Vortices with source, FQHE and nontrivial statistics in 2+1 dimensions
Horatiu Nastase***E-mail address: email@example.com and Francisco Rojas†††E-mail address: firstname.lastname@example.org
Instituto de Física Teórica, UNESP-Universidade Estadual Paulista
R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil
Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile
Blanco Encalada 2008, Santiago, Chile
We investigate vortex soliton solutions in 2+1 dimensional scalar gauge theories, in the presence of source terms in the action. Concretely, this would be applied to anyons, as well as the Fractional Quantum Hall Effect (FQHE). We classify solitons for renormalizable potentials, as well as some nonrenormalizable examples that could be relevant for the FQHE. The non-Abelian case, specifically for theories with global non-Abelian symmetries, is also investigated, as is the non-relativistic limit of the above theories, when we get a modification of the Jackiw-Pi model, with an interesting new vortex solution. We explore the application to the ABJM model, as well as more general SYM-CS models in 2+1 dimensions.
- 1 Introduction
- 2 Condensed matter motivation
- 3 Abelian vortex solutions
- 4 Non-Abelian vortex solutions
- 5 Non-relativistic limit
- 6 Embedding in the ABJM model and SYM-CS models in 2+1 dimensions
- 7 Discussion and conclusions
Soliton solutions have played a pivotal role in the development of field theories. For instance, understanding monopoles and instantons was crucial to obtaining the first complete non-perturbative low energy effective action by Seiberg and Witten in the case of 4 dimensional supersymmetric gauge theories [1, 2]. In 2+1 dimensions, vortex solutions have also played a key role in condensed matter theory, in particular for the theory of superconductivity, notably Abrikosov’s vortex lattice. But usually, and especially in the last example (the standard Abrikosov-Nielsen-Olesen, or ANO, vortex), the vortices were solitonic solutions of the equations of motion of an abelian gauge theory involving a scalar field.
Moreover, especially given the fact that there exists a duality between particles and vortices (in 4 dimensions there is a duality between particles and monopoles), which has been explained in a path integral context in , one can ask whether there are relevant vortex solutions that need a source term, just like the electron is a source for electromagnetism. In the case of string solitons of -brane type , it is well known that a source term is needed for both ”electric” and ”magnetic” solutions, albeit for magnetic solutions it drops out of the calculations. More relevantly, one way to describe an anyonic particle (with fractional statistics) is to add a delta-function magnetic field associated with the particle position, and such a magnetic field source can be obtained from adding a source term and a Chern-Simons term for the gauge field. In such a way anyons can be obtained in the Fractional Quantum Hall Effect, via an effective action of the Chern-Simons type, and also by an analysis of the possible wavefunctions.
It is then a relevant question to ask whether we can find more general vortex solutions in 3 dimensional field theories that have Chern-Simons terms, by including source terms for the vortices.111Note that vortex solitons in Chern-Simons theories, relevant to condensed matter theories and the FQHE, have been considered before, for instance in , see  for a review, but here we consider them in the presence of a source, with a different ansatz. We believe this is a novel approach. This has been considered briefly in , but here we will systematically explore the issue. We will carefully study the case of a general renormalizable potential, and find under what conditions do we have a solution. We will then consider some examples of nonrenormalizable potentials that have relevance to the physics of the FQHE. We will then consider the case of a potential for a scalar field with a non-Abelian global symmetry. Moreover, since the nonrelativistic limit of Chern-Simons-scalar theories is known to be relevant in condensed matter systems, we consider the non-relativistic limit of the models we have found, and find a very interesting new type of nonrelativistic vortex solution. We will finally see if we can embed the soliton solutions that we have found in interesting CS+scalar models in 2+1 dimensions, specifically the ABJM model and SYM-CS systems.
The paper is organized as follows. In section 2 we consider the motivation coming from condensed matter, namely anyonic physics and the FQHE. In section 3 we examine the generality of solutions of abelian models. In section 4 we generalize to non-Abelian models, and in section 5 we take the non-relativistic limits. In section 6 we embed in supersymmetric CS theories, and in section 7 we conclude.
2 Condensed matter motivation
The motivation for the present work comes from condensed matter physics, thus in this section we present the implications of having delta function sources together with Chern-Simons terms.
The explanation of why adding a delta function magnetic field situated at the position of a particle makes that particle anyonic is a standard one (see for instance section 2.1 in , as well as [8, 9]).
Consider an action for a Chern-Simons field coupled to a (vortex) current,
where the (vortex) current is, in the static case, simply a delta function,
Then the equations of motion are
and we see that we have a magnetic field localized only at the position of the delta function source. In the gauge the equations of motion have the solution
which generalizes to the case of separated sources as
Except at the positions of the sources (vortices), this is a pure gauge, but removing it by a gauge transformation would imply that a scalar field or wavefunction would acquire a phase,
The Hamiltonian of such particles (or rather, vortices, see later) in the presence of the singular gauge field would be
Moving one particle around another results then in an Aharonov-Bohm phase
which can be interpreted as a double exchange of two identical particles, thus having an anyonic exchange phase of
In conclusion, coupling the Chern-Simons term with a source results in anyonic particles at the position of the source. We want these particles to actually be vortices in a more general theory, in which we embed the Chern-Simons plus source term.
The gauge field above, obtained from a gauge transformation, is called a statistical gauge field, and is an emergent one. Indeed, the Chern-Simons action that describes it has no degrees of freedom. Note however that for the anyon argument it does not matter whether is the emergent or the electromagnetic gauge field, all we need is the presence of the gauge field flux, which will lead to the Aharonov-Bohm phase giving fractional statistics.
2.2 The Fractional Quantum Hall Effect
We can in fact have anyons in the Fractional Quantum Hall Effect. An effective action that describes the FQHE is written in terms of the electromagnetic field and the statistical (emergent) gauge field as (see for instance , and the original reference )
In the absence of the source term, we could integrate out via its equation of motion,
which means that, up to a gauge transformation, we have , and by replacing back in the action we get
The appearence of the fractional coefficient instead of the integer one suggests a fractional quantum Hall conductivity, which is indeed the case.
The charge carriers added to the effective action are anyonic quasi-particles, with integer charge under the statistical gauge field . We can also add an explicit electric charge coupling to the effective action,
The equation of motion for is
which can be solved by either or being a delta function.
If one chooses , which is a truly 2+1 dimensional field (statistical gauge field), to be a delta function,
then by substituting it back in the effective action we get an effective electric charge coupling (from the coupling to of ) of
i.e., these quasi-particles have fractional electric charge.
But we have also another possibility. Since is a 3+1 dimensional field restricted to 2+1 dimensions, we can solve by being the delta function, by considering the extension in the third spatial dimension, specifically a flux tube in it. In the absence of vortices, this interpretation is shaky, since we could only consider small current loops, that create the above flux tube in the center, but that also needs corresponding anti-fluxes nearby, generated by opposite loops. However, a vortex obtained when adding scalar fields to the system makes more sense: it will have the flux at the core, and the only constraint is to have an equal number of vortices and anti-vortices on the plane, so that there is no total flux escaping in the third spatial dimension.
Vortices also are necessary for the standard explanation of the FQHE by Robert Laughlin, using Laughlin’s wavefunction (postulated as an approximation to the true ground state, but verified experimentally to be approximately correct to a very high degree of accuracy) for electrons at , , of
Here is odd, , and this wavefunction corresponds to the FQHE at filling fraction (ratio of filled Landau levels) of . For (a single electron), we have
which corresponds to vortices situated at ( in polar coordinates becomes ).
On top of this ground state, the state with a ”quasi-hole” at the origin is
Thus the quasi-hole has a quantum of vorticity (it is a vortex), has an electric charge , and putting of them together makes up an electron, in the Laughlin ground state.
Thus a vortex with a scalar field profile, with a source term, could generate the real magnetic field at the origin, or the magnetic field, and could be identified with the quasi-hole needed by Laughlin. The scalar field can be thought of as a composite field in terms of the fundamental degrees of freedom.
3 Abelian vortex solutions
We have now defined the starting point of our analysis: we want a scalar field coupled to a gauge field, including a Chern-Simons term, and with a source term in the action. We will analyze the possible vortex solutions of such an action. In fact, the analysis of abelian vortex solutions with source was begun in , but most of the analysis was missed there, so here we will do the full analysis.
3.1 Set-up and general analysis
With the idea of being able to apply to the FQHE, we consider an effective action for the statistical field , electromagnetic field and a complex scalar field ,
where for the FQHE we would put , and possibly with a Chern-Simons term for the statistical gauge field and a source for it, making it an anyon,
The equations of motion of , after imposing (zero electromagnetic field), are
where . Adding to would only change the equation for (the second one above), by adding a term .
Note that we consider (zero electromagnetic field) since we are now interested in a solution that doesn’t have its topology associated with a nontrivial magnetic field , like for the usual ANO vortex, but rather it’s associated to a nontrivial statistical gauge field. Indeed, there is no evidence for a nontrivial ANO-type vortex structure in the FQHE, like in the case of the Abrikosov vortex lattice for superconductors.
We consider the usual one-vortex ansatz,
where is the polar angle, satisfying
The equations of motion for then become
We consider also an ansatz for the gauge field that solves the second equation of motion explicitly. We see that , where is the argument of (), i.e.
solves it, and it also solves the first equation of motion everywhere, including at , if (and only if) . Therefore if we can consider all the equations of motion to be satisfied at , whereas if , we consider that we have an idealization of some real situation, and the equation of motion at need not be satisfied.
Adding to , the extra term in the equation of motion vanishes on the ansatz , , if (so, if , for ).
Since the equation for the gauge field ansatz can be written as , and is also valid at infinity, this gives the usual charge quantization condition , so . Note however that the charge that is quantized is the topological statistical charge associated with , and not the magnetic charge associated with . The solution has statistical magnetic flux , not usual magnetic flux .
Finally then, one is left with an equation of motion for to solve in order to find the vortex solution. The equation,
takes the form of a classical mechanics problem for motion in an inverted potential ,
where time is replaced by radius and position is replaced by field .
The usual argument for vortices, that we need in order for the solution to make sense (since it gets multiplied by the phase that is ill defined at ), still holds.
Less clear is the case of the usual condition for the one-vortex solution, as (imposed such that , where ), and for an -vortex solution (so that ). We will see that now it makes sense to consider more general possibilities for the behaviour near . This analysis however would all be valid only for the case; otherwise the equations of motion at are not satisfied anyway, so we need to think of the system as an idealization of some real, but smooth, system, that will not need to satisfy the equation at .
The picture of a classical (point mass) motion in an upside down potential allows us to get a simple intuition for whether there are solutions or not. The motion in is frictionless, so energy (kinetic plus potential) is conserved. We start off at with a position and a velocity , and the question is whether the point mass can come to rest at with zero velocity (only potential energy, no kinetic). That is only possible if the potential has an extremum () at the position where we end up at infinity. See Figures 1 and 2 showing the relevant cases for our analysis.
For the condition at , we see that , with is excluded, since then the velocity at zero, , so motion with infinite initial kinetic energy can only lead to infinite final potential energy, which is impossible (and contradicts the finite energy condition for the soliton).
If , then we have a finite velocity at zero, thus a finite initial kinetic energy for motion in , which means that the final point can only be at a higher value of , which means a lower value of . There are basically two classes of potentials admitting this type of vortex solutions characterized by the sign of . Figures 1 and Figure 2 show the potentials for and respectively, denoted type 1 and type 2 in the following. Note that for (depicted in Figure 2) the potential needs to be sextic.
Finally, if , with , the velocity at zero vanishes, (though there can be acceleration, , if ), which means we move in with zero initial kinetic energy, thus the initial and final points must have the same value of , i.e. . We will however see (below eq. (3.18)) that this possibility doesn’t satisfy the equations of motion. In all cases must be an extremum of .
We must also consider that we want finite energy solitons. Because of this, the extremum where ends up at must have , otherwise the constant energy density would integrate to infinity.
We can then easily summarize the condition for a potential to admit this kind of solitons. must have an extremum () at a nonzero , and we need (so , since the point mass must have smaller potential energy than at infinity throughout its trajectory).
Note that in all these solutions, we have , which then implies that we need . Indeed, if the point mass moving in leaves at and reaches some at finite time, after which it comes back towards , it will necessarily reach again in finite time (corresponding to finite for ), so this possibility is excluded.
But in the above analysis we have assumed that we reach . However, that is actually not necessary, we can have solutions with finite support, so called “compactons” .222These solitons are solutions of a generalization of the KdV equation (see  for details). These would be solutions for which for . Since must be continuous, we need that as well, and since as we saw we wanted , the classical motion in must be symmetrical: we start off at a , perhaps with some velocity, then move up to some , and then come back to at . The possibilities are plotted in Figure 3. The cases in figures (a), (b) and (d), with negative leading term, will be called type 3 in the following, and cases in figures (c) and (e), with positive leading term in the potential , will be called type 4. Note that for all these cases also at , corresponding to motion with an initial velocity. The case of , with , will again be excluded by the equations of motion.
Note that the fact that (or a higher order derivative) is discontinuous at is not a problem. The only thing we need is that the equations of motion at are satisfied, which means that we need that the second derivative is continuous,
This condition means that we need that is an extremum of .
Note also that, since the field has compact support, any (non-divergent) solution will have finite energy upon integrating over . Thus, as opposed to the previous cases, there is no need to impose extra conditions on .
3.2 Renormalizable potentials
We now consider various possible potentials and analyze the solutions in more detail. We start the analysis with renormalizable scalar potentials, having in mind that the composite scalar is nevertheless a quantum field. We will treat the case that comes from a full quantum effective action, so can be nonrenormalizable, in the next subsection.
In 3 dimensions, the most general renormalizable potential is sextic. Since moreover, we want the potential to only depend on , we consider the general potential
giving the equation of motion
In the following we will assume that is such that the solution ends up in an extremum point () with , and we will not write it explicitly. This is needed, since we want to have a solution with finite energy, otherwise, if at , we would get an infinite contribution to the energy.
We will treat the potential case by case. An important condition is whether is positive (a mass term) or negative (an instability), so we will split the analysis according to this condition.
If , starts going down at , and is a local maximum for . Therefore we need to be in type 2 for the usual solitons, or type 4 for the compactons. Note that all of them require that the potential starts to go down after the initial upward trend, so we need either or , but both cannot be positive (nor zero).
We start off with the solution of the type 1, that has a nonzero velocity at , i.e. as .
Near the origin, , considering the first power law correction to the vortex behaviour , i.e.
the equation of motion fixes and , so we have
We could imagine a solution with , with , as . But that doesn’t satisfy the equation of motion, since then , whereas or higher, independent of whether or are zero. So a solution of this type is actually never possible, because of the equations of motion. In fact, the same argument shows that a compacton solution of this type is also always excluded.
We will only consider the compacton solutions at the end of this subsection, so we will ignore them for now.
Solution with massless ”Higgs”?
Near , in  it was considered a power law behaviour,
Then the equations of motion lead at first order to
Note that , with implies , which is needed in order to have a nontrivial minimum for the potential, and then implies .
Together, the above conditions give a constraint on the potential () and a constraint for the constant part of the field at infinity to be at the minimum of the potential, so that the scalar field solution tends to the nontrivial vacuum,
In this case, solving the equations of motion to the next order, to find , one finally obtains the solution
But also note that in this case, the ”Higgs”, i.e. the fluctuation transverse to the vacuum manifold, has zero mass, i.e. the second derivative of the potential at the minimum is also zero,
We see that in this case we need .
It would then seem like we are in the case of type 1 solution, but we are not, since in this case
and the potential is strictly increasing, there is no maximum in between and as needed (and is not larger than as needed).
Thus there is no solution with massless ”Higgs” in this case.
Solution with massive ”Higgs”
But the generic case is one with massive ”Higgs”. Let us be general, and ask for an arbitrary exponential subleading term (instead of a power law) as ,
Subsituting in the equation of motion, we obtain
From the vanishing of the constant piece, we again obtain that , the minimum (or rather, extremum) of the potential, i.e.
as in the previous case.
From the exponential term, imposing that we are not in the massless Higgs case, i.e. we have
which amounts to the condition on the coefficients of the potential
then we need (since if , so the first term on the left hand side of the equation of motion is dominant, and to match the right hand side, it must be a constant of , i.e. ).
Then, equating the coefficients of the exponential terms in the equations of motion, we obtain
where in the last equality we have used the equation for to be a minimum of the potential, (3.28).
If , from (3.28) we get , which when substituted in above leads to , which is impossible, since is real and positive.
Therefore we need , in which case the solution of the extremum condition (3.28) is
Note that we expect the smallest solution (with the minus sign) to be a maximum, and then the largest solution (with the plus sign) to be the minimum. That is so, since starts to increase from zero, so needs to encounter a maximum before a minimum.
From the condition that the above is real and positive (such that is real and positive) we get first , i.e.
Note that in this inequality we have excluded the case , since that was the massless Higgs case above.
We note that this is real under the same condition as .
We still need to impose that and .
a) If , implies , in which case both solutions are OK for , but we easily see that then only the upper one is OK for . That is as it should be, since the solution with the upper sign corresponds to the minimum, where the field at infinity should end up, whereas the lower sign corresponds to the maximum, which shouldn’t be allowed as solution at infinity.
Thus we have
This potential and a vortex solution obtained by numerically solving the equations of motion are shown in Fig. 4.
b) If , implies , in which case only the lower sign is OK for , but then only the upper one is OK for , so we have no good solution.
Since we are in the case of type 1 solution, where in the inverted potential problem one starts off with kinetic energy, and in the end it is all converted to potential energy, it means that we need to have , which leads to the condition
or in the end
Finally, we need to impose that the asymptotic value of is a minimum, not a maximum of the potential. For that, we need to impose that the mass squared, the second derivative of the potential at the mininum, is positive. We have
But note that, using the equation for the minimum (3.28),
so we have already imposed its positivity, and moreover, since is the mass of the ”Higgs”, i.e. of the physical excitation transverse to the vacuum manifold, and denoting , we have
Thus as usual for vortices, the decay of the scalar field at infinity is governed by the mass of the ”Higgs”. Also as usual, is only fixed by the ”shooting method”, which means we vary (defining the asymptotics at infinity) until the solution has the right asymptotics at .
We have already seen that we need if , and it is easy to see that if and or (purely quartic or purely quadratic potential) we also cannot have a nontrivial soliton solution, since we are in the case of type 1 or 2 (or type 3 or 4 for the compacton), which do not happen for purely quartic or purely quadratic potentials. Therefore in the following we will assume , and consider separetely the cases and .
In this case,
As seen in , considering the same vortex asymptotics near as before,
from the equations of motion we obtain and so that as ,
For the behaviour at , we can actually take a limit on the above case (with ).
We first see that the power law behaviour at infinity (massless Higgs) is impossible, since doesn’t have a good limit. We can also check directly that the power law ansatz doesn’t solve the equations of motion.
Then we can take the of the massive Higgs case. From the condition (3.28), we have a nontrivial vacuum at
We also have a trivial vacuum at (see also (3.33) for ), which is now a maximum of the potential. Thus we are now in the case of type 2 vortex in our classification.
The mass of the Higgs (around the nontrivial vacuum) is, from (3.39),
We need this to be positive, so that is a minimum, not a maximum. This again imposes , which in turn, from the form of , implies that .
Then from (3.35) again we have for the nontrivial exponent,
as well as the trivial .
So the nontrivial behaviour at infinity is again given by the general form (3.78), and the coefficient would be fixed by the shooting method, by imposing that we have the correct behaviour (3.44) at .
Note that we cannot impose that goes over to the trivial vacuum at . This was what was assumed in , but it is incorrect, since this would not correspond to one of the cases (1, 2, 4 or 5) that we have described.
The behaviour at is again given by (3.18), since that was independent of .
Now again there is no solution with massless Higgs, since doesn’t have a limit.
From the condition (3.28), we have an extremum at
which requires (since now), but then from (3.39),
so the extremum is actually a maximum. Indeed, in this case, the potential grows to a maximum and then drops without bound. Since there is no minimum of the potential, there is no nontrivial vacuum. In fact, since , , so the potential is unbounded from below.
Then at , we could only imagine that the field tends to the trivial vacuum, however then the equation of motion gives a contradiction,
which is consistent with the fact that there is no vortex case corresponding to this possibility, according to our general analysis.
This is a purely sextic potential, . This choice was analyzed in . On physical grounds we must choose , otherwise the potential is negative definite, and unbounded from below. There is no usual vortex case corresponding to this potential. Nevertheless, we consider a more unusual possibility, since in this case we can solve the equations of motion exactly.
The possible behaviour at consistent with a normal vortex ansatz is easily found to be
whereas at we must have
In this case we can actually solve exactly the equation of motion. It integrates to
but as explained in , for , the potential is unbounded from below, and for the physical case , there is no solution with the normal vortex asymptotics at zero, i.e. no solution with . Instead, there is a kind of vortex solution with , obtained by putting in the above, namely
We see that it has the the right asymptotics (3.51), but it has , and finite derivative at zero,
However, this probably doesn’t make sense for our vortex equations, since the rest of the equations are satisfied at , but contradicts the equations of motion at .
In this case, if (the case was already analyzed at type above), the only possibilities are vortices of type 2 or 3. Since we will not consider compactons until the end of the subsection, we consider type 2 only.
Again, near the region we have
Just as in the case, we will now analyze the behaviour.
Solution with massless ”Higgs”
As before, assuming a non-trivial vacuum with a power series decay for large of the form
When plugging this into the equations of motion we obtain the same conditions as in the case, namely,
which when combined yield
However, we can see that there is a substantial difference. Since is negative, so must be, otherwise becomes imaginary. Thus, we need , and from (3.58) we also need . Thus, the potential is unbounded from below. Also, once again, the second derivative of the potential at , i.e, the mass of the putative “Higgs” is exactly zero.
Unlike the case however, this solution is actually possible (if we ignore the fact that the potential is unbounded from below, hence unphysical). Indeed, now
so the potential is monotonically decreasing, and has at , which is therefore a good point for the behaviour. This satisfies all the conditions on the type 2 vortex solution.
Solution with massive ”Higgs”
The ansatz is given in (3.25) which we repeat here for convenience
Just as before, plugging this into the equation of motion for gives plus the conditions
at leading and subleading order respectively. Combining both yields
Since here , demanding positivity of gives . Note that contrary to the case, is not ruled out here. More precisely, combining both leading and subleading conditions with yields , and since , the positivity of is ensured.
Solving for yields
and we must impose that be positive.
a) If , then the requirement