A.8). The dashed blue lines show a static kink solution. The red line in the leftpanel adds a zero-energy translational excitation to the kink, while the red line in theright panel shows the excitation of the internal vibrational mode of the kink.be interpreted as the Schr¨odinger equation ˆ Hϕ (cid:48) = Eϕ (cid:48) with zero energy and ˆ H defined byEq. (A.5). We thus find that for every solution ϕ s to the equation of motion, there existsa zero-energy excitation δϕ ( t, x ) ∝ ϕ (cid:48) s .Focusing now on the specific case of kinks in ϕ -theory, consider ϕ s to be the statickink configuration of Eq. (2.9), with a = 0. The Hamiltonian whose eigenfunctions arethe excitation modes of the kink, then becomes:ˆ H = − d dx − (cid:0) x/ √ (cid:1) + 2 . (A.7)The eigenvalue equation for this specific Hamiltonian happens to be a well-known problemwith an analytic solution [59]. There are two eigenfunctions, corresponding to two distinctexcitation modes of the kink: δψ ( x ) = (cid:18) √ (cid:19) / (cid:0) x/ √ (cid:1) with ω = 0 ,δψ ( x ) = (cid:18) √ (cid:19) / sinh (cid:0) x/ √ (cid:1) cosh (cid:0) x/ √ (cid:1) with ω = (cid:112) / . (A.8)The zero-energy solution δψ is proportional to ϕ (cid:48) s , the spatial derivative of the kinksolution. We thus recognise it as the translational mode that shifts the kink along thespatial axis. The mode at non-zero excitation energy, δψ , corresponds to vibrations ofthe kink around its equilibrium shape, which do not affect the location at which the fieldvalue crosses zero. Both excitation modes are plotted in Fig. A.1. The vibrational modeat non-zero energy can be excited during kink collisions, and enables the formation ofresonance windows. Exercise A.1 (Kink excitations)
Use Ref. [59] to analytically obtain the eigenval-ues and eigenfunctions of the Hamiltonian operator defined in Eq. (A.7).
A.2 Static solutions
To analytically find static solutions to the equation of motion (2.8) in ϕ -theory, we assumethat there exists a stationary solution ϕ ( x ) that must then obey the equation of motionwith vanishing time derivatives: d ϕdx = ϕ − ϕ. (A.9)18 ciPost Physics Lecture Notes Submission Figure A.2: Sketch of the potential V ( X ) in Eq. (A.10). The green arrow schematicallyindicates an oscillating solution. Using the mapping between X ( T ) and ϕ ( x ), this corre-sponds to the static elliptic sine solution in Eq. (A.16) for the ϕ -theory.To get some feeling for this equation, notice that it is reminiscent of Newton’s second lawof motion, − dV /dX = m d X/dT , where the role of time T is played by the coordinate x , and the position X corresponds the field value ϕ . In other words, the configuration ϕ ( x ) may be read as the trajectory X ( T ) of a particle with mass m = 1, subject to thepotential: V ( X ) = − (cid:0) − X (cid:1) . (A.10)This potential is displayed in Fig. A.2. For initial conditions X (0) = X < X max and dX/dT = 0, the particle motion will consist of periodic oscillations between X and − X . These stable trajectories all correspond to static solutions of the ϕ -theory, aftersubstituting back X ( T ) → ϕ ( x ).We can find an analytic expression for the static field configurations by first integratingboth sides of Eq. (A.9): d ϕdx + ϕ − ϕ = 0 , (cid:90) dx dϕdx (cid:18) d ϕdx + ϕ − ϕ (cid:19) = (cid:90) dx dϕdx (0) , (cid:18) dϕdx (cid:19) + 12 ϕ − ϕ = C. (A.11)Here, C is a constant of integration. Using the intuition based on the particle trajectoryanalogy, we can choose boundary conditions such that ϕ = ϕ at the point where dϕ/dx =0. This yields C = ϕ / − ϕ /
4, and suggests separating the amplitude and spatialdependence of the field, by writing ϕ ( x ) = ϕ χ ( x ), with | χ ( x ) | ≤
1. In terms of these newvariables, Eq. (A.11) becomes: (cid:18) dχdx (cid:19) = 1 − χ − ϕ (cid:0) − χ (cid:1) ≡ b (1 − χ ) (cid:0) − k χ (cid:1) . (A.12)In the final line, we introduced the definitions: k = ϕ − ϕ ,b = 1 − ϕ . (A.13)19 ciPost Physics Lecture Notes Submission Because we know from the particle trajectory analogy that there will be no stable solutionsfor | ϕ | >
1, we always have 0 ≤ k ≤
1, and 1 / ≤ b ≤
1. Integrating Eq. (A.12) nowyields: (cid:18) dχdx (cid:19) = b (1 − χ ) (cid:0) − k χ (cid:1) , (cid:90) x (cid:48) dx | dχ/dx | (cid:112) (1 − χ ) (1 − k χ ) = (cid:90) x (cid:48) dx b, (cid:90) χ ( x (cid:48) ) χ (0) dχ (cid:112) (1 − χ ) (1 − k χ ) = bx (cid:48) . (A.14)In the final line, we assumed that dχ/dx is positive over the integration interval. It iseasily checked that the final solution we obtain below will be valid also along intervalswhere dχ/dx is negative, and that these intervals connect smoothly. Choosing χ (0) = 0without loss of generality, and making the substitutions χ ≡ sin( ψ ) and C (cid:48) ≡ arcsin χ ( x (cid:48) ),Eq. (A.14) reduces to: (cid:90) C (cid:48) dψ (cid:113)(cid:0) − k sin ψ (cid:1) = bx (cid:48) . (A.15)The integral on the left hand side is a standard integral, known as the incomplete ellipticintegral of the first, and is often denoted by F( C (cid:48) , k ) [76,77]. Using the known properties ofthe elliptic integral, Eq. (A.15) can be inverted, and yields the solution χ ( x (cid:48) ) = sn( bx (cid:48) , k ),with sn( bx, k ) the elliptic sine [76, 77]. The static solutions to the equation of motion for ϕ -theory can thus finally be written as: ϕ s ( x ) = ϕ sn( bx, k ) . (A.16)For ϕ <
1, these solutions are the elliptic sines shown in Fig. 2.3 of the main text.For very small values of ϕ , the elliptic sine approaches sin( x ), and the static solutioncorresponds to small sinusoidal oscillations around the unstable homogeneous solution ϕ = 0, which it approaches in the limit ϕ → ϕ →
1, notice that the period of the staticsolution ϕ s ( x ) is given by [77]: T = 4 F( π/ , k ) (cid:112) − . ϕ . (A.17)The period is plotted as a function of ϕ in Fig. A.3. As ϕ approaches one, the perioddiverges. In terms of the particle trajectory analogue, this situation corresponds to themassive particle starting out precisely in an unstable equilibrium state on one of thepotential maxima. If the particle at some point (spontaneously, or due to an infinitesimalperturbation) leaves the unstable maximum, it will traverse the minimum in the potentialand ends up at the opposite maximum at infinite time.In terms of the field ϕ s ( x ), the same limiting behaviour corresponds precisely to akink configuration. Taking the limit ϕ → k → b → / √
2, we see (cid:90) χ ( x (cid:48) )0 dχ − χ = x (cid:48) √ , (cid:20)
12 ln (cid:18) χ − χ (cid:19)(cid:21) χ ( x (cid:48) )0 = x (cid:48) √ ,χ ( x (cid:48) ) = tanh (cid:16) x (cid:48) / √ (cid:17) . (A.18)20 ciPost Physics Lecture Notes Submission Figure A.3: The dependence of the elliptic sine period T in Eq. (A.17) on the value of theamplitude parameter ϕ . The period tends to infinity as the amplitude approaches one.Since we took ϕ →
1, the final static solution ϕ s equals χ , and we recover the equationfor a kink, Eq. (2.9), centered at the origin. Exercise A.2 (Static solutions)
Reproduce the derivations of the elliptic sine andkink solutions in this appendix.
A.3 Numerical method
We would like to numerically integrate the equation of motion (2.2) and determine thetime evolution of any given initial field configuration. To do so, we are first of all forcedby the finite computing power of any numerical code to restrict our attention to a finiteinterval of space and time. We thus consider only positions − x max ≤ x ≤ x max , and times0 ≤ t ≤ t max . Furthermore, we necessarily have to consider a discrete grid within thiscontinuous space-time interval. We choose a simple ( N + 1) by ( M + 1) rectangular grid,with step sizes ∆ t = t max /N and ∆ x = 2 x max /M . Points on this grid can be denoted by( n, j ) with 0 ≤ n ≤ N and − M/ ≤ j ≤ M/
2. These coincide with the continuous space-time coordinates at t n = n ∆ t and x j = j ∆ x . To compute derivatives on the discretegrid, we employ a method of finite differences: ∂ϕ∂t (cid:12)(cid:12)(cid:12)(cid:12) ( t,x )=( t n ,x j ) ≈ ϕ nj − ϕ n − j ∆ t , ∂ ϕ∂t (cid:12)(cid:12)(cid:12)(cid:12) ( t,x )=( t n ,x j ) ≈ ϕ n +1 j − ϕ nj + ϕ n − j ∆ t ,∂ϕ∂x (cid:12)(cid:12)(cid:12)(cid:12) ( t,x )=( t n ,x j ) ≈ ϕ nj − ϕ nj − ∆ x , ∂ ϕ∂x (cid:12)(cid:12)(cid:12)(cid:12) ( t,x )=( t n ,x j ) ≈ ϕ nj +1 − ϕ nj + ϕ nj − ∆ x . (A.19)The equation of motion for field values ϕ nj on the discrete grid can now be written as: ϕ n +1 j = 2 ϕ nj − ϕ n − j + ∆ t ∆ x ( ϕ nj +1 − ϕ nj + ϕ nj − ) − ∆ t dUdϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ nj . (A.20)The initial conditions required to solve the complete dynamics then consist of the fieldconfigurations ϕ j and ϕ j at the initial two time slices. This can be thought of as specifyinga position (amplitude) and velocity (rate of change) for all classical degrees of freedom (thefield values) in the classical field theory. In general, the numerical simulation is expectedto be stable on the chosen rectangular grid as long as the temporal step size is smallerthan the spatial one [78].A peculiarity of the finite differences used to compute spatial derivatives, is that theycannot be reliably computed at the edge of space because no information is available aboutthe field values at | j | = M/ ciPost Physics Lecture Notes Submission field that reaches the edges of space will be reflected by the boundaries. To avoid seeingthese unphysical reflections in the final simulated dynamics, the output of the code shouldonly contain field values at positions | j | < M/ − N . In this way, even the mistake madein the spatial derivative at time step n = 0, or equivalently the wave that is reflected atthe edge of space at t = 0, has no time to propagate into the observed spatial interval.To ensure that the numerical code functions correctly and that the inherent numericalerror of the calculation does not get out of hand, several consistency checks can be con-sidered. First of all, static solutions like ϕ = ± | j | < M/ − N . This can be checkedby computing the change in total energy, which should be close to zero:∆ E tot ( n (cid:48) ) = E ( n = n (cid:48) ) − E ( n = 0) − n (cid:48) (cid:88) n =0 (cid:20) ∂ϕ∂t ∂ϕ∂x (cid:21) M/ − Nj = − M/ N with E ( n ) = M/ − N (cid:88) j = − M/ N (cid:32) (cid:18) ∂ϕ∂t (cid:19) + 12 (cid:18) ∂ϕ∂x (cid:19) + 14 (cid:0) − ϕ (cid:1) (cid:33) . (A.21)In these expressions, the partial derivatives should be interpreted in terms of the finitedifferences defined in Eq. (A.19). The conservation of energy (∆ E tot (cid:39)
0) can be checkedat every time step in the simulation, and provides a quantitative measure for the accuracyof the simulation.The final consistency check is to ensure that the field dynamics obtained in the numer-ical calculation does not change when decreasing the step sizes ∆ t and ∆ x . The actualvalues of the step sizes can then be chosen empirically such that they ensure a desirablebalance between having a manageable run time for the numerical code, and an adequatelevel of accuracy in the results. To quantify the latter, we should use both the indepen-dence of the result from the value of step sizes and the conservation of total energy. Tosimulate two-kink collisions in ϕ -theory, it turns out that ∆ t = 0 .
009 and ∆ x = 0 .
01 arereasonable values [43]. To simulate kink-impurity interactions the values ∆ t = 0 .
01 and∆ x = 0 .
02 seem more appropriate. 22 ciPost Physics Lecture Notes Submission
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