Abelian Higgs model

The Abelian Higgs model couples a complex scalar field to a $U(1)$ gauge field and permits magnetic vortex solitons in two spatial dimensions.

For $|D\Phi {|}^2-\lambda (|\Phi {|}^2-{\eta }^2{)}^2/4$, write $\Phi =(\eta +h/\sqrt{2})e^{i\theta }$ and absorb $\partial \theta /e$ into the vector field. The quadratic Lagrangian has a real scalar of mass $m_h=\sqrt{\lambda }\eta$ and a vector of mass $m_A=\sqrt{2}e\eta$. The two scalar plus two massless-vector polarizations in the symmetric phase become one radial scalar plus three massive-vector polarizations. The removed phase supplies the longitudinal mode; local vortices retain global winding and quantized flux.

// Target: cosmology.bigb

In two dimensions, scale ${\Phi }_{\lambda }(x)=\Phi (\lambda x)$ and $A_{j,\lambda }(x)=\lambda A_j(\lambda x)$. The magnetic field has scaling weight two, the gauge covariant derivative of the Higgs field has weight one, and the potential has weight zero. Their integrated energies therefore have the powers shown. Stationarity gives $E_B=E_P$, a Derrick virial identity. This balance allows a planar Abelian Higgs vortex to have finite size even though a scalar-gradient-plus-nonnegative-potential model has no comparable magnetic term.

This planar Abelian Higgs model replaces the usual magnetic energy by $B^2/h$. Use $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$. The magnetic term is singular at a zero of the Higgs field, so the admissible class must have finite energy; formal field equations are first derived where $h>0$. Its positive-flux Bogomolny equations are $(D_1+i{D}_2)\Phi =0$ and $B=h(1-h)/2$. On those solutions $B/h=(1-h)/2$ extends regularly across zeros.

Write $\Phi =e^{u/2+i\chi }$ away from its zeros and use the oriented planar Hodge star with $\ast d{x}^1=d{x}^2$. The first Bogomolny equation gives $A=d\chi -\frac{1}{2}\ast du$. A zero of multiplicity $n_j$ contributes $u=2n_j\log |x-p_j|+\text{smooth}$ and phase winding $2\pi n_j$. Combining the smooth magnetic field with these delta sources gives the displayed equation and the topological boundary condition $u\to 0$. Prescribed zero positions and multiplicities reduce multi-vortex construction to this scalar elliptic problem; a regular solution reconstructs both smooth fields.

Let $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$. The identityand completing the square give, when the current boundary integral vanishes,For flux $2\pi N>0$, the lower bound is $\pi N$, saturated by the two first-order Bogomolny equations. Division by $h$ is interpreted in the finite-energy sense at its zeros.

Use metric $(+--)$, $D_{\mu }={\partial }_{\mu }-i{a}_{\mu }$ and Lagrangian $-f_{\mu \nu }{f}^{\mu \nu }/4+\overline{D_{\mu }\varphi }{D}^{\mu }\varphi /2-(1-|\varphi {|}^2{)}^2/8$. The Euler-Lagrange field equations are $D_{\mu }{D}^{\mu }\varphi +(|\varphi {|}^2-1)\varphi /2=0$ and ${\partial }_{\mu }{f}^{\mu \nu }=\mathrm{Im}(\bar{\varphi }{D}^{\nu }\varphi )$. Varying $\bar{\varphi }$ gives a kinetic contribution $-D_{\mu }{D}^{\mu }\varphi /2$ and potential contribution $(1-|\varphi {|}^2)\varphi /4$; variation of $a_{\nu }$ gives ${\partial }_{\mu }{f}^{\mu \nu }-\mathrm{Im}(\bar{\varphi }{D}^{\nu }\varphi )$. For static fields in temporal gauge $a_0=0$, the positive Bogomolny vortex equations imply the scalar equation because $(D_1-i{D}_2)(D_1+i{D}_2)=D_1^2+D_2^2+B$. They imply the spatial gauge equations because $\mathrm{Im}(\bar{\varphi }{D}_1\varphi )=-{\partial }_2|\varphi {|}^2/2$ and $\mathrm{Im}(\bar{\varphi }{D}_2\varphi )={\partial }_1|\varphi {|}^2/2$, while $B=(1-|\varphi {|}^2)/2$. The Gauss equation then vanishes identically.

An Abelian Higgs vortex has quantized magnetic flux and a Higgs field whose phase winds at spatial infinity. Its vortex number is $N=(2\pi {)}^{-1}\int F$ under the convention $D=d-iA$.

For a unit Abelian Higgs vortex, the piecewise profiles $h=r/R$ and $a=(r/R{)}^2$ inside $r=R$, with both equal to one outside, are continuous finite-energy trial fields. In the normalization $E=\int [B^2/2+|D\varphi {|}^2/2+(1-|\varphi {|}^2{)}^2/8]$, their radial energy is the displayed function. Its minimum has $R^4=48$ and $E=\pi (2/3+1/\sqrt{3})$, an upper-bound estimate above the exact Bogomolny bound $\pi$. Derivative jumps at the matching radius are allowed in the variational energy space; the trial profile is not an exact solution.

When $|\Phi |\to 1$ and its gauge covariant derivative decays fast enough, on a large circle write $\Phi =\rho e^{i\chi }$ locally. The tangential derivative gives $A=d\chi -\mathrm{Im}(\overline{\Phi }D\Phi )/{\rho }^2$. The integrated error tends to zero, so Stokes theorem turns total magnetic flux into the phase change $2\pi N$. This integer is the winding number of the normalized Higgs field on the circle. For positive Bogomolny vortex equations it is the total zero multiplicity.

The vortex number is the total multiplicity of the zeros of the Higgs field, equivalently its phase winding at infinity or its magnetic flux divided by $2\pi$ in the standard normalization.

At critical coupling, the Abelian Higgs vortex moduli space is the space of gauge-equivalence classes of static $N$-vortex solutions. Vortex positions provide complex coordinates, and the field-theory kinetic energy induces an $L^2$ Riemannian metric whose geodesics approximate slow vortex motion.

Interchanging two Abelian Higgs vortices replaces $q=z_1-z_2$ by $-q$. Thus $w=q^2$ is a single-valued relative coordinate on the unordered Abelian Higgs vortex moduli space. Coincidence is smooth in $w$; a geodesic crossing from positive real $w$ to negative real $w$ changes the line of separation by $\pi /2$, explaining right-angle scattering of Abelian Higgs vortices. The asymptotic cone in the $q$ coordinate has angular period $\pi$ and must not be treated as a singularity of the actual coincidence metric.

A head-on geodesic through the smooth coincidence point of the two-vortex moduli space emerges along the orthogonal axis. Two identical critically coupled vortices therefore scatter through $90^{\circ }$ in the slow-motion approximation.

At critical coupling on a surface with conformal metric $g=\Omega dzd\bar{z}$, the vortex equations are

Away from zeros of the Higgs field, write $\varphi =e^{u+i\chi }$ with $u=\log |\varphi |$. The Bogomolny vortex equation $(D_1+i{D}_2)\varphi =0$ gives $a_1={\chi }_1+u_2$ and $a_2={\chi }_2-u_1$. Consequently $\mathbf{a}=\nabla \chi -J\nabla u$, where $J(v_1,v_2)=(-v_2,v_1)$, and $B=-\Delta u$ on a regular phase patch. If the contours of $u$ and $\chi$ are orthogonal, then $\nabla u\cdot \nabla \chi =0$, so $\mathbf{a}\cdot \nabla u=0$: the gauge potential is tangent to constant-amplitude contours. This does not imply that it is curl-free. At a vortex zero, $u\to -\infty$ and the phase is only locally defined. Including phase winding distributionally gives $\Delta u+(1-e^{2u})/2=2\pi {\sum }_j{n}_j{\delta }^{(2)}(x-X_j)$. The usual Taubes equation uses $2u=\log |\varphi {|}^2$, so its delta coefficients double.

With $D_i={\partial }_i-i{A}_i$ and $B={\partial }_1{A}_2-{\partial }_2{A}_1$, use the critical-coupling energy $E=\int [B^2/2+|D_i\varphi {|}^2/2+(1-|\varphi {|}^2{)}^2/8]d^{2}x$. For positive vortex number, integration by parts givesThe boundary divergence vanishes for the usual decaying vortex fields, and magnetic flux is $2\pi N$. The two squares vanish exactly at the Bogomolny vortex equations. Reversing both signs treats negative $N$. Energy coefficients and the numerical bound change together under other normalizations.

For metric $g=\Omega (d{x}^2+d{y}^2)$, set $F_{12}={\partial }_x{A}_y-{\partial }_y{A}_x$ and use the physical magnetic field $B=F_{12}/\Omega$. At critical coupling, define $j_i=\mathrm{Im}(\bar{\Phi }{D}_i\Phi )$. The identity $|D_x\Phi {|}^2+|D_y\Phi {|}^2=|(D_x+i{D}_y)\Phi {|}^2+F_{12}|\Phi {|}^2+{\partial }_x{j}_y-{\partial }_y{j}_x$ completes the energy into nonnegative squares plus half the magnetic flux and half the current boundary integral. For unit-magnitude vacuum boundary data, the combined boundary contribution is $\frac{1}{2}\oint (A+j)=\pi N$, since $j=d\arg \Phi -A$ there. In the more restrictive covariantly decaying sector the current integral separately vanishes and the flux is $2\pi N$. Equality for positive $N$ requires $(D_x+i{D}_y)\Phi =0$ and $B=(1-|\Phi {|}^2)/2$. The factor $\Omega$ appears instead if the symbol $B$ denotes the coordinate component $F_{12}$.

For $u=\log |\varphi {|}^2$ and zeros of multiplicities $N_r$ at $z_r$, the Taubes equation is

Choose positions $z_a$ and positive integer multiplicities $m_a$. Solve the Taubes equation with $h\to 0$ at infinity and $h=2m_a\log |z-z_a|+O(1)$ near each position. The planar existence theorem supplies such a solution for every finite multiset of positions; see the original existence theorem. Set $\chi ={\sum }_a{m}_a\arg (z-z_a)$, $\Phi =e^{h/2+i\chi }$ and $A=\nabla \chi -J\nabla h/2$, with $J(v_1,v_2)=(-v_2,v_1)$. The singular terms cancel in $A$, while $\Phi$ has precisely the selected zeros. These fields satisfy the Bogomolny vortex equations, with $B=(1-|\Phi {|}^2)/2$. Their vortex number is ${\sum }_a{m}_a$. Complex conjugation of $\Phi$ together with $A\mapsto -A$ gives antivortices.

If $u$ solves the Taubes equation for a metric $g$ and $\tilde{u}$ solves it for $e^{u}g$, then $u+\tilde{u}$ solves it for $g$. The vortex divisors add, so the total vortex number is $N+\tilde{N}$.

Integrating the Taubes equation on a compact surface of area $A$ gives $A\ge 4\pi N$. A nontrivial vortex with nonzero Higgs field requires the strict inequality $A>4\pi N$.

At the integrable curvature scale on the Poincare disc, the vortex Taubes equation reduces to the Liouville equation. Holomorphic self-maps of the disc then generate explicit vortex solutions.

For metric $8|dz{|}^2/(1-|z{|}^2{)}^2$, these smooth fields satisfy the Bogomolny vortex equations with one simple zero at the origin. Their curvature coefficient is $F_{12}=4/(1+|z{|}^2{)}^2$, and their physical magnetic field is $(1-|z{|}^2{)}^2/[2(1+|z{|}^2{)}^2]=(1-|\Phi {|}^2)/2$. The magnetic flux is $2\pi$, and the critically coupled energy in the conformal-surface vortex square completion normalization is $\pi$.

For the disc metric $8|dz{|}^2/(1-|z{|}^2{)}^2$ and a holomorphic map $g:D\to D$, the Higgs magnitudesolves the critically coupled vortex equations. Choosing $g(z)=z^{N+1}$ gives a radial vortex of winding $N$.