Dissecting zero modes and bound states on BPS vortices in Ginzburg-Landau superconductors.

# Dissecting zero modes and bound states on BPS vortices in Ginzburg-Landau superconductors.

A. Alonso Izquierdo, W. Garcia Fuertes, and J. Mateos Guilarte
Departamento de Fisica
Departamento de Fisica Fundamental, Universidad de Salamanca, SPAIN
###### Abstract

In this paper the zero modes of fluctuation of cylindrically symmetric self-dual vortices are analyzed and described in full detail. These BPS topological defects arise at the critical point between Type II and Type I superconductors, or, equivalently, when the masses of the Higgs particle and the vector boson in the Abelian Higgs model are equal. In addition, novel bound states of Higss and vector bosons trapped by the self-dual vortices at their core are found and investigated.

PACS: 11.15.Kc; 11.27.+d; 11.10.Gh

## 1 Introduction

Vortex filaments carrying a single quantum of magnetic flux were discovered by Abrikosov in the realm of the Ginzburg-Landau theory of Type II superconductors in Reference [2]. The same magnetic flux tubes reappeared in the relativistic context of the Abelian Higgs model in the paper of Nielsen and Olesen [3], where their stringy nature was emphasized. Analytic formulas are available in this Reference for the -vortex profile near the center of the core and far away from the origin, although the behaviour at infinity was refined in [4]. An important step forward in our knowledge of the mathematical properties of these extended structures was achieved by Bogomolny, who identified in the seminal paper [5] a system of first-order PDE such that their solutions are the ANO vortices at the transition point between Type II and Type I superconductivity phases, the critical value where the quotient of the scalar and vector particle masses is one. These Bogomolny-Prasad-Sommerfield, [6], or self-dual111Prasad and Sommerfield found magnetic monopoles at the BPS limit of the Georgi-Glashow model where the Higgs potential disappears but still the vacuum orbit is a two-dimensional sphere. “Self-dual”refers to the fact that the first-order PDE systems, either governing vortices or monopoles, come from two different dimensional reductions of the self-duality Yang-Mills equations., vortices have very interesting features: (1) The magnetic flux is a topological quantity related to the first homotopy group of the circle of degenerate vacua. (2) At the BPS limit, these line defects do not interact with one another. They are thus free to move and zero modes of BPS-vortex fluctuations exist.

The primary aim of this work is to investigate the self-dual vortex zero modes of fluctuation. The BPS vortex PDE equations admit multivortex solutions. Proof of the existence of this type of solitons was given by Jaffe and Taubes in Reference [7]. The BPS multivortex moduli space with magnetic flux equal to , with integer, is the space of unordered points in the plane [8]. The freedom in the locations of the centers preludes the existence of linearly independent zero modes of fluctuation, a fact proved by E. Weinberg in [9] with a shrewd generalization of the index theorem of elliptic operators. This computation, motivated by a physical problem, paved the way to extending the Atiyah-Singer index theorem usually observed in compact spaces, with or without boundaries, to open spaces where problems with the continuous spectrum arise, see e.g. [10, 11]. The self-dual vortex solutions with cylindrical symmetry aroused special interest. In this case the vortex first-order equations reduce to an ordinary differential equation system which is solvable near the origin and very far from the vortex core. Several interpolation methods have been developed, either numerically or through some functional series expansion, to obtain the full multivortex solution, which is never expressible in terms of elementary or special functions, see [12]. In a later development, the self-dual cylindrically symmetric vortex zero mode fluctuations were studied in detail starting with E. Weinberg seminal paper [9], see also Chapter 3 in the recent monograph [13]. Given the rle of the vortex zero modes in the analysis of the low-energy vortex dynamics as geodesic motion in the moduli space of BPS vortex solutions, see e.g. [8]-[16], better ansatzes for the analytical structure of the zero mode fluctuations of BPS vortices were proposed for this purpose in References [14, 15]. This task was fully achieved in the papers just mentioned for the solutions with a low number of magnetic flux quanta, e.g., . In the first half of this paper we perform a complete and detailed analysis of the structure of the zero modes of fluctuation around BPS cylindrically symmetric vortices. Relying on the Ruback-Burzlaff ansatz, we describe the vortex zero mode profiles with the same level of precision as the precision attained in the knowledge of the BPS vortices themselves. After identifying analytically the zero mode radial profile near the core and close to infinity we perform the interpolation between these two regimes by means of a shooting procedure implemented numerically. The angular dependence of the zero mode wave function is fixed analytically by Fourier analysis. The regularity of the wave function near the origin and exponential decay at infinity, all together guaranteeing normalizability, impose the existence of linearly independent vortex zero modes in concordance with the index theorem. The interest of this study is twofold: (1) It extends the work of several authors on this subject to BPS vortices with more than three quanta of magnetic flux. (2) Recently, in [17] and [18] two of us improved on the one-loop shift calculations of kink masses and domain wall surface tensions by controlling the inaccuracies induced by zero modes in the heat kernel/zeta function regularization procedure. The new method requires precise information about the zero mode wave functions such that the information gathered in this paper is necessary to improve the results obtained in [19, 20, 21, 22, 23] by diminishing the impact of zero modes in heat kernel expansions 222 Our method applies not only to conventional topological defects but also to instantons, see Reference [24]..

However, vortex zero modes exist and are influential not only in critical vortices between Type I and II superconductors. Jackiw et alli, see e.g. [25], discovered that the spectrum of the Dirac operator in a vortex background includes linearly independent fermionic eigenfunctions of zero eigenvalue, where is the vortex magnetic charge. From a mathematical point of view the existence of zero modes in the vortex-fermion system obeys an index theorem on a open space, the plane. Besides these topological roots underlying their existence, the vortex-fermion zero modes add quantum states in the middle of the mass gap of the Dirac spectrum which, in turn, induce the phenomenon of fractionary charge. Thus, the context in which fermionic zero modes in a classical vortex field are considered is completely different: there are no scalar and vector particles and the vortex external field does not fluctuate as in the Abelian Higgs model. Quite recently, this problem gained importance in condensed matter physics, for instance in the mathematics and physics of graphene, see e.g. [26], or, in class A chiral superconductors, see [27].

Soon after the discovery of vortex filaments in Type II superconductors, interest aroused in the investigation of fermionic bound states trapped at the vortex core by looking at the one-particle spectrum of the Bogolyubov-de Gennes equation near a magnetic flux line background, see [28]. This pioneer paper by de Gennes et al prompted a long search aimed at unveiling the nature of this type of bound states, although without complete success from the analytic point of view. Nevertheless, interesting effects of these bound states on the vortex core have been disclosed in a superfluid phase of the isotope, see [29]. As a secondary goal, we shall study here the bound states arising when scalar and/or vector bosons are trapped at the core of a self-dual vortex in the framework of the Abelian Higgs model, mutatis mutandis in the Ginzburg-Landau phenomenological theory of superconductivity. Contrarily to the bound states mentioned above the particles trapped by the BPS vortices are bosons rather than fermions. In the context of the Abelian Higgs model bound states of mesons by vortices were discovered by Goodman and Hindmarsh in Reference [30] in the mid nineties. Another papers where the rle of these bound states in the framework of topological defects in Cosmology is emphasized are [31, 32, 33]. Taking profit of the supersymmetric quantum mechanical structure linked to BPS topological defects we were able in the short letter [34] to offer a quite detailed description of such bizarre bound states. We shall develop in this work a more complete analysis of the meson bound states on BPS vortices and we shall discuss their properties by comparison with the well known BPS vortex zero modes. Our approach follows the pattern found in the kink. Fluctuations of the domain wall defects in this model are of three types: 1) translational (zero) modes where a meson travels together with the kink center of mass without disturbing the defect profile. 2) kink internal modes of fluctuation where a meson is trapped forming a meson-kink bound state that produces an oscillating in time deformation of the defect profile. The existence of this second type of fluctuations is due to the supersymmetric quantum mechanics of the kink stability problem. 3) Scattering of mesons through the wall.

We shall address types 1) and 2) of fluctuation concerning the BPS vortices in the Abelian Higgs, a much more difficult task. In fact, the search for vortex fluctuations with frequencies greater than 0 but lower than the threshold of the continuous spectrum only differs from the search for zero modes in the fact that the eigenvalue is unsettled a priori. The shooting procedure for obtaining the form factor of the bound state in the intermediate region is thus ineffective and we shall approximate the radial ODE by means of a discretization of the radial coordinate, transforming this ODE into a linear system of difference equations. The bound state eigenvalues will be identified via diagonalization of the matrix of the linear system, after which the eigenfunctions will be found numerically.

The paper is organized as follows: In Section §.2 the Abelian Higgs model is revisited with the aim of fixing our notational conventions. Section §.3 is devoted to describing in a detailed manner the critical regime between Type I and Type II superconductivity leading to the BPS system of first-order PDE governing the static solutions of finite energy density. Also, the second-order differential operator, which is usually referred to as the Hessian, determining the small fluctuations around the vortex solutions is discussed in this Section and its factorization as the product of two first-order PD operators is explained. In Sections §.4 and §.5 the general structure of the fluctuation spectrum of cylindrically symmetric BPS vortices is developed forming the main contribution of the paper. Section §.4 offers a comprehensive analysis of the BPS vortex zero modes of magnetic flux and unveils the general pattern of zero mode fluctuations of a cylindrically symmetric BPS vortex carrying quanta of magnetic flux. In Section §.5 a similar picture describing the features of several boson-vortex bound states also with low magnetic charge, is developed. Finally, in Section §.6 we draw some conclusions and speculate about some future prospects.

## 2 Topological defects carrying quantized magnetic flux in superconducting systems

We start from the action of the Abelian Higgs model that describes the minimal coupling between a -gauge field and a charged scalar field in a phase where the gauge symmetry is broken spontaneously. In terms of non-dimensional coordinates, couplings and fields, the action functional for this relativistic system in Minkowski space-time reads:

 S[ϕ,A]=∫d3x[−14FμνFμν+12(Dμϕ)∗Dμϕ−κ28(ϕ∗ϕ−1)2]. (1)

The main ingredients are one complex scalar field, , the vector potential , the covariant derivative and the electromagnetic field tensor . We choose the metric tensor in Minkowski space in the form with , and use the Einstein repeated index convention. In the temporal gauge , the energy of static field configurations becomes

 E[ϕ,A]=∫d3x[14FijFij+12(Diϕ)∗Diϕ+κ28(ϕ∗ϕ−1)2],i,j=1,2

which, in a non-relativistic context, is the free energy of a superconducting material arising in the Ginzburg-Landau theory of superconductivity, see formula (17) in [2] where the order parameter responds to the Cooper pairs density. The search for static configurations requires us to look at the extrema of the functional:

 V[ϕ,A]=∫∫dx1dx2[12F212+12(D1ϕ)∗D1ϕ+12(D2ϕ)∗D2ϕ+κ28(ϕ∗ϕ−1)2]

The critical points of are the static fields satisfying the second-order PDE system

 (D1D1+D2D2)ϕ(x1,x2)=12κ2ϕ(x1,x2)[ϕ∗ϕ(x1,x2)−1] ∂22A1(x1,x2)−∂2∂1A2(x1,x2)=−12i[ϕ∗(x1,x2)D1ϕ(x1,x2)−ϕ(x1,x2)(D1ϕ)∗(x1,x2)] (2) ∂21A2(x1,x2)−∂1∂2A1(x1,x2)=12i[ϕ∗(x1,x2)D2ϕ(x1,x2)−ϕ(x1,x2)(D2ϕ)∗(x1,x2)].

Solutions of (2) that comply with the asymptotic boundary conditions at the circle at infinity, i.e. when ,

 ϕ∗ϕ|S1∞=1,Diϕ|S1∞=0andF12|S1∞=0 (3)

have finite energy. In fact, choosing

 ϕ|∞=einθand(A1,A2)|∞=(−ie−inθ∂1einθ,−ie−inθ∂2einθ), (4)

where and is an integer, as representatives of (3), one checks that the configuration space of the static fields is the union of topologically disconnected sectors: . The fields in each sector are asymptotically constrained by the formula (4) where it is evident that is the winding number of the map from the circle at infinity in the -plane to the vacuum orbit determined by the phase of the scalar field at infinity. Thus, all the field configurations in the non-trivial sectors, , are endowed with a quantized magnetic flux:

 Φ=12π∫R2d2xF12=12π∮S1∞(A1dx1+A2dx2)=n∈Z.

Rotationally symmetrical solutions of (2) with finite energy for and a quantum of magnetic flux, , are vortices given that the vector field is purely vorticial. Choosing, e.g., , the ansatz, see [3],

 (5)

together with the asymptotic conditions, and , and the regularity conditions, and , convert the PDE system (2) into a second-order ODE system

 d2fdr2+1rdfdr−(1−β)2r2+κ22f(1−f2)=0 (6) d2βdr2−1rdβdr+(1−β)f2=0, (7)

and at the same time guarantees regular behavior at the origin and appropriate fall-off at infinity. There is no available analytical solution to this ODE system. The asymptotic form of these solutions, however, is known, see [4]. Linearization of equations (6), (7) around the scalar and vector fields vacuum values reveals that:

 β(r)\lx@stackrelr→∞≃1+cV⋅√re−r,f(r)\lx@stackrelr→∞≃⎧⎨⎩1−cH⋅1√re−κr% ifκ<21−cH⋅1re−2rifκ>2 (8)

where and are integration constants. The full Nielsen-Olesen vortex profiles can only be identified by numerical methods. One might also search for vortex solutions carrying quanta of magnetic flux. Vector and scalar mesons respectively produce repulsive and attractive forces of Yukawa type between charged objects of the same type. Thus, an effective potential arises prompting vortex solutions of unit magnetic flux to either repel, if (Type II superconductors), or attract each other when (Type I superconductivity materials). In the first case, the vortices are arranged in a triangular Abrikosov lattice, whereas in Type I superconductors the magnetic flux aggregates on slices piercing the material.

## 3 Self-dual/BPS vortices and their fluctuations

At the transition point between Type I and II superconductors no forces exist between the vortices, which thus, become very special. In order to investigate these critical vortices it is convenient to write in the form, see Reference [5]:

 V[ϕ,A]=12∫R2d2x[(F12±12(ϕ∗ϕ−1))2+|D1ϕ±iD2ϕ|2]∓ ∓12∫R2d2x[F12(ϕ∗ϕ−1)−i{(D1ϕ)∗D2ϕ−(D2ϕ)∗D1ϕ}]+κ2−18∫R2d2x(1−ϕ∗ϕ)2

where . Because we obtain

up to a total derivative term that integrates to zero over the whole plane if the fields tend to their vacuum values at infinity. The parameter , determined by the and electromagnetic couplings as , measures the quotient between the penetration lengths of the scalar and electromagnetic fields in the superconducting medium. Values such that characterize Type II superconductors (typically alloys) whereas type I superconductors (metals) correspond to , as explained above. In the QFT context the parameter is the quotient between the masses of the Higgs particle, , and the vector meson, , after the Higgs mechanism has taken place giving to the photon a finite mass.

The critical vortices are solutions of the first-order PDE’s

 D1ϕ±iD2ϕ=0,F12±12(ϕ∗ϕ−1)=0,

which, written in terms of the real and imaginary parts of the complex scalar field , read

 ∂1ϕ1+A1ϕ2±[−∂2ϕ2+A2ϕ1] = 0 ∂1ϕ2−A1ϕ1±[∂2ϕ1+A2ϕ2] = 0 (9) F12±12(ϕ21+ϕ22−1) = 0

Moreover, the BPS vortices are subjected to the asymptotic conditions (4). It is clear that at the energy of self-dual vortices saturates the Bogomolny topological bound: 333Recovering the physical dimensions, the magnetic flux and the energy per unit length of self-dual vortices would be: , , where is the vacuum value of the scalar field.. Additionally, it may be checked that self-dual vortices also solve the second-order PDE system (2). Proof of the existence of vorticial solutions of the PDE system (9) has been developed in Reference [7]. Given a positive integer , there exists a moduli space of self-dual vortices solving the PDE system (9) characterized by parameters, the centers of the magnetic flux tubes located at the zeroes of the scalar field counted with multiplicity , i.e., , where is the quantized magnetic flux of a vortex (the multiplicity) and is the total number of flux lines, see also Reference [8]. Behind this particular structure lies the fact that there are no forces between self-dual vortices () of one or several quanta of magnetic flux, which thus move freely throughout the --plane.

### 3.1 The first-order fluctuation operator: hidden Supersymmetric Quantum Mechanics

Knowing that the energy of self-dual vortices is a topological quantity, there is no doubt about the stability of these topological solitons. The main theme of this paper, however, is the analysis of field fluctuations around self-dual vortices. We shall concentrate on two special modes of fluctuations: 1) the vortex zero modes, those belonging to the kernel of the second-order fluctuation operator (the Hessian in variational calculus terminology), which arise because of the freedom of motion of the centers. 2) vortex internal modes of fluctuation corresponding to bound state normalizable eigenfunctions of the Hessian in the discrete spectrum and 3) scattering eigenfunctions in the continuous spectrum.

Let us denote the scalar field and the vector potential corresponding to a self-dual vortex solution of vorticity as:

The self-dual vortex fluctuations and built around the BPS vortex fields

 (A1(→x;n),A2(→x;n))=(V1(→x;n),V2(→x;n))+ϵ(a1(→x),a2(→x)) ϕ1(→x;n)=ψ1(→x;n)+ϵφ1(→x),ϕ2(→x;n)=ψ2(→x;n)+ϵφ2(→x) (10)

are zero modes of fluctuation if the perturbed fields (10) are still solutions of the first-order equations (9). To discard pure gauge fluctuations, we select the “background” gauge

 B(ak,φ,ϕV)=∂kak(→x)−(ψ1(→x)φ2(→x)−ψ2(→x)φ1(→x))=0 (11)

as the gauge fixing condition on the fluctuation modes. The system of four PDE equations (9)-(11) is satisfied if and only if the four-vector field

 ξ(→x)=(a1(→x)a2(→x)φ1(→x)φ2(→x))t

which assembles all the BPS vortex field fluctuations is annihilated by the first-order PDE operator:

 D=⎛⎜ ⎜ ⎜⎝−∂2∂1ψ1ψ2−∂1−∂2−ψ2ψ1ψ1−ψ2−∂2+V1−∂1−V2ψ2ψ1∂1+V2−∂2+V1⎞⎟ ⎟ ⎟⎠. (12)

Note that this operator is obtained by deforming the PDE system (9) together with the background gauge. Thus, the zero mode BPS vortex fluctuation fields belong to the kernel of the first-order PDE operator, , or, in components, are solutions of the PDE system:

 −∂2a1+∂1a2+ψ1φ1+ψ2φ2 = 0 −∂1a1−∂2a2−ψ2φ1+ψ1φ2 = 0 (13) ψ1a1−ψ2a2+(−∂2+V1)φ1+(−∂1−V2)φ2 = 0 ψ2a1+ψ1a2+(∂1+V2)φ1+(−∂2+V1)φ2 = 0

Important information about the particle spectrum in the Abelian Higgs model not only comes from the vortex zero mode fluctuations but also from vortex fluctuations demanding positive energy because these positive perturbations determine the dynamics of mesons in the different topological sectors. Thus, we shall investigate the spectral condition , where is a label in either the discrete or the continuous spectrum useful to enumerate the eigenfunctions and eigenvalues, and is the second-order vortex small fluctuation operator

 H+=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−Δ+|ψ|20−2D1ψ22D1ψ10−Δ+|ψ|2−2D2ψ22D2ψ1−2D1ψ2−2D2ψ2−Δ+12(3|ψ|2−1)+VkVk−2Vk∂k−∂kVk2D1ψ12D2ψ12Vk∂k+∂kVk−Δ+12(3|ψ|2−1)+VkVk⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠

coming from linearizing the field equations (in the background gauge) around the BPS vortices, see [21]. The fluctuation vectors belong in general to a rigged Hilbert space, such that there exist square integrable eigenfunctions belonging to the discrete spectrum, for which the norm is bounded:

 (14)

together with continuous spectrum eigenfunctions with ranging in a dense set. One observes that the second-order differential operator is one of the two SUSY partner operators obtained from as

 H+=D†D,H−=DD† (15)

which are isospectral in the positive part of the spectrum444The continuous part of the spectrum might cause some difficulties, which are explained in [9]. Explicitly, we find

 H−=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−Δ+|ψ|20000−Δ+|ψ|20000−Δ+12(|ψ|2+1)+VkVk−2Vk∂k−∂kVk002Vk∂k+∂kVk−Δ+12(|ψ|2+1)+VkVk⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠.

The Hamiltonians are superpartners in a supersymmetric quantum mechanical system built from the “supercharges”555Although written in the text as -matrices, both the supercharges and the SUSY Hamiltonian are -matrices of partial differential operators.

 Q=(00D0),Q†=(0D†00),

which is governed by the SUSY Hamiltonian:

 H=QQ†+Q†Q=(H+00H−).

The stability of the BPS -vortex solutions implies that the -spectrum consists of non-negative eigenvalues. Indeed in [9] within the framework of index theory in open spaces, see [10]-[11], E. Weinberg proved that there are linearly independent normalizable BPS vortex zero modes in the topological sector of magnetic flux , i.e., the dimension of the algebraic kernel of , henceforth of is . By inspection one sees that lacks zero modes (all the potential wells are non-negative) and the index theorem dictates:

 indD=dimKerD=limM→∞TrL2{M2D†D+M2−M2DD†+M2}=2n,

which means that has zero modes, see Appendix B in [13]. Analysis of the zero mode eigenfunctions begun in [9] and was further developed in References [14] and [15]. In the last two references the motivation to describe in detail the vortex zero modes came from the study of vortex scattering at low energies within the approach of geodesic dynamics in their moduli space, see e.g. [16].

The first goal in this paper is to seek a complete description of the vortex zero modes not fully developed in the previous references because interest was focused in sectors of very low magnetic charge. The second task that we envisage is the search for the eigenfunctions in the strictly positive spectrum of , in particular the bound states, i.e., the internal modes of fluctuation where the self-dual vortex captures scalar and/or vector mesons.

## 4 Zero mode fluctuations of BPS cylindrically symmetric vortices

The Nielsen-Olesen ansatz (5) generalized to the topological sector , in cylindrical coordinates also in field space, reads:

 ϕ(→x)=fn(r)einθ;rAθ(r,θ)=nβn(r). (16)

In this ansatz we assume the radial gauge , such that the vector field is purely vorticial. Plugging (16) into the first-order PDE system (9), the following ODE system emerges:

 dfndr(r)=nrfn(r)[1−βn(r)],dβndr(r)=r2n[1−f2n(r)]. (17)

The solutions for the radial profiles and are the self-dual vortex solutions 666Without loss of generality, we restrict and the signs in the first-order system (9) to be positive.. The asymptotic conditions (3) demand that and as . In fact, it is immediate to check that the asymptotic behaviour fits formula (8) when , i.e., the critical value where the mass of the Higgs boson is equal to (henceforth less than twice) the mass of the vector boson. The requirement of regularity at also fixes the behaviour of the solutions near the origin to be and , where and are integration constants. Choice of these constants must be tailored to fit the asymptotic behaviour. A shooting procedure implemented numerically allows us to solve the system (17) by interpolating the field profiles between their shapes in the neighborhoods of the origin and of infinity. In this way we construct the cylindrically symmetric self-dual -vortices.

Investigation of zero mode fluctuations around cylindrically symmetric self-dual vortices begins with rewriting the PDE system (13) in polar coordinates:

 ∂aθ∂r−1r∂ar∂θ+1raθ+fn(r)cos(nθ)φ1+fn(r)sin(nθ)φ2 = 0 (18) −1r∂aθ∂θ−∂ar∂r−1rar−fn(r)sin(nθ)φ1+fn(r)cos(nθ)φ2 = 0 (19) −1r∂φ1∂θ−∂φ2∂r−nβn(r)rφ2+fn(r)cos(nθ)ar−fn(r)sin(nθ)aθ = 0 (20) −∂φ1∂r+1r∂φ2∂θ−nβn(r)rφ1−fn(r)sin(nθ)ar−fn(r)cos(nθ)aθ = 0, (21)

where we recall that and .

### 4.1 Analytical investigation of the algebraic kernel of the first-order operator D

In this Section we shall prove a proposition characterizing the analytical behavior of the eigenfunctions which belong to the kernel of the operator acting on self-dual rotationally symmetric vortices. First, we state a helpful Lemma which unveils the existence of a useful symmetry in BPS vortex fluctuation space.

Lemma: Let us assume that is a zero mode of the second-order fluctuation operator . Then, is a second zero mode of , which is orthogonal and linearly independent of .

Proof: The -rotation in the internal space of scalar and vector field fluctuation planes, , , and , is a symmetry of the ODE system (18)–(21) because it replaces (18) by (19) and (20) with (21). This discrete rotation also transforms into . Thus, if belongs to the kernel of evaluated on self-dual cylindrically symmetric vortices, is also a zero mode around the same vortex solution. The orthogonality between and follows directly from the -inner product.

Regarding vortex zero modes the main result is as follows:

Proposition: There exist orthogonal zero mode -fluctuations of the self-dual cylindrically symmetric -vortex solution of the form

 ξ0(→x,n,k) = rn−k−1⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝hnk(r)sin[(n−k−1)θ]hnk(r)cos[(n−k−1)θ]−h′nk(r)fn(r)cos(kθ)−h′nk(r)fn(r)sin(kθ)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (22) ξ⊥0(→x,n,k) = rn−k−1⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝hnk(r)cos[(n−k−1)θ]−hnk(r)sin[(n−k−1)θ]−h′nk(r)fn(r)sin(kθ)h′nk(r)fn(r)cos(kθ)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (23)

where , and the zero mode radial form factor satisfies the second-order ODE

 −rh′′nk(r)+[1+2k−2nβn(r)]h′nk(r)+rf2n(r)hnk(r)=0 (24)

with the contour conditions and .

Proof: Because the discrete symmetry explained in the previous Lemma we are only interested in the construction of the zero modes , their orthogonal partners follow immediately.

The structure of the PDE system (18)-(21) suggests the ansatz

 ar(r,θ)=gnk(r)sin[(n−k)θ] ; aθ(r,θ)=gnk(r)cos[(n−k)θ] φ1(r,θ)=tnk(r)cos(kθ) ; φ2(r,θ)=tnk(r)sin(kθ),

where the radial and angular dependencies of the components of are separated, in the search for the BPS vortex zero modes. Plugging this ansatz into the set (18)-(21) of four ODE’s, only two independent but coupled first-order ODE’s remain:

 rdgnkdr(r)+rfn(r)tnk(r)+(1+k−n)gnk(r)=0 (25) rdtnkdr(r)+[nβn(r)−k]tnk(r)+rfn(r)gnk(r)=0, (26)

which govern the behaviour of the and functions. Continuity in the angular part of the solution requires to be an integer, , and the integrability of the fluctuations demands that

 ∥ξ0(→x,n,k)∥2=2π∫rdr[g2nk(r)+t2nk(r)]<+∞ (27)

Solving for the function in (25), we obtain

 tnk(r)=−g′nk(r)fn(r)−(1+k−n)gnk(r)rfn(r), (28)

and plugging this expression into (26) we end with a single second-order ODE for

 −r2g′′nk(r)−r[1−2n+2nβn(r)]g′nk(r)+[(1+k−n)(1+k+n−2nβn(r))+r2f2n(r)]gnk(r)=0 (29)

in terms of the self-dual vortex profiles and . Now we shall investigate the behaviour of this function:

• Regularity of the function at the origin: The origin is a regular singular point of the second-order differential equation (29). In this situation there exists a single analytic solution at although the second linearly independent solution of (29) has a singularity. The analytic solution admits a series expansion around of the form:

 (30)

is chosen as the minimum value that selects , i.e., is regular, and does not vanish, at . Plugging (30) into (29), and taking into account that and near the origin, we obtain the identity

 ∞∑j=0[−(−1+j−k−n+s)(1+j+k−n+s)c(n,k)jrj]+ +∞∑j=2[−2ne2(−1+j+k−n+s)c(n,k)j−2rj]+O(r2n+1)=0 (31)

from which we extract recurrence relations that determine the series expansion coefficients up to order . Annihilation of the term independent of in (31) unveils the indicial equation

 (1+k+n−s)(1+k−n+s)c(n,k)0=0. (32)

Because , the two characteristic exponents, the two values of compatible with (32), are:

Both possibilities are equivalent: simply redefine , . Thus, we shall stick to choice . In terms of the function , let us recall that , the norm (27) of the vortex zero modes is

 ∥ξ0(→x,n,k)∥2=2π∫∞0rdrρnk(r)=2π∫∞0drr2(n−k)−1[h2nk(r)+(h′nk(r))2f2n(r)]. (33)

Near the origin, the behaviour of the first summand of the integrand of (33) is

 r2(n−k)−1h2nk(r)≈(c(n,k)0)2r2(n−k)−1+O(r2(n−k)+1).

In order to avoid singularities in the integrand coming from a pole at the origin, we demand that

 2(n−k)−1≥0⇒k≤n−1.

The term proportional to in (31) is null if for the characteristic exponent A. Thus, the second coefficient in the expansion vanishes: . Moreover, the recurrence relations extracted from (31) for the odd terms proportional to , with ,

 (2i+1)(2k−2i+1)c(n,k)2i+1=2e2(2i−1)nc(n,k)2i−1

show that all the odd coefficients also vanish, at least up to , given that and the constants multiplying and in the equation above are non-null. The two-term recurrence relations between the even coefficients is read from the annihilation of the terms proportional to , , in (31):

 i(k−i+1)c(n,k)2i=e2(i−1)nc(n,k)2i−2. (34)

The relations (34) imply up to the coefficient. When the left member vanishes despite not being null. Thus, in a neighborhood of the origin the function adopts the form:

 hnk(r)=c(n,k)0+c(n,k)2k+2r2k+2+… (35)

with and arbitrary constants. Therefore, near the origin the second summand of the integrand in (33) behaves as:

 r2(n−k)−1(h′nk(r))2f2n(r)≈(c(n,k)2k+2)2(2k+2)2r2k+1+O(r2k+3)

Singularities in the integrand coming from a pole at the origin are prevented if the inequality

 2k+1≥0⇒k≥0

holds. Together with the inequality , the univaluedness of the BPS vortex zero modes , equivalent to being an integer, restricts the possible values of to being , as stated in the proposition.

• Asymptotic behaviour of the function : The following step is to investigate the asymptotic behaviour of the function far from the origin. Replacing the asymptotic vortex profiles when , and