Spin gauge theory, duality and fermion pairing

Dualities provide a powerful tool to understand phenomena which are not tractable by perturbation techniques. Dualities usually relate strongly coupled sector in one theory to the weakly coupled sector in another theory, in particle physics, string theory, statistical physics and also condensed matter physics. One class of duality transformations in quantum field theory involves exchanging a differential $p$-form $A_{p}$ in $D$-dimensions with a $D-p-2$-form $A_{D-p-2}$ by the Hodge duality of their exterior derivatives, ${\rm d}A_{p}={}^{*}{\rm d}A_{D-p-2}\,.$ For free fields in topologically trivial spacetimes, both are equivalent descriptions, but interactions or topological obstructions can break this duality. Dualities in interacting theories, when they can be constructed, can lead to deep mathematical and physical insights. In two and three dimensions, (anti)self-dual configurations of gauge fields interacting with scalars correspond to solitons. In four dimensions, where the dual of a 1-form is also a 1-form, the (anti)self-dual configurations of Yang-Mills gauge fields minimize the action of instantons. A particularly interesting duality in four dimensions is that between a scalar and a two-form, which persists when there is a gauge field coupled to the scalar. The scalar field is now compact and the dual two-form appears in a topological $B\wedge F$ interaction with the gauge field. If there is no other field in the theory, this is a duality between the strongly coupled Abelian Higgs model and topological mass generation mechanism in four dimensions.

The low energy, long wavelength properties of a condensed matter system can be captured in an effective field theory, Ginzburg-Landau theory being the original example. Effective field theories which describe topological states of quantum matter are topological field theories, or more generally, quantum field theories which include topological interaction terms. The application of topological quantum field theories to condensed matter systems has, in recent years, greatly improved our understanding of both [3, 4, 5, 6, 1, 2, 7, 8]. Although the two-form gauge field is ubiquitous in these theories of topological matter, the difficulties of coupling it to electrons has stood in the way of a deeper understanding of the field and its applicability in condensed matter physics.

This is most easily seen from the point of view of gauge symmetries. Vector gauge transformations, which can be called the fundamental or defining symmetry of the gauge theory of two-form fields, generalize the gauge transformations of electromagnetism to $B\to B+{\rm d}\beta$ , under which the field strength $H={\rm d}B\,$ remains unchanged. But unlike the U(1) symmetry of ordinary gauge theory, this appears to have no representation as a local unitary transformation of fermions, and the interaction above does not remain invariant under this transformation.

There is also a problem of nonlocality. Duality between $B$ and vector fields, for the example of the Abelian Higgs model, appears only through the field strength as $H={}^{*}(A+{\rm d}\phi)\,.$ So one might be encouraged to try an interaction with fermions in the form $H\wedge{}^{*}j\,$ where $j$ could be either the usual fermionic current or the axial current. But this is also not correct. The reason is that the duality relation, between $B$ on one hand and $(A,\phi)$ on the other, is not local. Thus in presence of fermions, the actions obtained by replacing one set of fields with the other are not equivalent and do not lead to the same equations of motion. The nonlocality can be understood if we remember that extended objects rather than point particles are essential to the definition of higher form gauge fields. For example, the two-form $B$ couples to worldsheets of strings rather than worldlines of particles, so one expects that the interaction between $B$ and fermionic fields representing point particles could involve nonlocality in some form.

Such an interaction was proposed in [9] between a two-form and a nonlocal pseudotensor current $J$ related to the curl of the fermion current

$$J=m\,{}^{*}{\rm d}\,\left({\Box}^{-1}\,J_{\psi}\right)\equiv m{}^{*}{\rm d}\,\int d^{4}y\,G(x,y)\,J_{\psi}\,,$$

(1)

where $J_{\psi}$ is the electron current, $m$ is a mass scale appropriate to the problem where this interaction might be relevant, and $G(x,y)$ is the Green function of the wave equation,

$$\Box G(x,y)=\delta^{4}(x-y)\,.$$

(2)

This current is identically conserved, ${\rm d}{}^{*}J=0$ . The velocity field of the Dirac fermion is given by $v^{i}=\psi^{\dagger}\alpha^{i}\psi$ , so the conserved charges $\Box J^{0i}=-m\epsilon^{ijk}\partial_{j}v_{k}$ can be identified with the vorticity field of the fermion. There is another way of looking at the conserved charges. In the absence of interactions, and in the non-relativistic limit in which the lower components of the Dirac fermions may be neglected for energies small compared to their mass, the static charge of the pseudovector current takes a simple form,

$$\left(J^{0i}\right)_{\rm NR}\propto{1\over 2}\left(\psi^{\dagger}\sigma^{i}\psi\right)\,.$$

(3)

The quantity on the right hand side is the spin density of the electron field, or the intrinsic magnetic moment density because it is multiplied by the electron charge. The proportionality becomes equality if $m$ is chosen to be the electron mass. Just as the interaction between charges and currents is mediated by the 1-form gauge field $A$ , the interaction between magnetic moments and their currents is mediated by the 2-form gauge field $B$ .

We can write an action for the electrons and the gauge fields incorporating this interaction,

$$S=\int\left[\bar{\psi}\left(i\not{\partial}+e\not{A}\right)\psi-m\bar{\psi}\psi+{g}B\wedge{}^{*}J-{1\over 2}F\wedge{}^{*}F+{1\over 2}H\wedge{}^{*}H\right]\,.$$

(4)

This action is invariant under the vector gauge transformations $B\to B+{\rm d}\beta$ . We should think of this as a low energy effective action, valid for energy scales well below some cutoff $\Lambda$ . The number of degrees of freedom can be worked out by first making the action local using Lagrange multipliers. The gauge field $A$ has two degrees of freedom as usual, while $B$ carries only one, since $B_{0i}$ are non-dynamical and the vector gauge symmetry takes away two more degrees of freedom (not three, since $\beta$ and $\beta+{\rm d}\chi$ are equivalent gauge parameters). There are two additional degrees of freedom in the Lagrange multipliers which stay around and may be thought of as auxiliary field degrees required for a local formulation of the non-local spin gauge interaction [9]. By formally treating the nonlocal interaction like any other coupling term, we can integrate out the fermions to find that the one-loop action contains a $B\wedge F$ interaction.

Such a term corresponds to topologically massive gauge theory, so the potential between two sources interacting via the gauge field ought to be Yukawa in the non-relativistic limit. But that is not what happens in this case. Calculated directly from the action in Eq. (4), the potential has two components. One is an ${r}^{-1}$ Coulomb potential which corresponds to a massless gauge field, and the other is a linear potential which is attractive irrespective of the charges or spin alignment of the fermionic sources [10]. In this paper we argue that the action of Eq. (4) can arise in a system described by local fields.

Since a linear potential is produced by a string, we start with a system which contains string-like objects as well as fermions. As we will see below, the nonlocal interaction appears naturally if we consider Abrikosov-Nielsen-Olesen (ANO) vortex strings in the Abelian Higgs model interacting with charged fermions. Such a system, called a boson-fermion model, was originally proposed as a model of high-$T_{c}$ superconductivity [11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 20], but also appear in models of superconductor-insulator transitions [22, 23, 24], of BCS-BEC crossover [25, 26], and of charged Bose liquids [27]. The system we consider has certain differences with the boson-fermion model, in particular we do not consider the $\phi\bar{\psi}\psi$ Yukawa interaction of the fermions with the scalar Higgs field. However for ease of convenience we will refer to the system of Abelian Higgs model with charged fermions as a boson-fermion model. These fermions are “unpaired” or “itinerant”, meaning that they are not part of Cooper pairs whose condensation is responsible for the superconducting transition. The main result of our paper is that these fermions show a kind of confinement being joined by a pair of flux strings to other fermions. The path to that argument requires that we bring together several results, not all of them well known.

We start by showing in Sec. II how the nonlocal interaction between fermions and the 2-form gauge field $B$ arises in the dual picture of ANO strings interacting with fermions via the electromagnetic gauge field. However, it turns out that the resulting action is not exactly what we are looking for. In Sec. III we calculate the static potential between two fermions and find that it also is not what we were hoping for, a mass term for the $B$ field prevents the appearance of a linear potential. We show in Sec. IV that if the strings condense in a kind of Higgs mechanism, then in a phase where the $B$ field is massless, we recover Eq. (4) and thus the fermions are bound by a linear potential. We end with a discussion on the physical implications of our results.

In the dual formulation of ANO strings in the Abelian Higgs model, the phase of the scalar field is written as the sum of singular and regular parts, then the singular field is dualized to the worldsheet of the string and the regular field is dualized to a 2-form [28, 29, 30, 31, 32, 33]. Let us carry out this procedure in presence of fermions which couple to the gauge field.

We start from the action ¹¹1While it is more economical to use the form notation, the calculations in this section and the next are more transparent in the index notation.

$$S=\int d^{4}x\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+|D_{\mu}\Phi|^{2}+V(|\Phi|^{2})+\bar{\psi}(i\not{\partial}-e\not{A}-m)\psi\right)\,,$$

(5)

where $\Phi$ is a complex scalar with electric charge $q$  and $\psi$ is a fermion with charge $e$ . We have not assumed any relation between the charge $e$ of the fermion and the charge $q$ of the scalar. The potential $V(|\Phi|^{2})$ has a degenerate minimum at $\Phi^{*}\Phi=v^{2}$ for some non-vanishing $v^{2}$ , but the exact form of $V$ is not important for our calculations. Vortex strings or magnetic flux tubes form in this model when the global U(1) symmetry is broken in the vacuum and the phase of $\Phi$ , which lives on the vacuum manifold $\Phi^{*}\Phi=v^{2}$ , becomes multivalued. The position of the ANO string is the locus of the zeroes of $\Phi$.

The corresponding partition function

$$Z=\int{\mathscr{D}}A_{\mu}{\mathscr{D}}\Phi{\mathscr{D}}\Phi^{*}{\mathscr{D}}\psi{\mathscr{D}}\bar{\psi}\;\exp iS\,$$

(6)

can be rewritten in presence of these vortex string configurations by using a polar decomposition of the complex scalar in field space as $\Phi=\dfrac{1}{\sqrt{2}}\rho\exp(i\theta)\,.$ Then in the presence of flux tubes, we can write $\theta=\theta^{r}+\theta^{s}\,,$ where $\theta^{s}$ corresponds to a given magnetic flux tube, and $\theta^{r}$ describes single valued fluctuations around this configuration. For a string with winding number $n\,,$ $\theta^{s}$ changes by $2\pi n$ for going around the string once, while $\rho$ vanishes along the core of the string. The ANO string world sheet $\Sigma$ is the collection of singular points ²²2The singular points of $\theta$ are at the zeroes of $\rho$ [29]. of $\theta\,$,

$$\epsilon^{\mu\nu\lambda\rho}\partial_{\lambda}\partial_{\rho}\theta=\Sigma^{\mu\nu}(x)=2\pi n\int d^{2}\sigma\,\epsilon^{ab}\frac{\partial X^{\mu}}{\partial\sigma^{a}}\frac{\partial X^{\nu}}{\partial\sigma^{b}}\delta^{4}({x}-{X}(\sigma))\,,$$

(7)

where we have included the vorticity quantum $2\pi$ and the winding number $n$ in the definition of the world sheet. The string carries quantized magnetic flux,

$$\oint_{\Gamma}A_{\mu}dx^{\mu}=\frac{2n\pi}{q}\,.$$

(8)

In the polar decomposition, the action takes the form

	$\displaystyle S=\int d^{4}x$	$\displaystyle\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho+\frac{1}{2}\rho^{2}(\partial_{\mu}\theta^{s}+\partial_{\mu}\theta^{r}+qA_{\mu})(\partial^{\mu}\theta^{s}+\partial^{\mu}\theta^{r}+qA^{\mu})\right.$
		$\displaystyle\qquad\left.+V(\rho^{2})+\bar{\psi}(i\not{\partial}-e\not{A}-m)\psi\right)\,.$		(9)

We first linearize the term $\frac{1}{2}\rho^{2}(\partial_{\mu}\theta+qA_{\mu})(\partial^{\mu}\theta+qA^{\mu})$ by introducing an auxiliary field $C_{\mu}$ into the partition function through the Gaussian integral

$$N\int{\mathscr{D}}C_{\mu}\exp{\left(-\frac{i}{2}\int d^{4}x\,\left[{C_{\mu}}+\rho(\partial_{\mu}\theta+qA_{\mu})\right]^{2}\right)}=1\,.$$

(10)

Here $N$ is a normalization factor independent of all fields – we will suppress $N$ and other such normalization factors for all functional integrals. We can now write the partition function as

	$\displaystyle\mathcal{Z}=\int$	$\displaystyle{\mathscr{D}}A_{\mu}{\mathscr{D}}\rho{\mathscr{D}}\theta^{s}{\mathscr{D}}\theta^{r}{\mathscr{D}}\bar{\psi}{\mathscr{D}}\psi{\mathscr{D}}C_{\mu}$
	$\displaystyle\qquad\exp$	$\displaystyle\left(i\int d^{4}x\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho-\frac{1}{2}{C_{\mu}C^{\mu}}-C^{\mu}\rho\partial_{\mu}\theta^{r}-C^{\mu}\rho(\partial_{\mu}\theta^{s}+qA_{\mu})\right.\right.$
		$\displaystyle\left.\qquad+V(\rho^{2})+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi-eA_{\mu}\bar{\psi}\gamma^{\mu}\psi\bigg{)}\right)\,.$		(11)

Integration over $\theta^{r}$ produces a $\delta(\partial_{\mu}(\rho C^{\mu}))$ , leading to the dualization

$$\rho C^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\sigma}\partial_{\nu}B_{\lambda\sigma}\,.$$

(12)

Then the condition $\partial_{\mu}(\rho C^{\mu})=0$ is automatically satisfied.

Putting this expression for $C^{\mu}$ and noting that $\varepsilon^{\mu\nu\rho\lambda}\varepsilon_{\mu\nu_{1}\rho_{1}\lambda_{1}}=-\delta^{\nu\;\rho\;\lambda}_{[\nu_{1}\rho_{1}\lambda_{1}]}$ we get the partition function as

	$\displaystyle\mathcal{Z}=$	$\displaystyle\int{\mathscr{D}}A_{\mu}{\mathscr{D}}\rho{\mathscr{D}}x^{\mu}{\mathscr{D}}\bar{\psi}{\mathscr{D}}\psi{\mathscr{D}}B_{\mu\nu}$
		$\displaystyle\exp\bigg{(}i\int d^{4}x\bigg{(}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho+\frac{1}{12\rho^{2}}H^{\mu\nu\lambda}H_{\mu\nu\lambda}-\frac{1}{2}B^{\mu\nu}\Sigma_{\mu\nu}$
		$\displaystyle\qquad\qquad-\frac{1}{6}q\varepsilon^{\mu\nu\rho\lambda}A_{\mu}H_{\nu\rho\lambda}+V(\rho^{2})+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi-eA_{\mu}\bar{\psi}\gamma^{\mu}\psi\bigg{)}\bigg{)}\,.$		(13)

Here we have written $H_{\mu\nu\lambda}=\partial_{\mu}B_{\nu\lambda}+\partial_{\nu}B_{\lambda\mu}+\partial_{\lambda}B_{\mu\nu}$ for the field strength of the 2-form gauge field $B$ and also replaced the integration over $\theta^{s}$ by an integration over the spacetime coordinates $x^{\mu}(\sigma,\tau)$ of the worldsheet of the ANO string. The Jacobian for this change of variables produces the action for the dynamics of the string [30, 34, 35, 36].

In the absence of strings we could eliminate $B_{\mu\nu}$ in favor of $A_{\mu}$ by summing over a perturbation series or equivalently by using the equations of motion, leaving only a massive gauge field [37, 38]. We cannot do that here because of the $B_{\mu\nu}\Sigma^{\mu\nu}$ interaction. So in order to diagonalize the system and eliminate the mixed $A-B$ term, we first formally integrate over $A_{\mu}$ . The relevant part of the partition function is

$\displaystyle\int{\mathscr{D}}A_{\mu}\exp i\int d^{4}x\bigg{(}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{6}q\varepsilon^{\mu\nu\rho\lambda}A_{\mu}H_{\nu\rho\lambda}-A_{\mu}\bar{\psi}\gamma^{\mu}\psi\bigg{)}\,.$

(14)

Into this we insert the following Gaussian integral

$$N\int{\mathscr{D}}\chi_{\mu\nu}\exp\left(-\frac{i}{4}\int d^{4}x\left\{\chi_{\mu\nu}\chi^{\mu\nu}-\varepsilon^{\mu\nu\rho\lambda}\chi_{\mu\nu}F_{\rho\lambda}+\frac{1}{4}\left(\varepsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}\right)^{2}\right\}\right)=1\,,$$

(15)

where $N$ is a constant (field-independent) normalization factor which is to lumped with other similar factors coming from other integrations. We now integrate over $A_{\mu}$ to be left with $\delta\left(\frac{1}{2}\varepsilon^{\mu\nu\rho\lambda}\partial_{\nu}\left(\chi_{\rho\lambda}-qB_{\rho\lambda}\right)-e\bar{\psi}\gamma^{\mu}\psi\right)$ , which can be resolved by setting

$$\chi_{\mu\nu}=\partial_{\mu}A^{m}_{\nu}-\partial_{\nu}A^{m}_{\mu}+qB_{\mu\nu}+e\varepsilon_{\mu\nu\rho\lambda}\partial^{\rho}\frac{1}{\Box}\bar{\psi}\gamma^{\lambda}\psi\,,$$

(16)

where we have introduced another vector field $A_{\mu}^{m}$ , known as the magnetic photon. The partition function now has the form

	$\displaystyle\mathcal{Z}=$	$\displaystyle\int{\mathscr{D}}A^{m}_{\mu}{\mathscr{D}}\rho{\mathscr{D}}x^{\mu}{\mathscr{D}}{\psi}{\mathscr{D}}\bar{\psi}{\mathscr{D}}B_{\mu\nu}$
		$\displaystyle\exp i\int d^{4}x\bigg{[}\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho+\frac{1}{12\rho^{2}}H^{\mu\nu\lambda}H_{\mu\nu\lambda}-\frac{1}{2}B_{\rho\lambda}\Sigma^{\rho\lambda}+V(\rho^{2})+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi$
		$\displaystyle-\frac{1}{4}\bigg{(}\partial_{\mu}A^{m}_{\nu}-\partial_{\nu}A^{m}_{\mu}+qB_{\mu\nu}\bigg{)}^{2}-\frac{1}{2}eqB^{\mu\nu}\varepsilon_{\mu\nu\rho\lambda}\partial^{\rho}\frac{1}{\square}\bar{\psi}\gamma^{\lambda}\psi-\frac{1}{4}e^{2}\bigg{(}\varepsilon_{\mu\nu\rho\lambda}\partial^{\rho}\frac{1}{\square}\bar{\psi}\gamma^{\lambda}\psi\bigg{)}^{2}\bigg{]}\,.$		(17)

If we now do a field redefinition $B_{\mu\nu}\to B_{\mu\nu}+\dfrac{1}{q}\left(\partial_{\mu}A^{m}_{\nu}-\partial_{\nu}A^{m}_{\mu}\right)$ , we can integrate over $A^{m}_{\mu}$ and find that $\partial_{\nu}\Sigma^{\mu\nu}=0\,,$ which shows that vortices form closed loops in this model as the world sheet current is conserved by itself [32]. Furthermore, we notice that the last term in the integrand above can also come from integrating over the gauge field $A_{\mu}$ in the partition function of ordinary quantum electrodynamics with no additional field,

$$\int{\mathscr{D}}A_{\mu}{\mathscr{D}}{\psi}{\mathscr{D}}\bar{\psi}\exp i\int d^{4}x\bigg{(}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\xi}(\partial_{\mu}A^{\mu})^{2}-eA_{\mu}\bar{\psi}\gamma^{\mu}\psi\bigg{)}\,.$$

(18)

Thus we can reinstate the QED part of the action to write the Lagrangian as

$$\boxed{\begin{aligned} \mathscr{L}&=-\frac{1}{4}M^{2}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-eA_{\mu}\bar{\psi}\gamma^{\mu}\psi-\frac{1}{2}eMB^{\mu\nu}\varepsilon_{\mu\nu\rho\lambda}\partial^{\rho}\frac{1}{\square}\bar{\psi}\gamma^{\lambda}\psi\\ &\qquad+\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho+\frac{v^{2}}{12\rho^{2}}H^{\nu\rho\lambda}H_{\nu\rho\lambda}-\frac{v}{2}B_{\rho\lambda}\Sigma^{\rho\lambda}+V(\rho^{2})+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi\,,\end{aligned}}$$

(19)

where we have rescaled $B_{\mu\nu}\to vB_{\mu\nu}$ , written $M=qv\,,$ and suppressed the gauge-fixing term. The nonlocal interaction between $B_{\mu\nu}$ and charged fermions cannot be removed as long as ANO strings are present in the system.

The first term is a mass term for the $B_{\mu\nu}$ field – in an alternative derivation of the dual action it appears not in this form but as the equivalent nonlocal Meissner term [39, 40] $\displaystyle{\frac{q^{2}v^{2}}{12}H^{\nu\rho\lambda}\frac{1}{\square}H_{\nu\rho\lambda}}\,.$ There is an advantage of writing the mass term in the nonlocal form. The action of Eq. (II) which was found by dualization was invariant under the vector gauge transformation $B_{\mu\nu}\to B_{\mu\nu}+\partial_{[\mu}\Lambda_{\nu]}\,.$ Written in the nonlocal form, the mass term and thus the action based on the Lagrangian of Eq. (19) remains invariant under the same transformation. In this paper we will work with the mass term in the local form as in Eq. (19), but we could have worked as well with the nonlocal mass term. Another way of handling the nonlocal mass term is by introducing an additional vector field [9], but we will not take that route in this paper.

The nonlocal “spin-gauge” coupling between the fermions and the $B$-field gives rise to a linear attractive potential between fermions, irrespective of whether they were positively or negatively charged [10]. However, as compared to the action considered there, several additional terms have appeared in the derivation from the boson-fermion model. The effective static interaction potential between nonrelativistic fermions in this system was calculated recently [39] by integrating out the gauge fields $A_{\mu}$ and $B_{\mu\nu}$ . It is worthwhile to revisit this calculation, as we will be concerned in this paper with a particular modification of the result found there.

To find the effective static potential between non-relativistic electrons, we first integrate over the gauge fields $A_{\mu}$ and $B_{\mu\nu}$  using the Lagrangian of Eq. (19) with $\rho=v$ . Introducing a gauge-fixing term we can integrate over $A_{\mu}$ ,

$\displaystyle\int{\mathscr{D}}A_{\mu}\exp\,$

$\displaystyle i\int d^{4}x\bigg{[}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2\xi}\left(\partial_{\mu}A^{\mu}\right)^{2}-eA_{\mu}\bar{\psi}\gamma^{\mu}\psi\bigg{]}\sim\exp\frac{i}{2}\int d^{4}k\;J^{\mu}(-k)\;\frac{1}{k^{2}}\;J_{\mu}(k)\,,$

(20)

where $J^{\mu}$ is the fermion current and we have used the fact that it is conserved, $\partial_{\mu}J^{\mu}=0\,.$ For the $B$ integration, we get

$\displaystyle\int{\mathscr{D}}B_{\mu\nu}\exp\,i\int d^{4}x\left(-\frac{1}{4}B_{\mu\nu}K^{\mu\nu\rho\lambda}B_{\rho\lambda}-\frac{1}{2}B_{\mu\nu}J^{\mu\nu}\right)\,,$

(21)

where we have written

$$K^{\mu\nu\rho\lambda}=\frac{1}{2}\left(\Box+M^{2}\right)g^{\mu[\rho}g^{\lambda]\nu}+\frac{1}{2}\left(g^{\nu[\rho}g^{\lambda]\sigma}\partial_{\sigma}\partial^{\mu}-g^{\mu[\rho}g^{\lambda]\sigma}\partial_{\sigma}\partial^{\nu}\right)\,,$$

(22)

and

$$J^{\mu\nu}=v\Sigma_{\mu\nu}-eM\varepsilon_{\mu\nu\rho\lambda}\partial^{\rho}\frac{1}{\Box}\bar{\psi}\gamma^{\lambda}\psi\,.$$

(23)

The inverse of $K^{\mu\nu\rho\lambda}$ is the propagator for $B_{\mu\nu}$ and is given in momentum space as

$$G_{\mu\nu\rho\lambda}(k)=-\frac{1}{(k^{2}-M^{2})}\left(g_{\mu[\rho}g_{\lambda]\nu}-\frac{1}{M^{2}}\left(g_{\mu[\rho}k_{\lambda]}k_{\nu}-g_{\nu[\rho}k_{\lambda]}k_{\mu}\right)\right)\,,$$

(24)

where

$$K^{\mu\nu\rho\lambda}G_{\mu\nu\rho^{\prime}\lambda^{\prime}}=\left(\delta^{\rho}_{\rho^{\prime}}\delta^{\lambda}_{\lambda^{\prime}}-\delta^{\rho}_{\lambda^{\prime}}\delta^{\lambda}_{\rho^{\prime}}\right)\,.$$

(25)

Thus the integration over $B_{\mu\nu}$ will result in

		$\displaystyle\int{{\mathscr{D}}}B_{\mu\nu}\exp\bigg{(}i\int d^{4}x\bigg{(}-\frac{1}{4}B_{\mu\nu}K^{\mu\nu\rho\lambda}B_{\rho\lambda}-\frac{1}{2}B_{\mu\nu}J^{\mu\nu}\bigg{)}\bigg{)}$
		$\displaystyle\qquad\sim\exp\bigg{(}\frac{i}{2}\int d^{4}k\,e^{2}\,J^{\mu}(-k)\Bigg{[}\frac{1}{(k^{2}-M^{2})}-\frac{1}{k^{2}}\Bigg{]}J^{\mu}(k)+\cdots\bigg{)}$		(26)

where we have used the expression for $J^{\mu\nu}$ in terms of the fermion current $J^{\mu}$. The dots stand for terms involving the string worldsheet $\Sigma$ [39].

Thus we see that the $\dfrac{1}{k^{2}}$ appearing due to the Coulomb potential cancels with a negative term appearing after the integration of the $B$ field so that the propagator corresponds to the Yukawa potential $\dfrac{\exp(-Mr)}{r}$ . The linear attractive potential which was found as a consequence of the nonlocal “spin-gauge” interaction term has disappeared. It is because of the mass term of the $B$-field that the effective static potential between nonrelativistic fermions calculated from this action is Yukawa. In a phase where the mass term is not present, the effective potential is a combination of Coulomb and linear potentials. Let us then consider the question whether there can be such a phase.

The photon becomes massive in the Anderson-Higgs mechanism which provides the phase transition from a massless gauge field to a massive one – the phase transition in the opposite direction is accompanied by symmetry restoration as the system is raised above the transition temperature. The 2-form $B_{\mu\nu}$ is a higher analogue of the usual 1-form gauge field $A_{\mu}$, so what we have in mind is “a higher analogue” of spontaneous symmetry breaking. Since $B_{\mu\nu}$ couples to worldsheets of vortex strings, it is the condensation of these strings which should produce the Higgs mechanism for the $B$-field. The idea of a phase transition due to vortex condensation at finite temperature is not a new one, more generally examples are known in quantum field theory and condensed matter physics of phase transitions driven by condensation of topological defects [41, 42, 43, 35].

It is well known that the Abelian Higgs model with flux strings has a dual description in terms of a two-form potential (sometimes called the disorder field in the context of superconducting phase transitions [53, 52, 51]). We have shown that the dual field has an effective nonlocal interaction with electrons in the system. The description of flux strings in terms of the Abelian Higgs model is an effective one for real type-II superconductors. The string interactions, as well as quantities like string tension or thickness, are determined by the properties of the underlying real system. The effective theory of infinitely thin flux strings in Abelian Higgs model cannot tell us what the terms in the string potential should be. The false vacuum for the string field may even be different from the false vacuum of the scalar field. However, we can say that the mass term for the $B$ field will not be present in the action in the false vacuum. We also see that the remaining terms which contain the $B_{\mu\nu}$ field have an enhanced symmetry – the vector gauge symmetry $B_{\mu\nu}\to B_{\mu\nu}+\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu}\,.$ While quantum corrections cannot be calculated without knowledge of the exact form of the string field potential in Eq. (29), this symmetry does not fix any relation among the parameters in the false vacuum, so those terms need not vanish. Thus if the flux strings condense in a stringy Higgs mechanism, there is a phase in which the static potential between non-relativistic electrons has a linear component.

A linear potential between pairs of particles would be interpreted as a flux tube. But there is a problem – the flux tubes we started out with are either infinite or end on magnetic charges [32], not electric charges. A possible solution to this problem comes from noting that the nonlocal interaction couples the 2-form field with the electron spin current, more precisely, the magnetic dipole moment of electron. In the background where flux lines coalesce into flux tubes, a pair of flux tubes can connect two dipoles and form a kind of Cooper pair. A cartoon representing a semiclassical view of the electron pair is shown in Fig. 1, the north pole of each dipole connected to the south pole of the other by a flux tube. In our construction the dipoles are electrons, thus point dipoles, and the flux tubes are infinitesimally thin. The dipoles can be antiparallel or parallel as in the figure, or they can be oriented in any which way with respect to each other.

Recently a model of superinsulators was proposed using Cooper pairs bound by strings of electric flux [55, 54, 56]. Our construction here is different from that one in two important aspects. One is that electric flux strings appear in dual superconductivity, which exhibits a dual Meissner effect and excludes electric fields from the bulk, constricting the flux into the strings (in analogy to QCD confinement [57, 58, 60, 59]). In our construction, we have the usual (type II) superconductivity arising from condensation of electric charges, which constricts the magnetic field into strings. The other difference is that in that model, electrically charged Cooper pairs are connected by the strings, the electric flux in a string ends on electric charges at the ends. In our construction, electrically charged fermions are connected by strings carrying magnetic flux, because charged fermions behave as point magnetic dipoles due to their spin. The magnetic field of a point dipole is constricted into a pair of strings in a superconductor, connecting it to another fermion as in Fig. 1. Bound states of three or more electrons are also possible, as shown in the example of Fig. 2.

We can estimate the energy of a localized pair as a function of its angular momentum, if we pretend that the electrostatic potential between two electrons joined by a magnetic flux tube remains the usual Coulomb potential. For a nonrelativistic string of length $L$ and tension $T$ , with electrons of mass $m$ and charge $e$ at the two ends, the energy is then

$$E=2m+TL+\frac{1}{2}I\omega^{2}+\frac{e^{2}}{L}\,,$$

(47)

where $I$ is the moment of inertia,

$$I=\frac{1}{12}TL^{3}+\frac{1}{2}mL^{2}\,.$$

(48)

Writing $J=I\omega$ for the angular momentum, we find the relation between the energy $E$ and the angular momentum $J$ of the string

$$E=2m+TL+\frac{6J^{2}}{L^{2}(TL+6m)}+\frac{e^{2}}{L}=2m+E_{b}\,,$$

(49)

where we have defined $E_{b}$ to be the binding energy. The same result can also be found by taking the non-relativistic limit of the relativistic string with massive end points [61].

Minimizing the energy with respect to the length, one can calculate $L$ as a function of the other parameters. Putting this back into the equation for $E$  we get a relation between the energy and the angular momentum. For type II superconductors the penetration depth is of the order $\sim$100 nm, which gives a representative string tension $T\sim 10$ eV². The corresponding plot of $E_{b}$ vs $J$ is shown in Fig. 3 . The size of the pair corresponding to the $J=0$ state is $\sim$19 nm. The energies are higher for bound states of more electrons – for example, $E_{b}=\frac{3}{2}TL+\frac{3e^{2}}{L}$ for the $J=0$ state of three electrons.

The possibility of having different numbers of fermions in bound states, with different geometries, suggests that our construction can also be useful as a toy model of quark confinement. In the usual string picture of quark confinement [58, 59, 60], the vacuum is thought of as a dual (color) superconducting vacuum in which magnetic monopoles condense, dual Meissner effect takes place, and (chromo)electric flux tubes end on quarks carrying (chromo)electric charge. Here we have found another description – that of magnetic dipoles, rather than monopoles, being connected by strings carrying magnetic flux. The electric charge of the fermion is screened by virtue of being in a superconductor.

We acknowledge discussions with C. Chatterjee and I. Dutta Choudhury during the early stages of this work. AL acknowledges discussions with J. K. Bhattacharjee.