Abelian vortices

It is clear, from the theorem proved in the previous section that to have a chance of finding classical solutions one needs to have other fields, besides scalar fields. The only other type of bosonic fields that play role in particle physics are gauge bosons. Thus, it is natural to seek classical solutions in Higgs theories. Let us consider the simplest Higgs theory that one can imagine, namely the Abelian ($U(1)$) gauge theory of a complex scalar boson. As the homotopy groups of

$$\pi_2(S_1)=\pi_3(S_1)=0,$$

we can only consider the fundamental group. As we discussed last time,

$$\pi_1(S_1)=Z,$$

so we expect mappings of circles into circles will provide nontrivial classical solutions. As the mapping should be of the circle of infinite radius into $U(1)$, one should either deal with a $D=2$ system, or with a $D=3$ system in which the solution is independent of the third direction. Then of course, unless the third dimension is compact, the energy of the classical solution becomes infinite, only the energy/unit length remains finite.

The Hamiltonian of the $U(1)$ Higgs theory has the form

$$H=\int d^2x \left[\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ (D_\mu\Ph... ...Phi^\star)+ \frac{\lambda}{4}(\Phi\Phi^\star-\eta^2)^2\right].$$

We chose the vacuum expectation value of the Higgs field to be $\eta$. The covariant derivative is

$$D_\mu=i\partial_\mu+ eA_\mu$$

and the gauge field tensor is the curl of the vector potential.

The time and $z$ components of the vector potential can be chosen to be zero in the appropriate gauge. Furthermore, one can require the transverse gauge condition for the remaining two components of the gauge field

$$\partial_i A_i=0.$$

This Hamiltonian is the same as the Landau theory for a superconductor. Then the Higgs field corresponds to the field of Cooper pairs. the only difference is that $e\to2e$ substitution should be taken.

The $U(1)$ gauge symmetry is broken by the potential term. The vacuum is a constant Higgs, $\Phi_0=\eta e^{i\alpha},$ where $\alpha$ is an arbitrary phase. The space of these vacuum solutions is, of course, just $U(1)$. Then we should seek nontrivial classical solutions in the form

$$\Phi(x)= e^{in\phi}\eta\chi(r),$$

where $\phi$ and $r$ are the two dimensional classical coordinates. $e^{in\phi}$ is the archtypical representative of the maps of the $n$th eqivalence class and it sweeps the target space ($n$ times) when $0\le\phi\le2\pi$. As we discussed earlier, to avoid singular behavior, $\chi(0)=0$ must be satisfied. That gives a finite energy to the configurations, because at least in some region of space $(\Phi\Phi^\star-\eta^2)^2>0$. In fact, it is easy to see that analyticity arguments require that $\chi(r)\sim r^{\vert n\vert}$ when $r\to0$. The arguments are very similar to those used in quantum mechancs for the behavior of the radial wave function $R_l(r)\sim r^l.$ The field is an regular function of the cartesian coordinates and $\cos n\phi=\cos\phi^n+...$ must be multiplied by $r^n$ as $x^n=r^n\cos^n\phi.$

The electromagnetic field, independent of time and $z$, satisfying in the $\mbox{\boldmath$\nabla$}\cdot \mbox{\boldmath$A$}=0$ gauge can be written as

$$\mbox{\boldmath$A$}=\mbox{\boldmath$\nabla$}\phi \, a(r)=\mbox{\boldmath$\hat \phi$}\frac{a}{r^2}.$$

Clearly, $a(r)$ must vanish at the origin (as $a\sim r^2$), otherwise $A$ is singular. The electric field is obviously zero. the magnetic field is

$$\mbox{\boldmath$B$}=\mbox{\boldmath$\hat \phi$}\times \mbox{\boldmath$\hat r$}\frac{d}{dr}\frac{a(r)}{r^2}$$

points in the direction of the $z$ axis. As we will soon see $\lim_{r\to\infty}a(r)=n/e$, thus the total magnetic flux in the perpendicular to the $z$ axis is

$$\oint dx_i A_i(x)=\oint d\phi r \frac{a(r)}{r}=\frac{2\pi n}{e}.$$

The magnetic flux is proportional to the topological quantum number. We will soon see that the energy of the vortex is also related to $n$.

We can easily rewrite the hamiltonian in terms of $\chi(r)$ and $a(r)$. We obtain

$$H=\pi\int dr\, r \left[\frac{a'^2}{r^2}+\chi'^2\eta^2+\frac{... ...\chi^2\eta^2}{r^2}+\frac{\lambda\eta^2}{2}(\chi^2-1)^2\right].$$

We discussed boundary conditions at the origin already. At $r\to\infty$ the self-interaction term requires that $\chi\to1$ (or -1). Then, if we wish to avoid a singularity in the term $(a-n)^2\chi^2/r^2$ we must have $a\to n$, as we earlier indicated.

The field equations obtained from the minimization of the Hamiltonian can only be solved numerically. They provide monotonic solutions for both $a$ and $\chi$, as shown on the figure. To find a lower bound on the energy one can use the linearization method of Bogomol'nyi. We can rewrite the original Hamiltonian as well, but it leads to the same result as the last form in terms of the functions $a$ and $\chi$. Let's assume $n>0$. Then we write

$$H = \pi\int dr\, r \left[\left(\frac{a'}{r}+\beta e\eta^2(\chi^2-1)\r... ...a-n)\chi}{r}\right)^2+\frac{(\lambda-2\beta^2 e^2)\eta^4}{2}(\chi^2-1)^2\right.$$

$$+ \left.\frac{(ea-n)^2\chi^2\eta^2(1-4\beta^2)}{r^2}-\frac{1}{r}2\beta\eta^2\frac{d}{dr}(ea-n)(\chi^2-1)\right]$$ (2)

The last term can be integrated and using the boundary conditions it gives

$$H=2\pi\beta\eta^2 n+...$$

Choosing $\beta>0$, but $\beta<1/2$, $\beta<\sqrt{\lambda}/(e\sqrt{2})$ all the terms not written down from the Hamiltonian are positive. This proves that the energy of this soliton is larger than $2\pi\beta\eta^2 n.$ What is the best bound one can get? Either $H\ge \pi\eta^2 n$ or $H\ge \sqrt{2\lambda}\eta^2 n/e$, whichever number is smaller. In either case the mass of the soliton is positive.

The limiting case is interesting. It corresponds to the case when the Higgs mass, $m_H=2\eta\sqrt{\lambda}$ and the gauge mass, $m_A=2\sqrt{2}e\eta$, coincide. This is the critical point where vortex-vortex forces vanish. Below that vortices attract each other (superconductor of the first kind) above it they repel each other (superconductor of the second kind). At the critical point the Hamiltonian reduces to

$$H=\pi\eta^2 n+\pi\int dr\, r \left[\left(\frac{a'}{r}+\frac{... ...ht)^2+\eta^2\left(\chi' +\frac{(ea-n)\chi}{r}\right)^2\right].$$

In other words, the energy is exactly $H_{\rm min}=\pi\eta^2 \vert n\vert$, and the fields satisfy linear field equations.