Dual map of a 3D superconductor

This section is the central part of the paper, in which we derive the dual formulation of the Ginzburg-Landau model. We shall argue that this description is one in terms of a genuine order parameter which involves a global rather than a local symmetry as is the case in the Ginzburg-Landau formulation [9]. For this reason the dual theory can be employed to arrive at a conventional Landau description of the superconducting phase transition [8,18,19], which will be the subject of Sec. ix.

The fundamental object of the dual description is the magnetic vortex, or Abrikosov flux tube. Such a topological defect is either closed, infinitely long, or---in the case of a finite system---starts and ends at the boundary of the specimen. A magnetic vortex can never terminate inside a superconductor. However, for reasons that shall become clear when we proceed, we will employ a construct that allows us to describe, at least theoretically, magnetic vortices that do terminate inside the superconductor. To this end we allow the system to contain a Dirac monopole [20] as a test particle. Recall that due to the Meissner effect, a magnetic field can penetrate a superconductor only by forming quantized flux tubes. Since also the flux lines emanating from the monopole are squeezed into a tube, a monopole produces precisely such a vortex.

Section ii revealed that a magnetic vortex is described by a plastic field which appears in the theory in the combination with the gauge field [20]. In other words, in the presence of a test tube, the Hamiltonian becomes

where we added a gauge-fixing term , and given H the superscript to indicate the presence of the test particle with its emanating tube. We recall that we consider the Ginzburg-Landau model (1) in the London limit, where the superconducting order field is written as , with w a constant. The mass term, with , is a result of integrating out the phase field . In three dimensions, the plastic field has the form [20,21]

which is the proper three-dimensional generalization of the 2D result (16). The field satisfies the equation on account of Stokes' theorem, implying that it indeed describes a monopole located at . The line starting at the point and running to infinity is the Dirac string accompanying the monopole. As will be clarified below, the location of the flux tube coincides with that of the Dirac string (see Fig. 1).

From (46) we infer that the operator describing the test tube is given by

We are interested in the expectation value of this operator:

Since the integral over is Gaussian, it can be evaluated by substituting the field equation,

back into the exponent. The gauge-field correlation function appearing here is

The gauge-dependent longitudinal part of the correlation function does not contribute to (49) since . The expectation value is therefore independent of gauge choice.

The local induction corresponding to the classical solution (50) in the presence of the test tube is

where

is the Yukawa potential. The term in (52) describes the Dirac string , the first term on the right-hand side corresponds to the screened Coulomb force generated by the monopole. The last term, which is only present when ---i.e., when there is a Meissner effect---describes the magnetic flux tube. A closer inspection of (52) reveals that the subtraction of the Dirac string from the field is necessary in order to obtain the physical local induction [20,21]. Indeed, if we calculate from the right-hand side of (52) the magnetic flux through a plane perpendicular to the Dirac string, we find

that precisely one flux quantum pierces the surface in the negative direction (see Fig. 1). Here, is an element of the surface orthogonal to the Dirac string, and we used Gauss' law to rewrite the first term on the left-hand side as

Equation (54) confirms Dirac's statement that the magnetic flux emanating from a monopole must be supplied by an infinitesimally thin string of magnetic dipoles and that in order to obtain the true local field of a genuine point monopole, this string has to be subtracted. While this string is indeed completely unphysical in the normal phase, it acquires a physical relevance in the superconducting phase [22] where it serves as the core of the Abrikosov flux tube.

Substituting the field equation (50) back into the theory, we obtain for the vacuum expectation value in the London limit

or

where is the monopole density. In deriving this we have omitted terms depending only on w.

It should be noted that the first factor in (56) diverges

representing the self-interaction of the Dirac string. This term canceled in (57). The first term in (57) contains for a diverging monopole self-interaction. This divergence is irrelevant and can be eliminated by defining a renormalized operator

The second term in (57) is the most important one for our purposes. It represents a Biot-Savart interaction between two line elements and of the magnetic vortex (see Fig. 2). It contains an ultraviolet singularity due to fact that in the London limit, where the mass of the superconducting order field is taken to be infinite, the vortices are considered to be ideal lines. For a finite mass the magnetic vortices have a typical width of the order of the coherence length . This mass therefore provides a natural ultraviolet cutoff to the theory. The last term in (57) then takes the form [22]

with the (infinite) length of the flux tube, and [23]

being the free energy per unit length. The combination defines the dimensionless Ginzburg-Landau parameter , which in the London limit is much larger than 1.

For a monopole-antimonopole pair, (60) amounts to a confining linear potential between the monopole and antimonopole in the superconducting phase. Let describe an antimonopole located at , with being a line running from infinity to . Collecting all terms, we find for such a pair

where is the flux tube connecting the monopole at with the antimonopole at , and is its length.

We remark that the two Dirac strings may initially run to any point at infinity. Due to the string tension, they join on the shortest path between the monopoles. The result (62) is central to our line of arguments. It shows that the correlation function behaves differently in the two phases [9,24]. In the superconducting phase, where the gauge field is massive, the ``confinement'' factor dominates and the correlation function decays exponentially for distances larger than :

This behavior is typical for an operator in a phase without massless excitations. On the other hand, in the high-temperature phase, where the gauge field is massless, the confinement factor in the correlation function (62) disappears, while the argument of the second exponential turns into a pure Coulomb potential. The correlation function remains, consequently, finite for large distances:

By the cluster property of correlation functions this implies that the operator describing the test tube develops a vacuum expectation value. This signals a proliferation of magnetic vortices. Indeed, according to (61) the free energy per unit length of a vortex vanishes at the transition point, where . It should be noted that it is the high-temperature phase and not the superconducting phase where develops an expectation value.

Before deriving the full-fledged dual theory let us rederive the correlation function (62) in a way that reveals some aspects of the nature of the dual theory. To this end we linearize the functional integral over the gauge field by introducing an auxiliary field . In the gauge , which corresponds to setting , we find

where now . Also the divergence is non-zero only at the location of the artificially introduced monopoles. In the absence of monopoles it follows that only the transverse part of couples to . We therefore restrict the integral over the auxiliary field to the transverse degrees of freedom. This is justified by considering the field equation for following from (65)

It tells us that apart from a factor i, the fluctuating field may be thought of as representing the local induction , which we know to be divergence-free in the absence of monopoles.

The integral over the vector potential is again easily carried out by substituting the field equation for ,

back into (65), with the result

We have incorporated a function enforcing explicitly the constraint . In the present formulation this constraint is an intrinsic part of the description of a fluctuating massive vector field. For a non-fluctuating field it is a consequence of the field equation of :

Applying to this equation, we obtain provided no monopoles are present and the mass is non-zero.

Expression (68) shows that the test tube described by the plastic field couples to the fluctuating massive vector field , with a coupling constant given by as in two dimensions. As T approaches the critical temperature from below, w goes to zero, and decouples from the test tube described by . After carrying out the integral over in (68) we recover the result (62).

The fact that the magnetic field has a finite penetration depth in the superconducting phase is reflected by the mass term of the field.

It is interesting to consider the limit in detail, where the massive vector field decouples from the magnetic vortex. This limit yields the constraint which can be solved by setting . The correlation function then takes the simple form

where is the monopole density. In the absence of monopoles, the theory reduces to that of a free massless mode that may be thought of as representing the magnetic scalar potential. This follows from combining the physical interpretation of the vector field (66) with the equation . Specifically,

Using the definition of the monopole density, , we see that in terms of the field the correlation function reads

This demonstrates that the operator describing the test tube, which was introduced in (48) in a real-space formulation involving the singular plastic field (47), is now represented as an ordinary field. Since we are in the normal conducting phase, where develops a non-zero expectation value, the presence of the phase indicates that this expectation value breaks a global U(1) symmetry, with the ensuing Goldstone field. This will be further clarified below.

Equation (72) reveals in addition that in the normal conducting phase the Dirac string looses its physical relevance, the right-hand side depending only on the end points and , not on the line . This fact is also apparent from our starting formula (57), where the last term and therefore any reference to disappears in the limit . It makes no sense to talk about magnetic vortices in this phase because they are condensed and do not exist as physical excitations. There is also no non-trivial topology to assure their stability.

We are now in a position to derive the dual theory of a 3D superconductor. This theory features a grand-canonical ensemble of fluctuating closed magnetic vortices, of arbitrary shape and length, which have a steric repulsion, i.e., a loop gas of magnetic vortices. We know that such an ensemble can be described by a disorder field theory, consisting of a complex theory. On the other hand, our study of a single magnetic vortex revealed that it couples with a coupling constant g to the fluctuating vector field . These two observations uniquely determine the dual theory in the London limit as being given by [25,26,8,18,19]

with

where the field is minimally coupled to the vector field . Equation (77) replaces the lattice Hamiltonian (45) near the critical point. It is a description of the superconducting state in terms of physical variables: the field describes the local induction, whereas accounts for the loop gas of magnetic vortices. There are no other physical objects present in a superconductor. The dual theory has no local gauge symmetry because the vector field is massive. In fact, the two observations are connected. The presence of a local gauge symmetry in a given theory may be looked upon as reflecting a redundancy in the description. Since the dual theory is formulated in terms of physical variables, there is no redundancy, and thus no local gauge symmetry.

Although (73) was derived starting from the London limit, it is also relevant near the phase transition. The point is that integrating out the size fluctuations of the scalar field would only generate higher-order interaction terms and a possible change of the mass and interaction parameter and u. But these modifications do not alter the critical behavior of the theory.

The energy (61) appears in the dual theory as a one-loop on-shell mass correction stemming from the graph depicted in Fig.\ 2, which we now interpret as a Feynman graph. The straight and wiggly lines represent the and field correlation functions, respectively.

A measure for the interaction strength of a massive vector field in 3D is given by the dimensionless parameter equal to the square of the coupling constant multiplied by the range of the interaction. For the dual theory this factor is , which is the inverse of the strength of the electromagnetic gauge field in the superconducting phase. This is a common feature of theories which are dual to each other.

Another notable property of the dual theory is that in the limit it changes into a local gauge theory [8],

as can be checked by rescaling the dual field in the Hamiltonian (77).

We next investigate what happens with the dual theory when we approach the critical temperature. Remember that w and therefore tends to zero in the limit where T tends to the critical temperature from below. From the first term in the Hamiltonian (77) it again follows that in this limit, so that we can write once more , and (77) becomes

This equation shows that , representing the magnetic scalar potential, cannot be distinguished from the phase of the disorder field. Indeed, let be this phase. Then, the canonical transformation absorbs the scalar potential into the phase of ; the first term in (77) decouples from the theory and yields a trivial contribution to the partition function. In this way, the dual theory reduces to a pure theory

It was already concluded that in the high-temperature phase the magnetic vortices proliferate as indicated by the fact that , giving a field theoretic description of the loop gas of these objects, develops a non-zero expectation value at the transition point. This transition is triggered by a change in sign of . In the London limit the Hamiltonian (77) then takes the simple form

with v the expectation value of the disorder field, , and where we now represented the phase of by to bring out the fact that describes the magnetic scalar potential. As we will demonstrate below, v has the value [9] of an inverse flux quantum, so that with our normalization choice of the phase of the field, Eq. (78) takes the canonical form.

The picture of the superconducting phase transition that emerges in the dual formulation of the Ginzburg-Landau theory is the following. When the critical temperature is approached from below, there is a proliferation of magnetic vortices. We recall that in the London limit parallel vortices repel each other, so that a single vortex prefers to crumple. Near we then have a spaghetti of vortices which fill the space completely at and above the transition temperature. Since inside the core of a vortex one has the normal phase, the system thus becomes normal conducting. Whereas in the Ginzburg-Landau formulation a magnetic vortex is described by a singular plastic field , in the dual formulation it is represented by the Noether current . This follows from comparing the terms coupling linearly to the fluctuating field. In the normal conducting phase the field develops a vacuum expectation value, and thereby breaks the global U(1) symmetry of the theory; is the ensuing Goldstone field. The Noether current becomes in the London limit , with representing the massless photon of the high-temperature phase. It should be noted that at the fluctuating local field decouples from because .