Vortices and magnetic impurities

Ginzburg-Landau vortices in superconductors attract or repel depending on whether the value of the coupling constant λ is less than 1 or larger than 1. At critical coupling λ = 1, it was previously observed that a strongly localised magnetic impurity behaves very similarly to a vortex. This remains true for axially symmetric configurations away from critical coupling. In particular, a delta function impurity of a suitable strength is related to a vortex configuration without impurity by singular gauge transformation. However, the interaction of vortices and impurities is more subtle and depends not only on the coupling constant λ and the impurity strength, but also on how broad the impurity is. Furthermore, the interaction typically depends on the distance and may be attractive at short distances and repulsive at long distances. Numerical simulations confirm moduli space approximation results for the scattering of one and two vortices with an impurity. However, a double vortex will split up when scattering with an impurity, and the direction of the split depends on the sign of the impurity. ∗jea29@kentforlife.net †S.Krusch@kent.ac.uk 1 ar X iv :1 80 8. 07 44 1v 1 [ he pth ] 2 2 A ug 2 01 8


I. INTRODUCTION
The Ginzburg-Landau model of superconductivity describes vortices in superconductors. In this paper, we investigate the scattering of vortices in the presence of magnetic impurities. Our aim is to provide an initial numerical study of such scattering processes for different choices of impurity. Previous studies of vortices in the presence of magnetic impurities have concentrated on the case of critical coupling. We carry out numerical simulations for critically coupled vortices, but also consider other values of the coupling constant λ.
Jaffe and Taubes showed in [1] that N vortices at critical coupling can be described by a 2N dimensional moduli space. The relativistic dynamics of vortices can be approximated by the moduli space approximation [2] which can be rigorously justified [3,4]. For small velocities, vortices move according to geodesic flow on the moduli space where the metric is induced by the kinetic energy. Samols found an implicit formula for the metric [5]. While no explicit solutions for vortices are known in flat space Witten noticed the vortex equation are integrable in hyperbolic space [6]. This allowed Strachan to evaluate the metric for two centred vortices [7] and some more general metrics have been found subsequently [8,9]. There are also interesting non-Abelian vortices [10,11]. Baptista showed in [12] how the moduli space metric can be calculated following Samol's method. The moduli space metric of non-Abelian vortices and their dynamics were discussed in [13,14].
The study of vortices in the presence of impurities has attracted interest in recent years. A notable example is the work of Tong and Wong in Ref. [15] concerning BPS vortices in the presence of electric and magnetic impurities. They argued that there still exists a moduli space of solitons after the addition of electric and magnetic impurities, and discussed the manner in which the moduli space dynamics is affected by each type of impurity. Ref. [15] has motivated several studies of vortices in product Abelian gauge theories which can be related to vortices in the presence of magnetic impurities. For example, existence theorems for solutions of vortices and anti-vortices in such models have been proven in Refs. [16,17], and similar ideas have been explored in an Abelian Chern-Simons-Higgs model in Ref. [18].
We have been particularly motivated by Ref. [19], in which the authors investigated the dynamics of vortices at critical coupling with magnetic impurities. They obtained solutions in flat space by numerically solving the Bogomolny equation for localised, axially symmetric impurities, and their numerics confirms the existence of a moduli space of vortex solutions. They also discussed vortices in hyperbolic space, where they calculated exact solutions and moduli space metrics for a delta function impurity. We will extend their results for localised axially symmetric impurities in flat space by carrying out numerical simulations of the full field equations. This enables us to consider vortices away from critical coupling and to numerically simulate vortex dynamics.
As yet, there has not been a comprehensive numerical study of the scattering of vortices with magnetic impurities. Previous numerical investigations into the scattering of two vortices in the Ginzburg-Landau model in Refs. [20][21][22][23] have explored the relationship between the scattering angle and impact parameter and any dependence of the scattering behaviour on the initial velocity given to the vortices. We carry out similar calculations for vortices scattering with magnetic impurities.
The paper begins with an introduction to vortices in the presence of magnetic impurities. We discuss the Bogomolny bound satisfied by vortices at critical coupling and obtain vacuum solutions for different values of the coupling constant λ and for different impurities. The energy of static vortex solutions in the presence of magnetic impurities can be evaluated for axially symmetric configuration which allows us to calculate a binding energy for vortices and impurities.
We also investigate the asymptotic interaction between vortices and impurities. A complete picture emerges when we calculate the energy as a function of the separation between vortex and impurities. We carry out numerical simulations of the scattering of vortices at critical coupling with magnetic impurities and analyse our findings. Finally we summarise the results and note opportunities for further work.

II. VORTICES WITH MAGNETIC IMPURITIES
The Lagrangian for gauged vortices in the Ginzburg-Landau model is Here, φ is a complex scalar field, and a µ is the U(1) gauge field. The field tensor f µν and the covariant derivative D µ φ are given by f µν = ∂ µ a ν − ∂ ν a µ and D µ φ = ∂ µ φ − ia µ φ.
To ensure finite energy, vortex solutions satisfy D µ φ → 0 and |φ| → 1 as |x| → ∞. The Lagrangian The coupling constant λ is a real parameter, which distinguishes between Type I and Type II superconductivity. If λ < 1, then we model Type I superconductivity when vortices attract, and if λ > 1 then we model Type II superconductivity when vortices repel. The value λ = 1 separating the two regimes is known as critical coupling.
We consider the deformation of the Lagrangian (1) that was proposed in Ref. [15] to include magnetic impurities, where σ is a fixed, static source term for the magnetic field B = f 12 . We restrict our attention to localised, axially symmetric impurities of the form σ(x, y) = ce −d(x 2 +y 2 ) , where c, d ∈ R, and d > 0.
The topological charge is an integer N giving the net number of vortices in a solution. It can be written in terms of the magnetic field as We begin by obtaining vacuum solutions in this model. The potential energy for vortices in the presence of magnetic impurities is given by At critical coupling, it is possible to obtain a Bogomolny bound on the energy. We first complete the square on the integrand of (6), to find The final term above integrates to zero. Using this fact in combination with the expression for the topological charge (5) we see that leading us to the Bogomolny bound Note that the same bound is satisfied by vortices in the original model (1), and the only difference in the equations is the additional σ in the second Bogomolny equation. Other possibilities of modifying the vortex equations while keeping the BPS structure are discussed in Refs. [24][25][26][27].
We can derive a lower bound on the energy when λ 1 in a similar way. We rewrite the integrand of (6) as Integrating (11) produces the same result as before, but with one additional term. The energy is bounded from below by When λ ≥ 1, the additional term is non-negative, and we have the bound (9) which is saturated at critical coupling. For the more general bound (12) to be meaningful, we must argue that (1 + σ − φφ) 2 d 2 x is bounded. This is true as long as σ and φ are non-singular and decay sufficiently quickly at infinity. We consider impurities of the form σ(x, y) = ce −d(x 2 +y 2 ) , which all decay very fast, and the asymptotics derived in Sect. II D ensures that φ also decays fast enough. We will also discuss the limit in which σ approaches a delta function. In this case, a square of a delta function is introduced into the Lagrangian (4), which is not defined. However, as noted in Ref. [19], it does make sense to substitute a delta function for σ in the equations of motion, and we can consider this to be a limit of impurities for which the energy is well-defined.

A. Symmetric solutions
We first study the effect of the impurity on the vacuum configuration. To simplify the problem, we assume circular symmetry which allows us to convert to polar coordinates r, θ, and fix the radial gauge a r = 0. Then we have φ(r, θ) = φ(r)e iNθ , and a θ (r, θ) = a θ (r), and will solve for the real profile functions φ(r) and a θ (r). For fields of this form, the energy (6) becomes To calculate φ(r) and a θ (r), we solve the reduced field equations via a finite difference method on grids typically of size 2001 with spacing ∆r = 0.01, subject to We consider impurities of the form σ(r) = ce −dr 2 , where c, d ∈ R, and d > 0.
Solutions for λ = 0.5 are shown in blue and those for λ = 1.5 are shown in red. Although the profile functions were calculated over r ∈ [0, 20], to highlight the more interesting features of the solutions we only display the ranges r ∈ [0, 5] for the profile functions and r ∈ [0, 3] for the energy density and magnetic field. We see that the effect of the impurity is localised, with the fields taking their usual vacuum values away from the impurity. In Ref. [19] it was noted that for vortices at critical coupling φ(0) → 0 as c → −∞, and φ(0) → ∞ as c → +∞. We have observed the same behaviour away from critical coupling, and this has been tested over a much greater range of c than those shown here. In Fig. 1(b) we see that a θ (r) ≥ 0 for c > 0 and a θ ≤ 0 for c < 0.
The energy density plot shows that there is a region of negative energy density, which is not the case in the absence of a magnetic impurity. We previously saw that it is possible to derive a lower bound on the energy for any λ as (12). This bound saturated when λ = 1 and can be negative for For c > 0, the energy density for λ = 0.5 is similar in shape to that for λ = 1.5 and the same impurity, however for c < 0 there is a significant difference in the energy density depending on λ.
For λ = 0.5, the region of negative energy density becomes more significant, which is especially clear for σ(r) = −8e −2r 2 . By contrast, for λ = 1.5, the region of negative energy density becomes much less significant. For σ(r) = −8e −2r 2 , the energy density is positive at the origin for λ = 1.5, whereas all other solutions have negative energy density at the origin. We see from Fig. 1(d) that the value of λ does not significantly alter the magnetic field B(r). Furthermore, changing the sign of c roughly reverses the sign of B(r).
We examine more carefully the effect of varying the coupling constant λ on the vacuum solutions in Fig. 2. This displays the profile functions, energy density and magnetic field for the vacuum solutions with σ(r) = 4e −r 2 , and λ = 0.5, 1.0, 1.5. We see in Fig. 2(a) that the value of φ(0) increases with increasing λ. Similarly, Fig. 2(b) indicates that the maximum of a θ (r) decreases with increasing λ. The energy density, as shown in Fig. 2(c), has a more significant region of negative energy for λ < 1, and for such values of the coupling constant, the total vacuum energy is negative. For λ > 1, the positive region of the energy density is more significant, and the total vacuum energy is positive. At critical coupling our numerical methods evaluate the energy as zero to five decimal places. Although we see from Fig. 2(d) that there are some differences in the magnetic field depending on λ, the value of the topological charge is unaffected, as we would expect.

B. Delta function impurities
For c < 0, the authors of Ref. [19] applied a singular gauge transformation to show in the case of critical coupling that a delta function impurity of the form −4παδ(z) for α ∈ N "behaves" like an (N + α)-vortex solution. By "behaves" we mean that the vortex and impurity solution looks identical to an (N + α)-vortex solution with the gauge field shifted down by α. In Fig. 3, we plot vacuum profile functions φ(r), a θ (r) for impurities of the form σ(r) = −4de −dr 2 . As d increases, these impurities approach the delta function with α = 1. We display vacuum solutions for λ = 0.5, 1.0, 1.5, and in each subfigure we also plot an N = 1 vortex profile function for the same λ as a solid black line. For a θ (r), the N = 1 profile function is shifted down so that it can be compared with the impurity vacuum solutions. As d increases, regardless of the value of λ, the vacuum solutions approach the vortex profile functions, with a singularity at the origin for a θ (r).
This suggests that, for negative c, a delta function impurity behaves like a vortex in the Type I and II regimes, as well as at critical coupling. The numerical evidence is compelling and encourages us to generalise the argument of Ref. [19] to any value of λ.
The key idea is to formally relate an axial vortex of degree N = n + m with an axial vortex of degree n in the presence of an impurity of strength m via a singular gauge transformation. As in the argument at critical coupling, we define h = 2 log φ, so e h = |φ| 2 . The quantity h is gauge invariant, and it is finite everywhere except for at the zeros of φ. We will rewrite the profile function equations (14) for an axial vortex of degree N in terms of h. First we consider the equation for φ from (14) with no impurity (σ ≡ 0). This can be written in terms of h as This is well-defined for r > 0, but the first four terms become singular as r → 0. For r ≈ 0, the Higgs field satisfies φ ≈ r N , so h ≈ 2N log r. The terms h + 1 r h are the radial part of the Laplacian. Since log r is the Green's function of the two dimensional Laplacian, we have regularised these terms by adding 4πNδ(r) on the right-hand-side of equation (15).   The remaining singular terms as r → 0 are Recall that the gauge field a θ satisfies the boundary conditions a θ (0) = 0 and a θ (∞) = N. Using the boundary condition a θ (0) = 0 and differentiating h for r ≈ 0 as h ≈ 2N/r, we see that these two terms cancel.
We have now rewritten the profile function equation (14) for the Higgs field φ of an axial vortex of degree N = n + m in terms of h. To relate this to an axial vortex of degree n in the presence of an impurity of strength m, we perform the singular gauge transformation a θ (r) →ã θ (r) = a θ (r) − m, for r > 0. Then equation (15) becomes where we have moved 4πmδ(r) to the left-hand-side and defined σ(r) = − 4πm λ δ(r). This is the radial equation of a vortex of charge n in the presence of an impurity σ(r) at the origin.
Equation (14) for the gauge field is Note that for fixed h this is a linear equation in a θ . The singular gauge transformation a θ (r) → where we added the appropriate impurity term. This is justified for r > 0 when σ = − 4πm λ δ(r) is a delta function. Hence, we have shown that for r > 0 the gauge fieldã θ satisfies the equation for an n vortex with an impurity at the origin. As r → ∞, the gauge potentialã θ → n, which is the topological charge. As a θ (0) = 0, we haveã θ (r) → −m as r → 0. However, to be well-defined at the origin, the gauge fieldã θ would need to satisfyã θ (0) = 0. So,ã θ is singular at the origin. In summary, if we impose axially symmetry then an N = n + m vortex configuration is related to an n vortex configuration in the presence of an impurity of the form σ(r) = − 4πm λ δ(r) by a singular gauge transformation.

C. Binding energy of vortices and impurities
In Fig. 4 we compare the energy of profile function solutions to equations (14) for λ = 0.5, 1.0, 1.5 and different values of the topological charge N. Fig. 4(a) gives the vacuum en- increasing with |c|. Note that for c < 0, the energy increases or decreases more steeply than for c > 0, and at roughly the same rate for both λ = 0.5 and λ = 1.5. However for c > 0, the decrease in energy for λ = 0.5 is steeper than the increase in energy for λ = 1.5. it is strictly decreasing. However for c < 0 there is a region in which the energy decreases with decreasing c for λ = 1.5, and increases for λ = 0.5. This is interesting as it indicates that for λ > 1 we can lower the energy of a vortex by including an impurity. As is the case in the absence of an impurity, the energy for λ < 1 is strictly less than E 1 λ=1 /π = 1 for any c, whilst the energy for λ > 1 is strictly greater than 1.
To determine whether a vortex and impurity will attract or repel, we consider the binding energy and λ = 0.5, 1.0, 1.5. Where the energy difference is positive, the impurity will repel a vortex, and where it is negative the impurity and vortex will attract. of a vortex and an impurity that is the difference between the energy of an impurity coincident to a vortex and one which is well-separated from the vortex. Let E N,σ denote the energy of an Nvortex profile function coincident to an impurity σ, and E N the energy of an N-vortex profile function with no impurity. We approximate the energy of a well-separated vortex and impurity by (E 1 + E 0,σ ): the sum of the energy of a single vortex with no impurity and the vacuum energy for the impurity σ. If E 1,σ − (E 1 + E 0,σ ) > 0, then the energy is lower when the vortex and impurity are well-separated and so they will repel. If E 1,σ − (E 1 + E 0,σ ) < 0 then the energy is lower when the vortex and impurity are coincident and so they will attract.
In Fig. 5, we plot the energy difference E 1,σ − (E 1 + E 0,σ ) against c for impurities of the form σ(r) = ce −r 2 , and λ = 0.5, 1.0, 1.5. At critical coupling, the energy difference is always zero since the vortex and impurity neither attract nor repel. The energy difference is also zero at c = 0 for any λ because there is no impurity in this case. For c > 0, the energy difference is positive for λ = 1.5, and negative for λ = 0.5, indicating that a vortex and impurity will repel for λ = 1.5 and attract for λ = 0.5. For c < 0, there are two regimes in which the impurity can either attract or repel a vortex, and the range of c for which each behaviour occurs depends on λ. For λ = 1.5, if c ∈ (−7.5, 0) then the energy difference is negative so the impurity attracts a vortex, and for c < −7.5 the energy difference is positive so the vortex and impurity repel. Similarly, for λ = 0.5, the energy difference is positive for c ∈ (−5.84, 0), and negative for c < −5.84. At the critical values of c between each regime, the energy difference is zero, and so the vortex and impurity neither attract nor repel. Since for c < 0 a delta function impurity behaves like another vortex, we would expect it to attract a vortex for λ < 1 and repel it for λ > 1. However we have found two different regimes of behaviour for c < 0 in which the impurity can either attract or repel a vortex. The critical value of c separating the two regimes depends on the coupling constant λ and the impurity parameter d. For each λ, we calculate these values by fixing d and evaluating the energy difference E 1,σ − (E 1 + E 0,σ ) for a range of c. The value of c at which the energy difference is zero is the critical value separating the two regimes for the chosen d. In Fig. 6, we plot the critical values of c against d for λ = 0.5, 1.5 in blue and red respectively. We will refer to these as the critical lines separating the two regimes of behaviour. Impurities for which (d, c) is located below the critical line will attract a vortex for λ = 0.5 and repel it for λ = 1.5. Conversely if (d, c) is located above the critical line, then the impurity repels a vortex for λ = 0.5 and attracts it for λ = 1.5.
We previously approximated a delta function impurity by σ(r) = −4de −dr 2 , which approaches a delta function of strength α = 1 as d increases. We also plot the line c = −4d corresponding to these impurities in Fig. 6. For small d, our approximation to a delta function impurity does not behave like a vortex: the impurities are located above the critical line and will repel a vortex for Since an impurity of this form only becomes more like a delta function as d increases, this supports our conclusion that a delta function impurity behaves like a vortex for c < 0.

D. Impurity asymptotics
As a final comment on the profile function equations, we discuss the impurity strength. This is calculated in Ref. [19] by linearising the Bogomolny equation for large r, but we will calculate the same quantities by linearising the profile function equations (14), following a similar argument for vortices given in Ref. [28]. We consider a vacuum solution in the presence of an impurity σ.
As r → ∞, we have the boundary conditions φ(∞) = 1, and a θ (∞) = 0. We linearise the profile functions at infinity by taking where α(r) and β(r) are small. Assuming that the impurity decays sufficiently rapidly as r → ∞, we obtain the linearised profile function equations or equivalently These are identical to the equations that were obtained for a vortex in Ref. [28] and their solutions are where K 0 (r) and K 1 (r) are Bessel functions. The interpretation is that for large r, the vortex, or We substitute the asymptotic forms (20) into the first equation above and find to leading order Since dK 0 dr = −K 1 (r), and we have already found that α(r) = qK 0 ( √ λr), we can write β(r) as Comparing this with the expression for β(r) in (23), we find q = m.
For any λ, we can calculate q and m by fitting the solutions (23) to numerical vacuum solutions of (14). In Table I, we give the values of q and m for different impurities and λ = 0.5, 1.0, 1.5.
Notice that the values of q and m are positive for c > 0 and negative for c < 0. We can see this by considering how the approach to the vacuum values of the profile functions in Fig. 1 changes with the sign of c and comparing this with the shape of the Bessel functions K 0 and K 1 . For c > 0 the profile functions approach the vacuum from above, asymptotically matching the shape of a Bessel function, but for c < 0 they approach from below, matching the shapes of −K 0 and −K 1 . Earlier we discussed the conjecture that impurities σ(r) approaching a delta function will behave like a vortex. In Table I we illustrate this at critical coupling with impurities with the values already identical to two decimal places for σ(r) = −16e −4r 2 .
In Ref. [28], the vortex asymptotics was used to understand how the attraction or repulsion between two vortices depends on the value of λ. Since we have found the large r behaviour of an impurity to be of the same form as that of a vortex, we adapt the expression for the intervortex potential of two vortices (see for example Ref. [28]) to obtain that for a vortex and an impurity. Let q σ , m σ be the charge and moment of an impurity σ, and q V , m V correspond to those of a vortex.
Then the static potential is given by Here s denotes the distance between the vortex and impurity. We plot the potential (27)  We note that this approximation is only accurate past some critical value of the separation between the vortex and impurity, s c which depends on the values of q and m, and we expect it to break down for s ≤ s c .
The force due to U(s) is given by When λ = 1, we know that q = m for both a vortex and an impurity, so these terms cancel and there is no force. If λ < 1, then K 1 (s) decays faster than K 1 ( √ λs), and so the behaviour is determined by the sign of q σ . For a vortex q v < 0, and for an impurity we saw that q σ < 0 if c < 0 and q σ > 0 if c > 0. Combining these, we find that for λ < 1 the net force is negative if c < 0 and  dominates. The type of force depends on the sign of m σ . We know that m V < 0, and we have that indicating that a vortex and impurity will repel for c < 0 and attract for c > 0. Comparing these predictions with the three regimes that we have found by considering the energy difference, we see that they are only accurate in one case -for impurities σ found below the critical line (shown in Fig. 6 for λ = 0.5, 1.5). This argument predicts a different behaviour for both c > 0 and impurities with c < 0 found above the critical line. However the case in which an impurity approaches a delta function is correctly predicted to behave like a vortex. The next subsection will provide an explanation why energy differences and asymptotics appear to lead to different results.

E. Static vortices and impurities
Next we obtain static vortex solutions in the presence of magnetic impurities on a 2D grid. We solve the gradient flow equations using a finite difference method that is first order in time, and fourth order in space. We use timestep ∆t = 0.001, and grid spacing ∆x = ∆y = 0.1, and typically solve on grids of size 401 × 401.
To generate initial conditions, we first use the vacuum profile functions obtained by solving (14) to create a vacuum solution on the 2D grid. We can combine vortex solutions by using the Abrikosov ansatz. This states that, given a vortex solution (φ(x), a µ (x)), we can obtain an approximate multi-vortex solution aŝ In Fig. 8, we plot |φ(x, 0)| 2 against x for critically coupled vortices positioned at a range of initial locations in the presence of the impurities (a) σ(r) = −e −r 2 , and (b) σ(r) = e −r 2 . We note that these solutions were also calculated in Ref. [19] using the Bogomolny equation (10). We have We see in both of the subfigures that a vortex positioned at the origin effectively "screens" the impurity. To numerically confirm the existence of a moduli space of solutions at critical coupling we evaluate the energy of vortex and impurity solutions for different values of the separation s between the vortex and impurity. Similar calculations for two vortices have been carried out in [30]. We calculate the energy E/π for a single vortex and impurity at critical coupling to be 1 to four decimal places regardless of the separation s. To illustrate the possible cases away from critical coupling, in Fig. 9 we display the energy E/π of a single vortex in the presence of an impurity σ as a function of s for λ = 0.5, 1.5 and three different impurities. This reveals a more complicated relationship between the impurity and vortex than that predicted by considering the sign of E 1,σ − (E 1 + E 0,σ ) and explains why the asymptotics disagreed with these results.
Figs. 9(a) and (d) correspond to the impurity σ(r) = 4e −r 2 . Here the energy calculation E 1,σ − (E 1 + E 0,σ ) predicted that the impurity would repel the vortex for λ > 1 and attract it for λ < 1, though the asymptotics suggested the opposite effect. Initially the energy decreases with increasing s for λ = 1.5 and increases with increasing s for λ = 0.5. However a close examination of the energy at larger values of s (see Fig. 9(d)) reveals that this behaviour reverses for s > 4: the energy begins to slightly increase with s for λ = 1.5 and to very slightly decrease with s for λ = 0.5, before settling on a constant value E| s→∞ . This is the behaviour predicted by the asymptotics, and in Fig. 9(d) we also plot the corresponding interaction potential (27) shifted by the value E| s→∞ as a black line. The asymptotic prediction agrees very well with E(s)/π for s > 5. The energy calculation disagreed with the asymptotic prediction because it only compared the energy of a well-separated vortex and impurity E| s→∞ to that of a coincident vortex and impurity E(0), and so it missed the subtle change in behaviour at larger values of s. The true behaviour is more complicated than that indicated by either prediction: when vortex and impurity are close, the impurity will attract the vortex for λ < 1 and repel it for λ > 1, but when they are further apart the impurity will repel the vortex for λ < 1 and attract it for λ > 1 In Figs. 9(b) and (e) we show similar plots for the impurity σ(r) = −4e −r 2 . In this case the energy calculation predicted that the impurity would attract the vortex for λ > 1 and repel it for λ < 1, and the asymptotic prediction disagreed. As before, we see that the energy prediction is correct for smaller values of s, but that the behaviour changes when s > 4, at which point the shifted interaction potential becomes a good fit for E(s)/π.
Figs. 9(c) and (f) correspond to the impurity σ(r) = −16e −4r 2 , which was one of the impurities for which the predictions of both the energy calculation and the asymptotics agreed. In this case they both predicted that the impurity should repel the vortex when λ > 1 and attract it when λ < 1.
However, as shown in the figures, the opposite is true for small s, with the behaviour changing for s > 2.5. Let E stat denote the energy at the stationary point of E(s). Unlike the previous cases, the change in energy of the behaviour at small s is less significant than that of the behaviour at large s, i.e. |E stat − E(0)| < |E stat − E| s→∞ |. So it is the large s behaviour which determines whether E| s→∞ is greater than or less than E(0). Since the large s behaviour has the most significant change in energy, the energy calculation agrees with the asymptotics. In the other cases, the small s behaviour had the most significant change in energy and so the predictions disagreed.
Recall that we earlier discussed a critical line separating the two different regions of behaviour for impurities with c < 0. For impurities on this line, the difference in energy between a coincident vortex and impurity and a well-separated vortex and impurity was zero. However this does not mean the vortex and impurity can be placed anywhere without affecting the energy. There is still a region where, using the example of λ > 1, for small s the energy increases with increasing s, and for larger s the energy decreases with increasing s until settling on a constant value E| s→∞ .
The energy calculation gives zero in this case because E(0) = E| s→∞ , but E(s) is not a constant function.

III. VORTEX DYNAMICS WITH MAGNETIC IMPURITIES
To study vortex dynamics in the presence of magnetic impurities, we return to the Lagrangian (4). The corresponding equations of motion are We solve these equations using a leapfrog method, with derivatives that are second order accurate We first consider the scattering of a single vortex at critical coupling with an impurity of the form σ(r) = ce −r 2 . It has already been noted in Ref. [19] that an impurity of this form with c > 0 will attract a slow moving vortex, and with c < 0 will repel the vortex. In the figures presented below, we take two different impurities σ(r) = e −r 2 and σ(r) = −e −r 2 , but we have verified that the general behaviour is the same for other impurities also. In our simulations, the initial vortex position is (x, y) = (−4, −b) for a range of b ∈ [0, 8], and the impurity is fixed at the origin.
In Fig. 11, we plot the scattering angle Θ in radians as a function of the impact parameter b for a single vortex at critical coupling scattering with the magnetic impurities σ(r) = e −r 2 and σ(r) = −e −r 2 . The initial velocity given to the vortex is v = 0.3. The sign of the scattering angle for σ = −e −r 2 is reversed in (b) for easier comparison with the other impurity. For both impurities, when impact parameter b = 0, the scattering angle Θ = 0: the vortex passes through the impurity in a head-on collision. As b increases, Θ also increases up to a maximum value occuring at b ≈ 1.5.
Then the scattering angle steadily decreases, returning to zero when the impurity is so far from the vortex as to no longer influence its trajectory. We note that the general shape of this plot is consistent with results obtained in Refs. [19,31] for vortices and impurities in hyperbolic space.
The scattering angle plots can be compared with Fig. 12 which shows the corresponding vortex trajectories for a selection of impact parameter values. The location of the impurity is indicated with a black dot. We see that in a head-on collision with either impurity, the vortex will pass through the impurity and continue on its original trajectory. Similarly in both cases, for impact parameter b = 7, the vortex is far enough from the impurity that its trajectory is unaffected, and it continues to travel along the x-axis as though the impurity was not there. In between these values, the vortex trajectories are altered by the presence of the impurity, bending towards the impurity for c > 0 and away from it for c < 0. We saw in Fig. 11 that the maximum scattering angle is found when b ≈ 1.5, after which the angle decreases with increasing impact parameter. In Fig. 12, we see that the most altered trajectories are those for b = 1, 2, and after this the trajectories start to flatten out.
Although we have seen that the vortex will travel through both impurities in a head-on collision without altering its initial trajectory, the details are different depending on the sign of c. with c > 0, as seen in Fig. 13(b), there is a localised lump of energy at the origin which is taller than the energy of the vortex alone. By contrast, for c < 0, as seen in Fig. 13(e), the energy forms a less localised ring, resembling an N = 2 vortex, at the origin, which is smaller than the energy of the single vortex was originally. In both cases, after scattering the vortex and impurity appear unchanged by their interaction, and the vortex continues on its original path.
We now consider the scattering of a 2-vortex ring at critical coupling with the same two impurities. In Fig. 14  In both cases, we initially see the vortex as a ring of energy density and the impurity as a region of negative energy density at the origin. As the vortex passes through the impurity for c > 0, seen in Fig. 14(b), a lump of energy taller than the vortex ring is formed at the origin. When the vortices emerge on the other side of the impurity in Fig. 14(c), we see the ring break up into two vortices along the x-axis. We observe a different behaviour for c < 0. Fig. 14(e) shows the vortex ring and impurity coincident. Here the energy forms a ring similar to an N = 3 vortex solution. When the vortices emerge on the other side of the impurity, the ring has broken up into two vortices along the y-axis. This is shown in Fig. 14(f). The vortices continue to move in the x-direction, but also separate to infinity in the y-direction. Finally, we investigate the scattering of two N = 1 vortices at critical coupling in the presence of a magnetic impurity. In these simulations, we begin with a static impurity at the origin and two vortices located at (±4, ±b), where b is the impact parameter between each vortex and the impurity, as illustrated in Fig. 10. The impact parameter between the two vortices is 2b. We boost the vortices towards each other in the x-direction with initial velocity v = 0.3.
In Fig. 15 we plot the scattering angle as a function of impact parameter for two vortices scattering in the presence of impurities σ(r) = e −r 2 and σ(r) = −e −r 2 . For impact parameter b = 0, the vortices scatter at right angles, as is the case in a head-on collision of two vortices in the absence of an impurity. As the impact parameter increases, the scattering angle decreases. For σ(r) = −e −r 2 , the vortex trajectories bend away from each other and the impurity. This can be seen in Fig. 16(b). For σ(r) = e −r 2 , the direction of the vortex trajectories changes after a certain value of the impact parameter. We see in Fig. 16(a) that the trajectories for b = 1, 2 bend away between the vortices is more significant than the attraction between each vortex and the impurity.
Once the vortices are sufficiently separated, the attraction to the impurity becomes the strongest effect.
Snapshots of the energy density during the head-on collision of two vortices through an impurity located at the origin are given in Fig. 17. The initial velocity given to the vortices is v = 0. show the initial configurations of two vortices on either side of an impurity located at the origin.
For c > 0, we see in Fig. 17(b) that the vortices and impurity all meet at the origin, forming a large lump of energy density. The same situation for c < 0 is seen in Fig. 17(e). Here the energy forms

IV. CONCLUSIONS
We have numerically investigated the dynamics of vortices in the presence of magnetic impurities of the form σ(r) = ce −dr 2 . We began by introducing the model and the Bogomolny bound satisfied by vortices at critical coupling. We reproduced the vortex configurations at critical coupling obtained in Ref. [19] by a different method. Our method also allows us to solve for vortices away from critical coupling, and we presented vacuum solutions for λ = 0.5 and λ = 1.5. We discovered that for c < 0 a delta function impurity behaves like another vortex regardless of the coupling constant λ. We illustrated this numerically by comparing vacuum profile functions for impurities approaching a delta function to the ordinary N = 1 vortex profile functions. We also showed that the differential equations of an N = n+m vortex are related to the differential equations of an n vortex and an impurity of charge m by a singular gauge transformation.
We determined how the attraction or repulsion between a vortex and an impurity depends on the coupling constant λ and the impurity parameters c and d by considering the difference in energy between a well-separated vortex and impurity and a coincident vortex and impurity. We found that there are three different regimes: (i) c > 0 where an impurity will attract a vortex for λ < 1 and repel it for λ > 1; (ii) c < 0 with the impurity repelling a vortex for λ < 1 and attracting it for λ > 1; and (iii) c < 0 with the impurity attracting a vortex for λ < 1 and repelling it for λ > 1. We calculated the critical line separating the two different types of behaviour for c < 0 and compared this to the line c = −4d along which impurities approach a delta function. We found that for sufficiently large d these impurities should fall into category (iii) and thus do behave like vortices.
We also investigated the attraction or repulsion between an impurity and a vortex by using the vortex asymptotics at large r, but found that this only agreed with the predictions of the energy calculation for case (iii). The disagreement between the two predictions was resolved by considering the energy as a function of the separation s between the vortex and impurity. This revealed that whether a vortex and impurity attracts also depends on the separation s, which explained the discrepancies. The final section was concerned with the scattering of vortices with magnetic impurities at critical coupling. Our aim was to provide a numerical study of the scattering processes, similar to those already conducted for vortices in the absence of impurities [20][21][22][23]. We first considered the scattering of a single vortex with an impurity for different impact parameters. In a headon collision, the vortex will travel through the impurity and continue on its original trajectory, though the details differ depending on the sign of c. The scattering angle increases with impact parameter until it attains a maximum value (at impact parameter b ≈ 1.5 for the two impurities we considered), after which it begins decreasing to zero. When c > 0, the vortex trajectory bends towards the impurity and when c < 0 it bends away. The general shape of the scattering angle plot was found to be similar to a scattering angle plot given in Ref. [19] for vortices and impurities in hyperbolic space. We also considered the head-on scattering of a 2-vortex ring with an impurity and found that the ring breaks up into two single vortices as it passes through the impurity. For c > 0, the ring breaks up along the x-axis, and for c < 0 it breaks up along the y-axis.
Finally, we considered the scattering of two vortices in the presence of an impurity. In a head-on collision, we found that the vortices pass through the impurity and scatter at right angles, as they would in the absence of an impurity, see for example Refs. [20,21]. The details of the scattering process are slightly different depending on the sign of c. As the impact parameter increases, the scattering angle decreases. Initially the repulsion between two vortices has the strongest effect on the vortex trajectories, even in the presence of an impurity with c > 0. However we found that there is a certain value of the impact parameter (b ≈ 2.54 for the impurity we considered) past which the relationship between the vortex and the impurity controls the direction of the vortex trajectories. So for an impurity which attracts a vortex, the trajectories will bend towards the impurity, and for an impurity which repels a vortex, the trajectories will continue to bend away from the vortex. For large enough b, the vortex trajectories are no longer affected by each other, and the scattering angles are identical to those for a single vortex scattering with the impurity.
There remain many possibilities for further work on this subject. Thus far, we have only simulated vortex dynamics at critical coupling, but it would be interesting to investigate the cases when λ < 1 and λ > 1 also. When a 2-vortex ring scatters with an impurity, it breaks up into two vortices in a different way depending on the sign of c. The scattering of higher charge multi-vortex rings with an impurity should be investigated to see whether a similar behaviour is observed for them.
Finally the scattering of vortices with different impurities should be studied. In particular, it would be interesting to consider impurities which approach a delta function, as we have observed that these should behave like vortices.
Here we focused on relativistic dynamics of Abelian vortices which has applications in collisions of cosmic strings [32]. Vortices in real superconductors move according to first order dynamics. Manton proposed an elegant Schrödinger-Chern-Simons dynamics in [33] which is conservative and Galilean invariant and also has a description in terms of moduli space dynamics, see [34] for a rigorous justification. The moduli space approximation predicts that vortices close to critical coupling move around each other [33,35] which has been verified numerically in [36].
It is not known yet how impurities will affect this dynamics.