Axion-packed superconductors

Physica B 152 (1988) 288-294 North-Holland, Amsterdam

AXION-PACKED SUPERCONDUCTORS

Laurence JACOBS Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

A theory of superconductivity which describes the behavior of a large class of materials is introduced. The new theory generalizes the Ginzburg-Landau theory and shares with it many properties. In a sense made precise in the paper, the generalization is unique. Depending on the magnitude of one of the parameters of the theory, the prediced effects could be dramatic.

1. Introduction

A fact which is seldom mentioned in standard introductions to the physics of superconductors, but it known to experts in the field, is that the essence of this startling phenomenon lies not in the detailed models which describe the dynamics of matter fields in a superconductor, but rather in the fact that electromagnetic gauge invarance is spontaneously broken in these systems. Of course, the calculation of physical parameters, like critical temperatures or fields, requires specific models, but to demonstrate the existence of, for example, the Meissner effect in a superconductor the detailed dynamics of the matter fields are, for the most part, irrelevant [1].

When confronted with a new and unexplained instance of the phenomenon of superconductivity, it is natural to consider the possibility that a generalization of the dynamics of the electromagnetic fields in a superconductor may ac- commodate the observed effects. The purpose of this paper is to present one such generalization, and to explore its predictions.

Fortunately (or unfortunately, depending on one's taste), there is very little one can do to generalize the dynamics of abelian gauge fields while staying within the confines imposed by general principles in quantum field theory. Any such generalization must, of course, be capable of leading to a reasonable scenario for spontaneously breaking the gauge symmetry through interaction with the matter fields.

In an abelian theory, the case of interest here, there are only two independent local invariants which can be constructed from the gauge field tensor, F,v , namely, F~'"F,,~, and F*'VFj,~, where F~,~ is the dual tensor. In terms of the electric and magnetic fields, the first corresponds to the familiar Maxwell expression:

ql -= E2 - B2 , (1.1)

while the second can be written as:

q2 - E- B . (1.2)

It is easy to see that q2 does not affect the field equations, since it can be written as the diver- gence of a (pseudo) vector current:

(1.3)

There is no way to avoid this conclusion without adding more degrees of freedom to the theory. However, if we consider adding an extra field, there is a way to add q2 to the Lagrangian in a way that the above conclusion no longer holds. Furthermore, this modification is essentially unique. Let me assume for a moment that we have at our disposal a field, X(x) , which is a given, external field, summarizing part of the interac- tions between an underlying crystal structure and the gauge fields, coupling linearly to the gauge fields through a term:

q2 =- X02 - X E " B . (1.4)

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L. Jacobs / Axion-packed superconductors 289

The appearance of a pseudoscalar field in the description of the physics of a material is not as peculiar as it may seem at first. Indeed, such a field is necesary to include the effects of certain types of crystalline non-uniformities. Axial fields have been introduced, for example, to describe semiconductor heterojunctions [2]. More gener- ally, a pseudoscalar field in a crystal is probably necessary to describe the interaction of an electromagnetic field with a crystal which has no inversion symmetry; in such systems, the polarization tensor may have non-vanishing crossed terms coupling to E . B . Leaving the analysis of specific examples for future work, in this paper I would like to explore some of the general consequences which follow from this additional term. As I shall show below, these consequences can be profound.

An interaction like that given by (1.4) has been considered in the context of the solution to a notorious problem in quantum field theory [3], where the field X was termed an axion field. It has since become commonplace to use this term to refer to any field which couples to gauge fields as in (1.4).

Below I shall discuss specific forms for the matter-field contribution to the theory, but for now it is not necessary to do so, since most of the properties of the theory do not depend on these details. What I have in mind for the purpose of the present discussion is a phenomeno- logical Lagrangian of the Ginzburg-Landau type, with the matter dynamics corresponding to those of a complex scalar field, ~b, which couples minimally to the gauge potential. Microscopic theories which lead to this effective theory can be written down, but I will not do so here.

The model I shall be working with is thus described by the Lagrangian:

L = 1 f d3x (E 2 _ B 2 + c x E . B ) + L m , (1.5)

where c is a constant, and

netic potential with

B = V × A ,

0,4 E = -7 ,4 o Ot

(throughout we use units where Planck's constant and the speed of light are unity).

The field equations which follow from (1.5) are given by:

v . e = Jo - c n . V x ,

OE v x s = J + a--7 - c ( e x v x - z £ ) ,

(1.6)

(1.7)

together with the Bianchi identities:

OB V x E = - - -

Ot '

V . B = O ,

and the field equations for the scalar field, ~b, which we leave unspecified for now. In the above, J0 and J are the usual currents which result from the interaction of the matter field with the gauge fields.

Much of what will follow is a consequence of eqs. (1.6) and (1.7).

2. The gauge-field equations

The introduction of q2 in the theory, as seen from (1.6) and (1.7), is thus equivalent to the addition of an extra electric charge density proportional to B "Vx, and an extra electric current proportional to E x VX - B)~. Notice that this is a Hall current, perpendicular to the electric field. That both of these terms vanish for constant X is simply a restatement of the fact that g . B

The theory defined by (1.5) is invariant under ordinary U(1) gauge transformations, with

L m = Lm(A0, A, ~b)

is the matter Lagrangian. A" is the electromag-

A.(x)--~ A~(x) + O,,A(x) ,

a(x)--> a (x ) + A(x ) , (2.1)

290 L. Jacobs / Axion-packed superconductors

where a(x) is the phase of the scalar field, 4~, and A(x) is an arbitrary real function. This symmetry is spontaneously broken by the ground state. According to the general remarks made in the Introduction, therefore, this theory describes a superconductor. In fact, this theory shares many of the properties of the Ginzburg-Landau (GL) picture. However, as we shall see below, the two theories also differ in important respects.

A fact which immediately follows from our general considerations is that our theory, like GL, predicts the existence of vortices. This is a direct consequence of the fact that the first homotopy class of the gauge group, 7rl[U(1)], coincides with the group of integers. The vortices in this theory, however, differ from the ordinary Abrikosov vortices of GL mainly in one respect. (In the following section I show how these features appear explicitly when Z m is specified.)

Notice first that the modified Gauss law, eq. (1.6), leads, upon integration, to a relationship between the total electric charge of the configuration and the projected magnetic flux:

Qx = c f d3x B "V X . (2.2)

This brings us to an old argument which showed that Abrikosov vortices could not support a static electric charge [4]. The argument is very simple. Recall that in the usual GL picture all fields are short-ranged (the gauge fields become massive as a result of the spontaneous symmetry breaking). Therefore, when X = 0, the integral of V. E vanish unless E is singular somewhere. But this would imply that the energy per unit length of the vortex diverges. Therefore, for Abrikosov vortices one must have Q = 0. This argument does not hold in Our theory.

The possibility of associating an electric field with our vortices affects their dynamics in important ways. Two immediate consequences are that the angular momentum stored in the field is non-zero, unlike the case of configurations with Abrikosov vortices, and that the magnetic moment of our vortices is not just proportional to the magnetic flux, but has an additional contribution. Also, clearly the intervortex potential will

have an extra repulsive contribution at short distances coming from the electric field.

I have been somewhat imprecise when refer- ring to these objects as vortices. In reality, all that we can say in general is that they are topologically stable objects. Their detailed structure depends on the properties of the axion field, X. In order to be more precise, let me specialize the theory somewhat by specifying X. A choice which meets our constraints is for X to be essentially constant along all but one direction, say the z-direction. What I have in mind is therefore a kind of antiferromagnetic domain wall separating x-y planes. In such systems one would have X changing smoothly from a value - e at z = - a to a value +e at z = 6, with small 8. Inside this region VX = (axlOz)~ ~ O, with an average magnitude of order za = ~/6. Eq. (2.2) leads to a striking effect: the electric charge per unit length of the resulting vortex is quantized as a consequence of the quantization of the magnetic flux:

27r - n , ( 2 . 3 )

Q =/zq, = ~z e

with integer n, and ~ = czl. Recall that Q is an ordinary conserved charge. In fact, the equation of motion for the phase of the scalar field, a(x), is just the equation for charge conservation:

a J0 - - + V . J = 0 . ( 2 . 4 ) at

The peculiarity is that a Noether charge gets quantized through a topological constraint. I know of no other instance of this effect. For the static case of interest here, the field equations (1.6) and (1.7) reduce, in the region where VX ~/~, to

v . E = Jo - (2.5)

V x n = J - / z E x ~z. (2.6)

Before proceeding to a specific example for the matter Lagrangian, L m, note that eqs. (2.5) and (2.6) can be derived from a two-dimensional field theory when the gauge fields are constant


along the z-direction, which is precisely the case of interest here. In two dimensions (d = 2 + 1), the term corresponding to q2 can be written as:

/xe"OVA F~, . (2.7)

This is the famous Chern-Simons second charac- teristic [5]. In the context of a (2 + 1)-dimensional theory, the vortices we have described here have been discussed in the literature and some of their properties have been given [6, 7]. The main thing to notice from (2.7) is that it implies that the gauge fields in the theory have a mass even in the absence of any symmetry breaking; indeed, even in the absence of matter fields. This topological mass [5], which appears in a gauge-invariant manner, can only exist in an odd number of space-time dimensions. To see that/~ is (2.7) is a mass term for the gauge fields, write the field equations as:

8Hm &(x) = 6(_J0(x) ) , (2.11)

where H m is the Hamiltonian for L m. But the voltage at x is just the change in energy density per change in charge density, so

V(x) = - & ( x ) . (2.12)

Therefore, a steady current in the system, consis- tent with static fields, implies that V(x) = 0. This can only be true if the resistance of the system to an applied current is zero.

We are now ready to show that the above picture can be realized in a specific model. This we do in the following section, where we will speci fy L m and demonstrate that the resulting theory admits static field configurations corresponding to an arbitrary number of vortices.

(2.8)

or, equivalently,

([--] -F 2 ~ p, )Fp = 0 , (2.9)

where F~ o, = %o,F 12 is the dual of the field tensor, a (pseudo) vector in (2 + 1) dimensions. This implies that the penetration length in our superconductor is a function of/x.

Several other features of the general model given in (1.5) can easily be deduced. In particu- lar, it is not difficult to see that, as anticipated, the theory displays the Meissner effect and al- lows static configurations which imply that the resistance of the system to an applied current is zero. This last point can be derived rather ele- gantly. Recall that the charge density, J0, is defined as the functional derivative;

6L m J0 - 6a ' (2.10)

where & is the time derivative of the phase of the scalar field. This equation states that -Jo(x) is canonically conjugate to a(x). Therefore, &(x) can be obtained from Hamilton's equation:

3. The matter Lagrangian

A simple choice for L m is the Ginzburg- Landau (or abelian Higgs) potential:

}[2

(3.1)

where, ~b -= F(x) e i~(x) is a scalar field, D~'~b is its covariant derivative, and ~ is a constant.

The full field equations thus take the form:

g"O ° + ~- ~ = (3.2)

D~,D ~ + -~- (~b*~b - ~) ~b = 0 , (3.3)

where the current in (3.2) is given by the usual expression:

j~ = ie[~b(D~b) * - 4~*(D'4,)1 " (3.4)

For / z = 0 these equations reduce, as they should, to the ordinary Ginzburg-Landau equations.

292 L. Jacobs / A x i o n - p a c k e d superconductors

The vortex fields can be obtained from a direct generalization [4, 6] of the Nielsen-Olesen static cylindrical Ansatz [8]:

Ao( p, z) = I t (r ) ,

A(p, z) G(r) = e o , (3.5) r

~b(p, z) =/3(r) e i"e ,

with n an integer, and all fields constant along the z-direction. In the above we have rescaled the radial coordinate, p = r/e~. The equations for the Ansatz functions can be written as:

l d ( d h ) r dr r-drrr-0)g - h f 2 = 0 ' (3.6)

d ( l d g ) rdrr r drr toh - g f 2 = 0 , (3.7)

I d(r ) r dr

n 2 A 2 r-~ (g2 _ rEhZ)f= -2 f ( f 2 _ 1),

(3.8)

where 0)--/x/~, and the dimensionless functions f(r), g(r), and h(r) are related to the rescaled Ansatz functions F, G, and H, by

H(r) = - nh(r) ,

G(r) = n[g(r) + 1], (3.9)

F(r) = f(r) .

f(r)---> 1 + f~ e - a r ,

g(r)--->g~ e -n-+r , (3.11)

- h(r ) ---> +-- g~ e-~+-r, r

with constant f0, go, f~, g~, and where

0)2 1/2 0)

~/+ = 1 + --4"/ -+ --2 " (3.12)

That there are two sets of solutions which differ in their behavior far from the vortex core is related to the fact that in this theory the gauge field, A" propagates two massive modes, one of which is present before the gauge symmetry is broken. We will return to this point shortly, but first note that in this theory there is no a priori constraint on the sign of the topological mass term, since it appears linearly in the Lagrangian; changing the sign of/~ maps one set of solutions into the other. Choosing for definitiveness/~ I> 0, it follows that

l~<r/+ < ~ , O<'o_ ~< 1. (3.13)

I shall refer to the two kinds of solutions as p-type vortices (+solution), and n-type vortices (-solution). Further details of the symmetric solutions can be found in the literature [7].

The magnetic flux for the above configuration is given by:

f ij 2zr • ( B ) = I dZx• OiA ~ = T n , (3.14)

One can show [7] that finite-energy (per unit length along the z-direction) solutions to the field equations exist which satisfy the relevant boundary conditions, with

f(r)----> fo rlnl ,

g( r ) - -~ - i +g0 r2 , (3.10)

to h(r)--* ~ go r2 ,

as r--->O, and, as r--->-0%

so the integer n appearing in (3.5) is the vorticity of the configuration. The remarkable relation between the electric charge of a vortex and its flux, anticipated on general grounds in eq. (2.2), is now explicitly given by:

2qr Q = / x ~ = - - / z n . (3.15)

e

Of course, because of the cylindrical symmetry of our Ansatz, the above configurations corre- spond to a vortex of vorticity n centered at r = 0. However, the Ansatz can be generalized [9] to


generate configurations corresponding to any number of vortices a finite distance apart.

The coherence length, from eqs. (3.11), is = 1/A, while the penetration length is 1/~, in

dimensionless units. (Recall that in the GL limit, /z = 0, 7/+ = 7/_ = 1.) The regimes of interaction of Abrikosov vortices can be understood from this point of view: the contribution of the matter field to the intervortex potential is always attractive with range l/A; that of the gauge field is repulsive with unit range. When A< 1, Ab- rikosov vortices attract each other, with the configurations corresponding to n superimposed vortices being energetically favored over one where the vortices are separated; this is a type I system. When A > 1, vortices repel each other, forming a stable lattice. This corresponds to a type II system. When A = 1, i.e. when the range of both fields is equal, Abrikosov vortices do not interact [9]. One expects the same sort of argument to apply in a heuristic way to our charged vortices, but for a repulsive range determined by /z (the situation here is somewhat more complicated due to the fact that, unlike Abrikosov vortices, our vortices have a non-vanishing angular momentum, but for the purposes of a qualitative description of the vortex dynamics, this picture suffices). This expectation is borne out by ex- plicit calculation [7].

When /z is small, the energy necessary to create a p-type vortex will not be much larger than that necessary to create an n-type vortex. The density of p-type vortices in the sample will thus be comparable to that of n-type vortices. In this case, the possible geometries of the vortex lattices in our theory can be very complex. Of course, determining the structure of the vortex lattice requires detailed calculation. Work along these lines is in progress [10] and, hopefully, results will be available in the near future.

When /z # 0, the two types of solutions we have described live in separate regimes, with I/T÷ < 1 and I/n_ > 1. Thus, in a System with /x # 0, the crossover between type I and type II behaviors will occur, respectively, for A > 1 (p- type vortices), or A < 1 (n-type vortices). For a given solution, then, a Type II system with/z ~ 0 will be characterized by having either a smaller

coherence length (p-type vortices), or a larger one (n-type vortices) than the corresponding neutral, /.t = 0 case. Calling the value of A at which the crossover between type I and type II regions occurs for the two types of vortices, respectively, )tpc and A,c, with

~pc -- "~'nc ~ ¢.0 ,

we will have three distinct interaction regions depending on the value of A relative to the crossover values. If A < A.c, all vortices are in an attractive regime and the material behaves like a type I superconductor; conversely, if A > Apc, all vortices are in a repulsive regime as in a type II system, but characterized by two different effective coherence lengths. Finally, for A,¢ ~< A ~< Ape, p-type vortices will attract each other, n-type vortices will repel each other, and the interaction between p-type and n-type vortices will be repulsive. Of course ~ and hence 7/, are functions of temperature (which in this effective theory enters only as a parameter), so it is possible that all the interaction regimes described above can exist in the same material but at different temperatures. If these vortices exist in physical systems they can thus be easily distinguished experimentally from their neutral counterparts.

If/~ is large, the probability of creating p-type vortices will be exponentially suppressed, and only the less massive, n-type, vortices will be present in the sample. In this case, the difference with ordinary vortices will appear mainly through electric, rather than magnetic, effects.

4. Conclusions

I have described a generaliztion of the Ginz- burg-Landau theory of superconductivity which predicts dramatic, observable effects in certain classes of materials. Although I have not investi- gated specific examples of such materials, and therefore cannot estimate the magnitude of these effects, it seems clear to me that such an investi- gation is warranted. If the magnitude of the crossed term in the polarization tensor of crystals withough inversion symmetry, or in antifer-

294 L. Jacobs / Axion-packed superconductors

romagnetic systems, is not too small, the effects that I have described in this paper may be expec- ted to be dominant.

In recent experiments on high-T¢ materials, certain unusual patterns have been observed under decoration with very small particles whose interaction with the material is electrostatic rather than magnetic [11]. Although there may be a more pedestrian explanation for these re- suits, it is possible that they may be due to the presence of electrically charged vortices such as those we have described here.

It is possible to derive q2 from a microscopic theory in several ways. An example follows from the anomaly equation [12] in a theory describing the interaction of fermions with an effective, non-constant complex mass [13]. It should be interesting to study this kind of microscopic theory, or others, to be able to estimate the magnitude of the various parameters, and their temperature dependence, in specific examples.

Finally, it would be very interesting to study what form the Kosterlitz-Thouless [14] scenario takes in a theory such as the one presented here, where the added interaction should lead to a rich and complex phase structure.

Acknowledgements

I am happy to acknowledge several very help- ful conversions with Francisco Guinea during this Workshop, who also brought ref. [2] to my attention. I also thank David Bishop for describing to me the results of their experiments on

high-T c materials, particularly the patterns seen on decoration with very small particles.

This work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract DE-AC02-76ER03069.

References

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[3] R. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440; S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. For a recent review, see J.E. Kim, Phys. Rep. 149 (1987) 1.

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2291; J. Schonfeld, Nucl. Phys. B185 (1981) 157. For a review and for more recent references on the relevance of such terms in the quantum theory, see R. Jackiw, in: Relativity, Groups and Topology II, B.S. DeWitt and R. Stora, eds. (Elsevier, Amsterdam, 1984).

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