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4.7. Topological Defects

In a quantum field theory, small oscillations around the vacuum appear as particles. If the space of possible vacuum states is topologically nontrivial, however, there arises the possibility of another kind of solitonic object: a topological defect. Consider a field theory described by a continuous symmetry group G which is spontaneously broken to a subgroup H subset G. Recall that the space of all accessible vacua of the theory, the vacuum manifold, is defined to be the space of cosets of H in G; M ident G / H. Whether the theory admits topological defects depends on whether the vacuum manifold has nontrivial homotopy groups. A homotopy group consists of equivalence classes of maps of spheres (with fixed base point) into the manifold, where two maps are equivalent if they can be smoothly deformed into each other. The homotopy groups defined in terms of n-spheres are denoted pin.

In general, a field theory with vacuum manifold M possesses a topological defect of some type if

Equation 135 (135)

for some i = 0, 1,.... In particular, we can have a set of defects, as listed in table 3. In order to get an intuitive picture of the meaning of these topological criteria let us consider the first two.

Table 3. Topological defects, as described by homotopy groups of the vacuum manifolds of a theory with spontaneously broken symmetry.

Homotopy Constraint Topological Defect

pi0(M) neq 1 Domain Wall
pi2(M) neq 1 Cosmic String
pi2(M) neq 1 Monopole
pi3(M) neq 1 Texture

If pi0(M) neq 1, the manifold M is disconnected. (A zero-sphere is the set of two points a fixed distance from the origin in R1. The set of topologically equivalent maps with fixed base point from such such a sphere into a manifold is simply the set of disconnected pieces into which the manifold falls.) Let's assume for simplicity that the vacuum manifold consists of just two disconnected components M1 and M2 and restrict ourselves to one spatial dimension. Then, if we apply boundary conditions that the vacuum at - infty lies in M1 and that at + infty lies in M2, by continuity there must be a point somewhere where the order parameter does not lie in the vacuum manifold as the field interpolates between the two vacua. The region in which the field is out of the vacuum is known as a domain wall.

Similarly, suppose pi2(M) neq 1. This implies that the vacuum manifold is not simply-connected: there are non-contractible loops in the manifold. The order parameter for such a situation may be considered complex and if, when traveling around a closed curve in space, the phase of the order parameter changes by a non-zero multiple of 2pi, by continuity there is a point within the curve where the field is out of the vacuum manifold. Continuity in the direction perpendicular to the plane of the curve implies that there exists a line of such points. This is an example of a cosmic string or line defect. Since we shall mostly concentrate on these defects let us now give a detailed analysis of the simplest example, the Nielsen-Olesen vortex in the Abelian Higgs model.

Consider a complex scalar field theory based on the Abelian gauge group U(1). The Lagrangian density for this model is

Equation 136 (136)

where phi(x) is the complex scalar field, the covariant derivative, Dµ, and field strength tensor Fµnu are defined in terms of the Abelian gauge field Aµ as

Equation 137 (137)
Equation 138 (138)

and the symmetry breaking (or "Mexican hat") potential is

Equation 139 (139)

Here e is the gauge coupling constant and eta is a parameter that represents the scale of the symmetry breaking. In preparation for our discussion of the more general cosmological situation, let us note that at high temperatures the Lagrangian contains temperature-dependent corrections given by

Equation 140 (140)

where C is a constant. For T > Tc, where Tc is defined by

Equation 141 (141)

we see that

Equation 142 (142)

is minimized by <phi> = 0. However, for T < Tc, minimizing the above expression with respect to phi yields a minimum at

Equation 143 (143)

Thus, for T > Tc the full symmetry group G = U(1) is restored and the vacuum expectation value of the phi-field is zero. As the system cools, there is a phase transition at the critical temperature Tc and the vacuum symmetry is spontaneously broken, in this case entirely:

Equation 144 (144)

The vacuum manifold is

Equation 145 (145)

The group U(1) is topologically a circle, so

Equation 146 (146)

The set of topologically equivalent ways to map one circle to another circle is given by the integers. (We can wrap it any number of times in either sense.) The first homotopy group is therefore the integers. Calculating homotopy groups in general is difficult, but for S1 all of the other groups are trivial. We therefore have

Equation 147-150 (147)




The Abelian Higgs model therefore allows for cosmic strings.

To see how strings form in this model, consider what happens as T decreases through Tc. The symmetry breaks and the field acquires a VEV given by

Equation 151 (151)

where alpha may be chosen differently in different regions of space, as implied by causality (we shall discuss this shortly). The requirement that <phi> be single valued implies that around any closed curve in space the change Delta alpha in alpha must satisfy

Equation 152 (152)

If for a given loop we have n neq 0, then we can see that any 2-surface bounded by the loop must contain a singular point, for if not then we can continuously contract the loop to a point, implying that n = 0 which is a contradiction. At this singular point the phase, alpha, is undefined and <phi> = 0. Further, <phi> must be zero all along an infinite or closed curve, since otherwise we can contract our loop without encountering a singularity. We identify this infinite or closed curve of false vacuum points as the core of our string. We shall restrict our attention to the case where n = 1, since this is the most likely configuration.

The first discussion of string solutions to the Abelian Higgs model is due to Nielsen and Olesen. Assume the string is straight and that the core is aligned with the z-axis. In cylindrical polar coordinates, (r, theta), let us make the ansatz

Equation 153 (153)

The Lagrangian then yields the simplified equations of motion

Equation 154 (154)

where a prime denotes differentiation with respect to r. It is not possible to solve these equations analytically, but asymptotically we have

Equation 155 (155)

This solution corresponds to a string centered on the z-axis, with a central magnetic core of width ~ (eeta)-1 carrying a total magnetic flux

Equation 156 (156)

The core region over which the Higgs fields are appreciably non-zero has width ~ (lambda1/2 eta)-1. Note that these properties depend crucially on the fact that we started with a gauged symmetry. A spontaneously broken global symmetry with nontrivial pii(M) would still produce cosmic strings, but they would be much less localized, since there would be no gauge fields to cancel the scalar gradients at large distances.

Strings are characterized by their tension, which is the energy per unit length. For the Nielsen-Olesen solution just discussed, the tension is approximately

Equation 157 (157)

Thus, the energy of the string is set by the expectation value of the order parameter responsible for the symmetry breaking. This behavior is similarly characteristic of other kinds of topological defects. Again, global defects are quite different; the tension of a global string actually diverges, due to the slow fall-off of the energy density as we move away from the string core. It is often convenient to parameterize the tension by the dimensionless quantity Gµ, so that a Planck-scale string would have Gµ ~ 1.

From the above discussion we can see that cosmology provides us with a unique opportunity to explore the rich and complex structure of particle physics theories. Although the topological solutions discussed above exist in the theory at zero temperature, there is no mechanism within the theory to produce these objects. The topological structures contribute a set of zero measure in the phase space of possible solutions to the theory and hence the probability of production in particle processes is exponentially suppressed. What the cosmological evolution of the vacuum supplies is a concrete causal mechanism for producing these long lived exotic solutions.

At this point it is appropriate to discuss how many of these defects we expect to be produced at a cosmological phase transition. In the cosmological context, the mechanism for the production of defects is known as the Kibble mechanism. The guiding principle here is causality. As the phase transition takes place, the maximum causal distance imposed on the theory by cosmology is simply the Hubble distance - the distance which light can have traveled since the big bang.

As we remarked above, as the temperature of the universe falls well below the critical temperature of the phase transition, Tc, the expectation value of the order parameter takes on a definite value (~ eta eialpha in the Abelian Higgs model) in each region of space. However, at temperatures around the critical temperature we expect that thermal fluctuations in <phi> will be large so that as the universe cools it will split into domains with different values of alpha in different domains. This is the crucial role played by the cosmological evolution.

The Kibble mechanism provides us with an order of magnitude upper bound for the size of such a region as the causal horizon size at the time of the phase transition. The boundaries between the domains will be regions where the phase of <phi> changes smoothly. If the phase changes by 2pin for some n neq 0 when traversing a loop in space, then any surface bounded by that loop intersects a cosmic string. These strings must be horizon-sized or closed loops. Numerical simulations of cosmic string formation indicate that the initial distribution of strings consists of 80% horizon-sized and 20% loops by mass. If we consider the temperature at which there is insufficient thermal energy to excite a correlation volume back into the unbroken state, the Ginsburg temperature, TG, then, given the assumption of thermal equilibrium above the phase transition, a much improved estimate for the initial separation of the defects can be derived and is given by

Equation 158

where lambda is the self coupling of the order parameter. This separation is microscopic.

Interesting bounds on the tension of cosmic strings produced by the Kibble mechanism arise from two sources: perturbations of the CMB, and gravitational waves. Both arise because the motions of heavy strings moving at relativistic velocities lead to time-dependent gravitational fields. The actual values of the bounds are controversial, simply because it is difficult to accurately model the nonlinear evolution of a string network, and the results can be sensitive to what assumptions are made. Nevertheless, the CMB bounds amount roughly to [118, 119, 120, 121]

Equation 158 (158)

This corresponds roughly to strings at the GUT scale, 1016 GeV, which is certainly an interesting value. Bounds from gravitational waves come from two different techniques: direct observation, and indirect measurement through accurate pulsar timings. Currently, pulsar timing measurements are more constraining, and give a bound similar to that from the CMB [122]. Unlike the CMB measurements, however, gravitational wave observatories will become dramatically better in the near future, through operations of ground-based observatories such as LIGO as well as satellites such as LISA. These experiments should be able to improve the bounds on Gµ by several orders of magnitude [123].

Now that we have established the criteria necessary for the production of topological defects in spontaneously broken theories, let us apply these conditions to the best understood physical example, the electroweak phase transition. As we remarked earlier, the GWS theory is based on the gauge group SU(2)L × U(1)Y. At the phase transition this breaks to pure electromagnetism, U(1)em. Thus, the vacuum manifold is the space of cosets

Equation 159 (159)

This looks complicated but in fact this space is topologically equivalent to the three-sphere, S3. The homotopy groups of the three-sphere are

Equation 160-163 (160)




(Don't be fooled into thinking that all homotopy groups of spheres vanish except in the dimensionality of the sphere itself; for example, pi3(S2) = Z.)

Thus, the electroweak model does not lead to walls, strings, or monopoles. It does lead to what we called "texture," which deserves further comment. In a theory where pi3(M) is nontrivial but the other groups vanish, we can always map three-dimensional space smoothly into the vacuum manifold; there will not be a defect where the field climbs out of M. However, if we consider field configurations which approach a unique value at spatial infinity, they will fall into homotopy classes characterized by elements of pi3(M); configurations with nonzero winding will be textures. If the symmetry is global, such configurations will necessarily contain gradient energies from the scalar fields. The energy perturbations caused by global textures were, along with cosmic strings, formerly popular as a possible origin of structure formation in the universe [124, 125]; the predictions of these theories are inconsistent with the sharp acoustic peaks observed in the CMB, so such models are no longer considered viable.

In the standard model, however, the broken symmetry is gauged. In this case there is no need for gradient energies, since the gauge field can always be chosen to cancel them; equivalently, "texture" configurations can always be brought to the vacuum by a gauge transformation. However, transitions from one texture configuration to one with a different winding number are gauge invariant. These transitions will play a role in electroweak baryon number violation, discussed in the next section.

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