Nonrelativistic Chern-Simons Vortices on the Torus
Nikolas Akerblom, Gunther Cornelissen, Gerben Stavenga, Jan-Willem van Holten
NNonrelativistic Chern-Simons Vorticeson the Torus
N. Akerblom , ∗ , G. Cornelissen , † ,G. Stavenga , ‡ , and J.-W. van Holten , § Nikhef Theory Group, Amsterdam, The Netherlands Department of Mathematics, Utrecht University, The Netherlands Theoretical Physics Department, Fermilab, Batavia, IL, USA
October 26, 2018
Abstract
A classification of all periodic self-dual static vortex solutions of theJackiw-Pi model is given. Physically acceptable solutions of the Liouvilleequation are related to a class of functions which we term Ω-quasi-elliptic.This class includes, in particular, the elliptic functions and also contains afunction previously investigated by Olesen. Some examples of solutions arestudied numerically and we point out a peculiar phenomenon of lost vortexcharge in the limit where the period lengths tend to infinity, that is, in theplanar limit.
NIKHEF/2009-030FERMILAB-PUB-09-590-T ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - t h ] J a n ontents γ ω and γ ω commute . . . . . . . . . . . . . 102.2 Case B: The matrices γ ω and γ ω anticommute . . . . . . . . . . . 112.3 The abstract underlying group . . . . . . . . . . . . . . . . . . . . . 14 In this paper we study periodic, static vortex solutions of the Jackiw-Pi model[1, 2]. This is a 2 + 1-dimensional nonrelativistic conformal field theory whosefield content consists of a complex scalar field Ψ with non-linear-Schr¨odinger typeaction minimally coupled to a U(1) Chern-Simons gauge field A µ . Let us begin byreviewing the elements of this model. We take as our starting point the action [6] S [Ψ , A µ ] = (cid:90) dx (cid:90) d x (cid:110) − ε ij ( A ∂ i A j + A i ∂ j A + A j ∂ A i )+ i Ψ ∗ D Ψ −
12 ( D Ψ) ∗ · ( D Ψ) − g | Ψ | (cid:111) . (1) For reviews see [3, 4, 5]. ε = − ε = +1, whilst D µ = ∂ µ − ieA µ is the gauge covariant derivativeand bold type indicates its spatial 2-vector part. We use the generic notation x for the time coordinate, and apply the summation convention for repeated indices.In the following we define ρ = Ψ ∗ Ψ , J i = − i ∗ D i Ψ − Ψ D i Ψ ∗ ) ,B = ∂ A − ∂ A , E i = ∂ A i − ∂ i A . (2)The field equations derived from the action (1) then read B = eρ, E i = eε ij J j ,iD Ψ = − D Ψ + g | Ψ | Ψ . (3)The chiral derivatives D ± = 1 √ D ± iD ) , (4)satisfy the identities 12 D = D − D + − e B = D + D − + e B. (5)Using equations (3), the Schr¨odinger equation for Ψ can then be written as iD Ψ = − D − D + Ψ + (cid:18) g + e (cid:19) | Ψ | Ψ = − D + D − Ψ + (cid:18) g − e (cid:19) | Ψ | Ψ . (6)By a similar argument, the hamiltonian takes the form H = (cid:90) d x (cid:18) | D Ψ | + g | Ψ | (cid:19) = (cid:90) d x (cid:18) | D ± Ψ | + 12 ( g ± e ) | Ψ | (cid:19) . (7)Hence there are two possibilities for constructing stationary zero-energy solutions:( I ) D + Ψ = 0 and g + e = 0 , ( II ) D − Ψ = 0 and g − e = 0 . (8)By stationary we mean that physical observables such as the particle density andcurrent are time-independent. This is achieved by separating space and time vari-ables as Ψ = e iω √ ρ, (9)3ith ρ non-negative and time independent: ∂ ρ = 0. Any time dependence there-fore resides in the gauge-dependent phase ω .Substitution of either of the Ans¨atze ( I ) or ( II ) for Ψ and the coupling con-stants ( e, g ) simplifies the Schr¨odinger equation (6) to iD Ψ = ( eA − ∂ ω )Ψ = ∓ e | Ψ | Ψ ⇒ eA = ∂ ω ∓ e ρ. (10)In addition, the real and imaginary parts of either condition D ± Ψ = 0 lead to thereal equations eA i = ∂ i ω ± ε ij ∂ j ln √ ρ, (11)and as a result eB = eε ij ∂ i A j = ∓ ∆ ln √ ρ, ∆ = ∂ + ∂ . (12)It follows directly that ρ satisfies one of the Liouville equations( I ) D + Ψ = 0 ⇒ ∆ ln √ ρ + e ρ = 0 , ( II ) D − Ψ = 0 ⇒ ∆ ln √ ρ − e ρ = 0 . (13)The solutions of these equations are respectively of the form [7]( I ) ρ f = 4 e | f (cid:48) | (1 + | f | ) , ( II ) ρ f = 4 e | f (cid:48) | (1 − | f | ) , (14)where f ( z ) is an analytic function of the complex coordinate z = x + iy, (15)and for physical reasons we make the hypothesis that f have at most isolatedsingularities (which then automatically are poles; see the discussion in Section1.2).Furthermore, boundedness of ρ requires | f | < II ). This immedi-ately implies that there are no relevant non-trivial solutions in case ( II ).However, case ( I ) leads to a rich spectrum of vortex-type solutions, dependingon the boundary conditions. For instance, if we take two-dimensional space tobe the plane R ↔ C , it is necessary to require that at infinity ρ tends to zerosufficiently fast. The problem of writing down all static vortex solutions in thisplanar case was solved in a beautiful paper by Horvathy and Yera [8].4or physical applications, e.g. in condensed matter systems, it is also of in-terest to study static vortex solutions in a finite volume with periodic boundaryconditions; in that case one requires ρ ( z + ω i ) = ρ ( z ) (16)for given R -linearly independent complex numbers ω , ω . This corresponds tostudying the Jackiw-Pi model in the case where space is a two-dimensional flattorus C / ( Z ω + Z ω ). Apparently, the first one to investigate this situation wasOlesen, who gave a remarkable solution on a square torus [9].If one thinks about how periodic boundary conditions are customarily employedin physics, the step from the plane to the torus seems innocuous. However, herethis is not at all so. The change in topology, in fact, has a rather dramatic effecton the allowed rationalized vortex charge q := | vortex charge | π × e = (total magnetic flux) × e π = e π (cid:90) ρ d x. (17)First of all, it can be shown that q , irrespective of whether we are working on theplane or the torus, must be a non-negative integer (see Appendix B). But, whilefrom the classification of Horvathy and Yera it is immediate that on the plane thecharge is always even: q = 2 n, n ∈ N (vortex charge on the plane) , (18)this is not the case for the Jackiw-Pi model on the torus. Indeed, Olesen’s solutionis an example of a static vortex on the torus with charge q = 1 (Olesen’s solution on the torus) . (19)Moreover, the correspondence between the solutions on the torus and those on theplane is somewhat involved. Olesen’s solution, for instance, vanishes in the limitwhere the period lengths tend to infinity, and in Section 3 we give an example ofa solution for which the charge q is halved as we pass from the torus to the plane!The adagium that the limit of a periodic solution, as the periods tend to infinity,gives a planar solution, fails dramatically in the case of Olesen’s solution. In complex coordinates (15) the non-linear wave equation (13) for positive chiralityfields of type ( I ) reads ¯ ∂∂ ln ρ + e ρ = 0 , (20) For a proof of equations (17) see [2]. ∂ := ( ∂ − i∂ ) / √ ∂ := ( ∂ + i∂ ) / √ I ) of this equation was discovered a long time agoby Liouville [7], who was led to the study of equations (14, I & II ) in connectionwith his researches on the theory of surfaces with constant intrinsic curvature (seealso [11, 12]; in [13] solutions with vanishing boundary conditions on a rectanglewere investigated). We shall frequently call ρ f defined by eq. (14, I ) “the densityassociated with f .”For our purposes, on physical grounds, we make the hypothesis that f is to haveat most isolated singularities. This is because we want to interpret ρ f as a soliton(a vortex) and this interpretation is upset when f has a non-isolated singularity. We also demand that ρ be bounded. In fact, we impose the stronger conditionthat the total particle number in the spatial domain (cid:90) ρ d x, (21)proportional to the total magnetic flux carried by vortices, is finite. In the case ofa periodic ρ , boundedness automatically follows from continuity , as we can interpret ρ as living on a compact space (the torus), and in this case, boundedness is all weneed for the integral (21) to make sense. For vortices on the plane, one obviouslyneeds to supplant this with a suitable decay condition at infinity, see [8].It can be shown: Lemma 1 (Horvathy-Yera [8]) . Let ρ f be the density associated with a complexfunction f having at most isolated singularities. If ρ f is bounded, then the onlypossible singularities of f are poles, i.e. f is meromorphic in the plane. In the plane case this extends to infinity, so that f is a meromorphic functionon the sphere, that is, a rational function : Theorem (Horvathy-Yera [8]) . Let the density ρ f associated with f be a vortexsolution of the Liouville equation on the plane. Then f is a rational function, i.e.there are polynomials P ( z ) and Q ( z ) , such that f ( z ) = P ( z ) Q ( z ) . On a surface of constant curvature the conformal factor of the metric in isothermal coordi-nates satisfies the Liouville equation; that is, if the metric is ds = ρ ( dx + dy ) with ρ > ρ satisfies equation (13) and e is equal to the Gaussian curvature K of the surface. In thissituation, the case K < II ). Itis known that equation (13, I ) has no nowhere vanishing solution on the torus [10]. Thus, bynecessity, all our torus solutions given below have zeros. Indeed, we may conjecture that if f has a non-isolated singularity, then its associated density ρ f is unbounded. In the plane case the integral extends over R , whereas in the periodic case it is taken oversome elementary cell; say, the closure of the fundamental region: { t ω + t ω | ≤ t , t ≤ } . oreover, the converse is also true. In the case of the torus, Lemma 1 still holds (since it is a local statement), butboundedness of ρ f is automatic, and there is no corresponding statement aboutthe behavior of f “at infinity.”We now state the analogous classification in the case where ρ is periodic, or,as one could also say, lives on a torus: Theorem 1.
Let ρ be a smooth periodic solution of the Liouville equation (13) withperiods ω and ω . It follows that ρ = ρ f for some complex function f (Liouville,[7]) meromorphic in the plane (Lemma 1) which falls into one of the following twocases: Case A
There are complex numbers µ , µ with | µ i | = 1 , such that f ( z + ω i ) = µ i f ( z ) , (22) that is, f is an elliptic function of the second kind with multipliers µ i of unitmodulus. For the reader’s convenience, we repeat the results of [14] for suchfunctions in Appendix C. Case B
There are complex parameters z , . . . , z n in the fundamental region of thelattice Z ω + Z ω , and complex constants a , . . . , a n , such that f ( z ) = − ϕ ( z ) − ϕ ( z ) + 1 O ( z ) , (23) where ϕ ( z ) = (cid:34) a + n (cid:88) k =1 a k d k ζdz k ( z − z ) (cid:35) σ ( z − z ) n (cid:81) nk =1 σ ( z − z k ) e ζ ( ω / z , (24) with z = ω n + n (cid:80) nk =1 z k , and O ( z ) = ℘ ω , ω ( z ) + bc ℘ ω , ω ( z ) + d , (25) for a suitable choice of parameters b , c , d , given in equations (65) and (66) .Moreover, the converse is also true: If f falls into one of the two cases above, itsassociated density ρ f is a periodic solution of the Liouville equation. This result is derived in the next section.
Remark on special functions.
Our conventions for the appearing special functionsare as follows: 7 ℘ ω ,ω indicates the Weierstrass p-function associated with the lattice Ω = Z ω + Z ω . • ζ = ζ ω ,ω and σ = σ ω ,ω are the Weierstrass zeta- and sigma-functions.The properties of these functions are given in many textbooks; see, for example,[15]. A word of caution: In the older literature, e.g. in the standard reference [16], ℘ = ℘ ω ,ω often denotes the Weierstrass p-function with half -periods ω , ω (andsimilarly for ζ and σ ). We now proceed to classify all periodic vortices on a given flat torus (Theorem 1).To this end, let a lattice Ω ⊂ C be given and suppose it is spanned by ω , ω , thatis, Ω = Z ω + Z ω . (26)As follows from our earlier discussion in Section 1.1, the task is to find all smoothsolutions ρ of the Liouville equation (13) such that ρ ( z + ω ) = ρ ( z ) for all ω ∈ Ω . (27)Suppose we are given such a ρ . Then, from [7] and Lemma 1 we know that thereis a complex function f , meromorphic on the plane, such that ρ = ρ f = 4 e | f (cid:48) | (1 + | f | ) , (28)where the prime (cid:48) denotes the derivative with respect to the complex variable z .Let ω ∈ Ω be arbitrary and define the function g ( z ) := f ( z + ω ) . (29)From equation (27) and the fact that g (cid:48) ( z ) = f (cid:48) ( z + ω ), it follows that ρ f ( z ) = ρ g ( z ) for all z ∈ C . (30)In Appendix A we prove: Lemma 2. Let f and f be non-constant meromorphic functions on the planeand suppose that their associated densities ρ f and ρ f are equal: ρ f = ρ f . Another proof has been given by de Kok [17]. hen there exists a matrix γ = (cid:20) a bc d (cid:21) ∈ SU(2) , such that f ( z ) = γ · f ( z ) := (cid:20) a bc d (cid:21) · f ( z ) := af ( z ) + bcf ( z ) + d . (31) Also, the converse is true [6, 18], even under the weaker hypothesis that (cid:20) a bc d (cid:21) ∈ U(2) ; that is, if f = V · f for some V ∈ U(2) , then ρ f = ρ f . Now, from Lemma 2 it follows that for any ω ∈ Ω, there is a matrix γ ω ∈ SU(2),such that f ( z + ω ) = g ( z ) = γ ω · f ( z ) . (32)This matrix is not unique in SU(2), but it is unique in PSU(2 , C ). We shall call ameromorphic function on the plane Ω-quasi-elliptic if it satisfies condition (32).A trivial corollary to Lemma 2 is that ρ = ρ f is periodic with respect to thelattice Ω = Z ω + Z ω if and only if there exists matrices δ ω i ∈ U(2), such that f ( z + ω i ) = δ ω i · f ( z ) for i = 1 , γ = (cid:20) a bc d (cid:21) ∈ SU(2) there is naturally associated a certaintransformation T ( γ ) ∈ PSU(2 , C ) from the Riemann sphere (cid:98) C to itself, namely T ( γ ) : (cid:98) C → (cid:98) C , z (cid:55)→ az + bcz + d . (33)Since, obviously, T ( γ ω +˜ ω ) = T ( γ ω ) T ( γ ˜ ω ) = T ( γ ˜ ω ) T ( γ ω ) for all ω, ˜ ω ∈ Ω , (34)equation (32) tells us that any Ω-quasi-elliptic function effects a group homomor-phism T : Ω → G, ω (cid:55)→ T ( γ ω ) , (35)from the lattice Ω to some abelian subgroup G of PSU(2 , C ). We recall thatPSU(2 , C ) = SO(3), the group of orientation preserving isometries of the sphere. Because Ω = Z ω + Z ω is a free module with generators ω and ω , G is anabelian group with at most two generators T ( γ ω ) and T ( γ ω ). For M = (cid:20) α βγ δ (cid:21) ∈ GL(2 , C ) and any complex function f we define M · f ( z ) := αf ( z )+ βγf ( z )+ δ . For completeness we mention the elementary rule T ( γ γ ) = T ( γ ) T ( γ ) for all γ , γ ∈ U(2). , C ) with two generators.There are various ways to classify such subgroups. We will work in PSU(2 , C )directly, and lift the two generators of the group from PSU(2 , C ) to SU(2). Onecan also use the isomorphism with SO(3), or consider rotations as quaternions.We shall comment on this later.By an earlier remark (immediately below equation (32)), we have the implica-tion T ( γ ) = T (˜ γ ) ⇒ γ = ± ˜ γ for all γ, ˜ γ ∈ SU(2) . (36)Then, since the generators T ( γ ω ) and T ( γ ω ) of G commute, T ( γ ω ) T ( γ ω ) = T ( γ ω ) T ( γ ω ) , it is easy to see that γ ω and γ ω either commute or anticommute. We will referto these cases as Case A and Case B, respectively and treat them in turn in thefollowing two sections. γ ω and γ ω commute Since γ ω and γ ω commute, they can simultaneously be put into diagonal form.More precisely, there exists a matrix U ∈ SU(2), such that γ ω i = U † (cid:20) √ µ i
00 1 / √ µ i (cid:21) U ( i = 1 , , (37)where the µ i are complex numbers of unit modulus: | µ i | = 1.Let f be Ω-quasi-elliptic and define the function g ( z ) = U · f ( z ) . (38)It follows that g ( z + ω i ) = µ i g ( z ) ( i = 1 , , (39)i.e. the function g is a so-called elliptic function of the second kind. There existsa complete classification of all such functions (cf. Appendix C). Thus, f will be ofthe form f = U † · g (40)with g some elliptic function of the second kind, and, by Lemma 2, the densitiesassociated with these functions are the same: ρ f = ρ g . (41)10onversely, if g is a quasi-elliptic function of the second kind with multipliers µ i satisfying | µ i | = 1, then its associated density ρ g is periodic. Indeed, for anysuch function g there are matrices γ ω i = (cid:20) √ µ i
00 1 / √ µ i (cid:21) ∈ SU(2) ( i = 1 , , (42)with g ( z + ω i ) = γ ω i · g ( z ) ( i = 1 , , (43)and the claim immediately follows from the corollary to Lemma 2. γ ω and γ ω anticommute If our matrices γ ω and γ ω anticommute, we can diagonalize one of them andanti-diagonalize the other. Specifically, there is a matrix U ∈ SU(2), such that γ ω = U † (cid:20) − i i (cid:21) U, γ ω = U † (cid:20) − λλ − (cid:21) U, (44)for some complex λ with | λ | = 1. Now put M := (cid:20) i λ (cid:21) , (45)whence γ ω = U † M † (cid:20) − i i (cid:21) M U, γ ω = U † M † (cid:20) ii (cid:21) M U, (46)which is to say γ ω = V † (cid:20) − i i (cid:21) V, γ ω = V † (cid:20) ii (cid:21) V (47)for some V = M U ∈ U(2).Let us briefly digress to remark on the subgroup G of PSU(2 , C ) generated by T ( γ ω ) and T ( γ ω ).If we define a := T ( γ ω ) : z (cid:55)→ (cid:20) − i i (cid:21) · z , b := T ( γ ω ) : z (cid:55)→ (cid:20) ii (cid:21) · z, (48)and c := a ◦ b : z (cid:55)→ (cid:20) − (cid:21) · z, (49)11e get the composition table of the famous Vierergruppe V = Z × Z : ◦ a b c a b ca a c bb b c ac c b a , (50)where 1 denotes the identity transformation z (cid:55)→ z . Our subgroup G is isomorphicto Z × Z !Coming back to our classification problem, it follows that any Ω-quasi-ellipticfunction f is of the form f = V † · g, (51)where g is a function meromorphic in the plane satisfying g ( z + ω ) = − g ( z ) , g ( z + ω ) = 1 /g ( z ) . (52)Conversely, from the corollary to Lemma 2 it is plain that the density ρ f associatedwith any such f is periodic, for there are matrices M , M ∈ U(2), such that f ( z + ω i ) = M i · f ( z ) for i = 1 ,
2. Moreover, ρ f = ρ g .We now proceed to classify all meromorphic functions in the plane which satisfythe period condition (52). Suppose g ( z ) is some such function satisfying equation(52) and let g ( z ) be any other such function. Put f ( z ) := g ( z ) /g ( z ) . (53)Then f ( z + ω ) = f ( z ) , f ( z + ω ) = 1 /f ( z ) . (54)If we define ϕ ( z ) := U † · f ( z ) (55)with U := (cid:20) − (cid:21) , (56)it follows that ϕ ( z + ω ) = ϕ ( z ) , ϕ ( z + ω ) = − ϕ ( z ); (57)therefore, ϕ ( z ) is some multiplicative quasi-elliptic function with µ = 1, µ = − a , . . . , a n ∈ C , (58)and parameters z , . . . , z n ∈ { t ω + t ω | ≤ t , t < } (59)12n the fundamental domain of the lattice Ω = Z ω + Z ω , such that ϕ ( z ) = (cid:34) a + n (cid:88) k =1 a k d k ζdz k ( z − z ) (cid:35) σ ( z − z ) n (cid:81) nk =1 σ ( z − z k ) e ζ ( ω / z , (60)where z = ω n + 1 n n (cid:88) k =1 z k . (61)Therefore, g is of the form g ( z ) = (cid:2) U · ϕ ( z ) (cid:3) g ( z ) = − ϕ ( z ) − ϕ ( z ) + 1 g ( z ) . (62)Conversely, any such function g satisfies the conditions (52).It remains to give some g satisfying equation (52). Inspired by Olesen’s specialsolution [9], we make the Ansatz g ( z ) = O ( z ) := ℘ ω , ω ( z ) + bc ℘ ω , ω ( z ) + d . (63)We have the general formulas [19] ℘ ( z + ω ) = e + ( e − e )( e − e ) ℘ ( z ) − e , ℘ ( z + ω ) = e + ( e − e )( e − e ) ℘ ( z ) − e , (64)with ℘ ≡ ℘ ω , ω , e := ℘ ( ω ), e := ℘ ( ω ), and e := − ( e + e ). Using theseformulas and demanding that g satisfy (52), we can choose the parameters b , c ,and d in our Ansatz (63) appropriately. With the help of a computer algebrasystem (Mathematica) we have found that b = − e + c ( − e + e )1 + c , d = c ( − e + e − c e )1 + c , (65)with c = (cid:115) − e + 2 (cid:112) ( e − e )(2 e + e ) e + 2 e (66)will do, as long as e + 2 e (cid:54) = 0. Indeed, e + 2 e = 0 only in the limit where ourtorus degenerates into a cylinder and this is excluded. This concludes our proof ofTheorem 1. It turns out to be immaterial which branches we choose for the square roots. In this sense,the choice of parameters is essentially unique. .3 The abstract underlying group We now explain how to refine our classification from a different perspective, usingthe isomorphism PSU(2 , C ) ∼ = SO(3) ∼ = H with the different model groups of spacerotations, and unit quaternions H . Let us denote by G the subgroup (in any ofthese models) generated by γ ω and γ ω . Then G is an abelian group of rotations,which is intrinsically attached to the vortex solutions of the torus Jackiw-Pi model.We call the abstract isomorphism type of this group the type of the vortex solution. We denote by Q the set of rational numbers. As usual, we call a real num-ber irrational if it is not rational. We call two real numbers linear dependentover Q (abbreviated “LD”) if one is a rational multiple of the other (and linearindependent otherwise).Suppose our rotations are around the same axis, one through an angle 2 πθ , theother through an angle 2 πθ (cid:48) . If one of θ and θ (cid:48) , say θ , is rational with denominator m , then its associated rotation generates a cyclic subgroup Z m of G of order m . If then θ (cid:48) is irrational, we find that G ∼ = Z m × Z (where it is possible that m = 1, in which case G is infinite cyclic: G ∼ = Z ). If both θ and θ (cid:48) are rationalwith denominators m and n , say, then G is a cyclic group of order the leastcommon multiple lcm( m, n ) of m and n , that is, G ∼ = Z lcm( m,n ) , a finite cyclicgroup (possibly trivial , which corresponds to genuinely elliptic functions). Finally,if θ and θ (cid:48) are both irrational and linearly independent over Q , the correspondingrotations generate a group G ∼ = Z × Z , but if they are linearly dependent over Q ,they generate a group G ∼ = Z .Suppose now that G consists of two commuting rotations around different axes.It is easy to show (e.g., using the unit quaternion picture, in which a rotationaround an axis (cid:126)v = ( v , v , v ) through an angle 2 θ is represented by cos θ +sin θ ( v i + v j + v k )) that the only pair of commuting rotations are two rotationsof 180 ◦ around two orthogonal axes, and then, abstractly, the group G is the Vierergruppe . Also, up to an isometry of space, we can assume that the axes arein a fixed position, so this group G can be conjugated in SO(3) into standard form.Thus, we see that Case A corresponds to rotations around the same axis,whereas Case B corresponds to the Vierergruppe of two rotations around twodifferent axes.We have summarized the preceding discussion in Table 2.3. In this table, wedenote by ord( µ ) the multiplicative order of a complex number µ in C ∗ , i.e., thesmallest positive integer N for which µ N = 1 (and we put ord( µ ) = ∞ if no suchinteger exists). We call two complex numbers µ and µ multiplicatively dependent (abreviated “MD”) if there exist integers N and N such that µ N = µ N . Wedenote a space rotation around an axis (cid:126)v through an angle θ by R (cid:126)v ( θ ). Note againthat in this table m and n are integers, so Z lcm( m,n ) can be the trivial group (if m = n = 1), and Z m × Z can be an infinite cyclic group ∼ = Z (if m = 1).14n SU(2) in SO(3) type Case A: commuting Same rotation axes • ord( µ ) = m and ord( µ ) = n (cid:104) R (cid:126)v (2 π/m ) , R (cid:126)v (2 π/n ) (cid:105) Z lcm( m,n ) • ord( µ ) = m and ord( µ ) = ∞ (cid:104) R (cid:126)v (2 π/m ) , R (cid:126)v (2 πθ ) (cid:105) , θ / ∈ Q Z m × Z • ord( µ ) = ord( µ ) = ∞ MD (cid:104) R (cid:126)v (2 πθ ) , R (cid:126)v (2 πθ (cid:48) ) (cid:105) , θ, θ (cid:48) / ∈ Q LD Z • ord( µ ) = ord( µ ) = ∞ not MD (cid:104) R (cid:126)v (2 πθ ) , R (cid:126)v (2 πθ (cid:48) ) (cid:105) , θ, θ (cid:48) / ∈ Q not LD Z × Z Case B: anticommuting Orthogonal rotation axes (cid:104) R (cid:126)v ( π ) , R (cid:126)w ( π ) (cid:105) ( (cid:126)v ⊥ (cid:126)w ) Z × Z Table 1: Possible “types” of vortex solutions on the torus ( m, n are integers).
A brief glance at Theorem 1 will convince the reader that, in particular, the den-sities associated with elliptic functions furnish examples of periodic vortices (take µ , µ = 1 in Case A). The type of these solutions is trivial.A function f is elliptic with respect to the lattice Ω = Z ω + Z ω precisely ifit can be expressed as f ( z ) = R ( ℘ ( z )) + ℘ (cid:48) ( z ) R ( ℘ ( z )) , (67)where R , R are rational functions and ℘ ≡ ℘ ω ,ω .It is easy to see that if we put ω i → t ω i ( i = 1 ,
2) and take the limit t → + ∞ ,then (compare [20], pp. 85 ff.) f ( z ) → R ( z − ) − z − R ( z − ) . (68)That is, in the limit where we remove the periodic boundary conditions (the planarlimit), f tends to a rational function. Since any rational function can be writtenin the form (68) for some rational function R , any rational function can arise inthis way as the limit of an elliptic function. Thus, in this way we obtain all staticvortices on the plane. An elliptic solution with flux loss.
Let ρ f t ( t >
0) be the density associatedwith the function f t ( z ) := ℘ (cid:48) t,it ( z ) ℘ t,it ( z ) . (69)(We are dealing with the torus C / ( Z t + Z it ).) Figure 1 shows a plot of this densityfor t = 1. Numerical integration suggests that for the rationalized charge q torus .8 1 1.2 1.4 1.6 0.4 0.6 0.8 1 1.20510 0.8 1 1.2 1.4 1.6 Figure 1: Plot of the density (in units of 1 /e ) associated with the function ℘ (cid:48) t,it ( z ) /℘ t,it ( z ) for t = 1 in the cell 0 . ≤ x ≤ .
7, 0 . ≤ y ≤ .
3. Large val-ues of the density have been clipped.associated with this solution ( t > q torus = e π (cid:90) F ρ f t d x = 4 ( F := [0 , t ] × [0 , t ]) . (70)Now, the planar limit of f t is f t ( z ) → − z for t → + ∞ , (71)and it is well known that the charge associated with this is q plane = e π (cid:90) R ρ z (cid:55)→− /z d x = 2 = 12 q torus . (72)We therefore have the surprising result that, in passing from the torus to the plane,some charge of a vortex can get lost. An elliptic solution with no flux loss.
That this need not always happen isshown by the example of the density associated with g ( z ) = ℘ t,it ( z ). Here, thecharge in the planar limit is the same as on the torus, namely = 4.16 .2 Relatives of Olesen’s solution In [9] Olesen investigated a periodic vortex with charge q = 1. In our language,this solution is associated with the function O ( z ) (equation (63)) for the squarelattice Ω = Z t + Z it with t >
0. In Figure 2 we have plotted this density for t = 1. Figure 2: Plot of Olesen’s density (in units of 1 /e ) on a 2 × , × [0 , O ( z ) on arbitrary tori C / ( Z ω + Z ω ). For instance, Figure 3 shows the density for a sequence of lattices Ω = Z + Z it , where successively t = . , . ,
1. Note how the drempel-like structure deforms to the lump of Figure 2 as the rectangle approaches a square. From nu-merical integration we know that all these vortices have charge q = 1 and the sameappears to be true for tori where the fundamental region is a true parallelogram.What is the planar limit of the density associated with O ( z )? It is easy tosee that for a square fundamental region, O ( z ) approaches a constant as the pe-riod lengths tend to positive infinity, that is, in this limit, the associated densityapproaches 0. It appears likely that the same is also true for more general funda-mental regions. “Drempel” is a Dutch word which, amongst other things, denotes a speed bump. Summary and discussion
In this paper we have studied the Jackiw-Pi model with periodic boundary con-ditions, which amounts to solving the Liouville equation on the torus. Physically,these solutions describe a two-dimensional periodic lattice of charged vortices withquantized magnetic flux. As first discussed in [1, 2] the existence of vortex solu-tions requires a delicate tuning of the coupling parameters: the electric chargeand the strength of the self-interaction. Surprisingly, it seems that this tuning isnot destroyed by quantum fluctuations [21, 6, 18]; on the contrary, the tuning isprecisely the condition for which the β -functions of the model vanish and there isno scale-dependence of the parameters, at least at one-loop order.On the torus, the spectrum of fluxes of the vortices differs from the planarcase; it is richer in that it allows both odd and even integer fluxes. This is possiblebecause periodic functions on the plane do not vanish at infinity, as required forthe solutions on the infinite plane. However, it also implies that the limit of thetorus to the infinite plane is singular and can change the flux associated with acertain solution. We have presented explicit examples of this phenomenon. Thisobservation may be relevant also in other field theories with soliton solutions, e.g.the Skyrme model as an effective theory for the bound states in QCD.It is amusing to note that our physical classification of vortices on the torushas a purely mathematical consequence having to do with the geometrical contentof the Liouville equation: We can interpret our density ρ as the conformal factorof a metric on a punctured torus, with punctures exactly at the zeros of ρ . Ourclassification theorem then gives all sufficiently smooth metrics of constant Gaus-sian curvature K = e > From our physicalarguments in Appendix B it also follows that the properly normalized integral (17)of the conformal factor over the torus is always a non-negative integer.In reference [22] a topological interpretation of the charge of vortex solutionson the plane was given. It would clearly be interesting to obtain an analogousinterpretation for the theory on the torus and we believe that the remarks inAppendix B could constitute the first steps in that direction.
Acknowledgments
We are indebted to P. Horvathy for correspondence and comments, and to C.Hill, S. Moster, E. Plauschinn and B. Schellekens for helpful discussions. Two ofus (N. Akerblom and J.-W. van Holten) have their work supported by the DutchFoundation for Fundamental Research on Matter (FOM). NA also thanks the Max-Planck-Institute for Physics (Munich) for hospitality during the final stage of this On an unpunctured torus, there are no such metrics, compare footnote . A Proof of Lemma 2
Here we prove Lemma 2 of Section 2. We only need to supply the proof of the“ ⇒ ”-direction; for the “ ⇐ ”-direction see [6, 18]. For clarity, let us repeat thestatement (in slightly altered notation): Lemma 2 (“ ⇒ ”). Let f and ˜ f be non-constant meromorphic functions on theplane and suppose that their associated densities ρ f and ρ ˜ f are equal: ρ f = ρ ˜ f ,where ρ f ( z ) = 4 e | f (cid:48) ( z ) | (1 + | f ( z ) | ) , (73) and analogously for ρ ˜ f .Then there exists a matrix γ = (cid:20) a bc d (cid:21) ∈ SU(2) , such that ˜ f ( z ) = γ · f ( z ) := (cid:20) a bc d (cid:21) · f ( z ) := af ( z ) + bcf ( z ) + d . (74) Proof.
Stereographic projection π : S → (cid:98) C w gives a bijection between the sphere S and the extended complex w -plane (cid:98) C w . In this way, the round metric on thesphere, ds S , induces a distance function d U on (cid:98) C w , for which the distance betweentwo points w , w ∈ (cid:98) C w is given by d U ( w , w ) = inf Γ (cid:90) | Γ (cid:48) ( t ) | | Γ( t ) | dt, (75)where the infimum is over all curves Γ : [0 , → (cid:98) C w with Γ(0) = w , Γ(1) = w .The orientation preserving isometry group of the sphere, SO(3), is mapped by π tothe orientation preserving isometries of (cid:98) C w equipped with the distance d U , whichis PSU(2 , C ).For a meromorphic function on the plane C z , define a quasi-distance d f by d f ( z , z ) := d U ( f ( z ) , f ( z )) . (76)(We call this a quasi-distance since, although it is positive and satisfies the triangleinequality, it is degenerate in the sense that points z , z at distance zero are notnecessarily equal, but rather satisfy f ( z ) = f ( z ).)19he hypothesis of the theorem concerning equality of densities implies that forevery z , z ∈ C z , we have d f ( z , z ) = d ˜ f ( z , z ) . (77)We now define a map ι : (cid:98) C w → (cid:98) C w by ι ( w ) := ˜ f ( f − ( w )) . (78)First of all, this is well-defined. Indeed, if f ( z ) = f ( z ) =: w , then the definition(76) implies that d f ( z , z ) = 0. Further, equation (77) implies that d ˜ f ( z , z ) = 0,and, again by definition (76), we obtain ˜ f ( z ) = ˜ f ( z ). Our claim is that ι ( w ) isan isometry of (cid:98) C w equipped with d U .It is surjective, since f and ˜ f are not constant. Indeed, for any two points w , w ∈ C w , we have d U ( ι ( w ) , ι ( w )) = d ˜ f ( f − ( w ) , f − ( w )) = d f ( f − ( w ) , f − ( w )) = d U ( w , w ) . (79)Also, ι is orientation-preserving since f is meromorphic.Hence, ˜ f = T ( f ) for some orientation preserving isometry T of (cid:98) C w , that is T ∈ PSU(2 , C ), whence there is a matrix γ = (cid:20) a bc d (cid:21) ∈ SU(2) , (80)such that ˜ f = γ · f. (81) Remark.
It is clear that this lemma can be used for determining the precise struc-ture of the moduli space of self-dual static vortices of the Jackiw-Pi model onthe plane. For, according to Horvathy and Yera [8], any such vortex with fluxΦ = 4 πN/e is given by a density ρ f , where f is a rational function f ( z ) = P ( z ) Q ( z ) , deg P < deg Q = N. (82)Therefore, every such solution has 4 N moduli but, obviously, they are not allindependent. Rather, by our result, the moduli space is some kind of quotient C N / PSU(2 , C ) . The invariant theory of PSU(2 , C ) is well-studied, see e.g. [23]. We leave theproblem of working out the physical implications in detail for the future [24].20 Quantization of flux
We comment here on the quantization of flux of static vortex solutions of theJackiw-Pi model.For the theory on the plane, this quantization is best seen a posteriori fromthe results of Horvathy and Yera [8]. For the time being, an analogous result onthe torus is, however, not available [24]. That is, given a solution from the classi-fication Theorem 1 we cannot say at the moment, without resorting to numericalintegration, what its associated flux is.Therefore, we now proceed to give a more general argument supporting theclaim that the flux is also quantized in the torus case.The boundary conditions of the Jackiw-Pi model on a spacetime of the form R × T , where T = C / Ω for some lattice Ω, are somewhat subtle. Naively, onewould write the gauge potential A as a 1-form on the torus, which would lead to (cid:90) T B = (cid:90) T dA = (cid:90) ∂T = ∅ A = 0 , (83)in contradiction to the solutions with a non-vanishing magnetic flux. The resolu-tion to this puzzle is of course analogous to the Dirac monopole, where we needmultiple gauge patches to describe the solution; in other words, A in reality is asection of a bundle.However, because we are dealing with a torus, we can also pull back the gaugeconnection to the plane, where the gauge potential can be written as a 1-form.The boundary conditions are then implemented by periodicity of the fields ρ , E ,and B , which translates to the equationsΨ( x + ω i ) = e iθ i ( x ) Ψ( x ) ,A ( x + ω i ) = A ( x ) + dθ i ( x ) , (84)where Ω = Z ω + Z ω , that is, our lattice is generated by ω and ω .Now we can use gauge transformations in the plane to set the phase θ tozero, and then we are left with a single phase θ . It is easy to show that undertranslation by ω we have e iθ ( x ) = e iθ ( x + ω ) , (85)and thus θ ( x + ω ) = θ ( x ) + 2 πn . This means that the total magnetic fluxthrough the torus is (cid:90) F B = (cid:90) F dA = (cid:90) ∂F A, (86)where F := { t ω + t ω | ≤ t , t < } (87)21s the fundamental domain of the torus in the plane. The boundary integral is theintegral along the parallelogram where the two sides in the direction of ω cancel,due to periodicity of A in ω . However, the sides in the direction of ω do notcancel, due to the non-periodicity caused by θ . The difference between the twosides is given by (cid:90) ∂F A = (cid:90) ω dθ = 2 πn. (88)Therefore, the total magnetic flux is quantized in units of 2 π . The topology of theprincipal U(1) gauge bundle over the torus is that of a twisted 3-torus with twist n . C Elliptic functions of the second kind
For easy reference we repeat here the results of [14], p. 154 concerning ellipticfunctions of the second kind (=multiplicative quasi-elliptic functions) specializedto the needs of the present paper (see also [25, 26]).
Definition.
Let Ω = Z ω + Z ω be a lattice. A function f which is meromorphicin the plane is said to be an elliptic function of the second kind with multipliers ofunit modulus , if there exist complex numbers µ , µ , with | µ | , | µ | = 1, such that f ( z + ω i ) = µ i f ( z ) ( i = 1 , . (89) Theorem (Lu [14]) . A function f which is meromorphic in the plane is an ellipticfunction of the second kind with multipliers µ , µ of unit modulus if and only ifthere are complex constants a , . . . , a n ∈ C , (90) and parameters z , . . . , z n ∈ { t ω + t ω | ≤ t , t < } , (91) such that f ( z ) = (cid:34) a + n (cid:88) k =1 a k d k ζdz k ( z − z ) (cid:35) σ ( z − z ) n (cid:81) nk =1 σ ( z − z k ) e λ z , (92) where λ = 1 πi ( γ η − γ η ) , (93) and z = 12 nπi ( γ ω − γ ω ) + 1 n n (cid:88) k =1 z k . (94) Here, η i := ζ ω ,ω ( ω i / and γ i := log µ i ( i = 1 , ). (The branch of log µ i can bechosen arbitrarily.) eferences [1] R. Jackiw and S.-Y. Pi, “Classical and quantal nonrelativistic Chern-Simonstheory,” Phys. Rev.
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5, (b) t = .
75, and (c) tt