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Formation of Topological Defects Formation of Topological Defects Tanmay Vachaspati Physics Department, Case Western Reserve University, Cleveland OH 44106-7079, USA. In these lectures, I review cosmological phase transitions and the topological as- pects of spontaneous symmetry breaking. I then discuss the formation of walls, strings and monopoles during phase transitions including lattice based studies of defect formation and recent attempts to go beyond the lattice. The close connec- tion of defect formation with percolation is described. Open problems and possible future directions are mentioned. 1 Motivation and Introduction An exciting development in cosmology has been the realization that the uni- verse may behave very much like a condensed matter system. After all, the cosmos is the arena where very high energy particle physics is relevant and this is described by quantum field theory which is also the very tool used in condensed matter physics. In condensed matter systems we have seen a rich array of phenomenon and we expect that the cosmos has seen an equally rich past. A routine ob- servation in condensed matter (and daily life!) is that of symmetry breaking which is the basis of all schemes in particle physics to achieve unification of forces. Symmetry breaking would lead to phase transitions in the early uni- verse, making their study crucial to our understanding of the cosmos. There are many cosmic theories that hinge on processes happening during phase tran- sitions. These include a large number of inflationary models, baryogenesis and structure formation. arXiv:hep-ph/9710292v1 9 Oct 1997 Many of the concepts that have gone into the explosion of cosmology in the last two decades are linked to each other. As an example, consider the birth of the inflationary idea. The success of the electroweak model led particle physi- cists to attempt to unify the strong force with the electroweak forces by pos- tulating Grand Unified Theories (GUTs). In such theories, a number of phase transitions occur and, based on mathematical results on the topology of group manifolds, it is known that GUTs always contain topological defects known as magnetic monopoles. The occurence of phase transitions in cosmology then tells us that monopoles must have formed in the early universe 1. In fact, standard cosmology then predicts an over-abundance of magnetic monopoles in our present universe that is clearly in conflict with observations 3. This head-on confrontation of particle physics and cosmology led Guth 2 to come 1 up with the idea of an inflationary universe - an idea that is now deeply embed- ded in cosmology as it not only solves the monopole problem but also several other unrelated problems that standard (pre-inflationary) cosmology did not address. Magnetic monopoles are one example of topological defects that can arise during phase transitions. But topological defects can occur in other varieties as well. They can be one dimensional, in which case they are called “strings”. If they are two dimensional, they are called “domain walls”. Magnetic monopoles and domain walls are two kinds of topological defects that have unpleasant cosmological consequences unless they are kept very light or are somehow eliminated at some early epoch (for example, by inflation). Cosmic strings are believed to be more benevolent and may even have been responsible for structure formation in the universe if they were formed at the GUT phase transition. The study of topological defects is fascinating for several reasons. As in the example of magnetic monopoles and inflation above, they can provide im- portant constraints on particle physics models and cosmology. On the other hand, if a GUT topological defect is found, it would provide a direct window 35 on the universe at about 10− secs. In a manner of speaking, a part of the early universe is trapped in the interior of the defect - much like dinosaur fossils trapped in amber. GUT topological defects would also shed light on the problem of how galaxies were formed. Their discovery would give us im- portant constraints on the symmetry structure of very high energy particle physics - this is non-perturbative information. In addition, it would give us information about phase transitions in the early universe, giving confidence in our understanding of the thermal history of the universe. Also, the topologi- cal defects would most likely have tremendous astrophysical impact since they are generally very massive and have unusual gravitational and electromagnetic properties. In the early 80’s, people would talk of the marriage of particle physics and cosmology. I actually think there is something wrong with this picture since a third party is also involved. This is condensed matter physics. When it comes to understanding cosmological phase transitions, we are forced to consider the corresponding advances in condensed matter physics since many of the ideas are the same. Topological defects have been studied by condensed matter physicists for many, many years. If we want experimental input in our theories of the very early universe, we must go to the condensed matter laboratory. This is a growing area of research - condensed matter experiments are being inspired by cosmological questions, and, cosmologists are revising their theories based on experimental input. 2 In these lectures, my aim is to discuss the formation of topological de- fects and to bring the reader to a point where the most relevant questions are apparent. The existing work on topological defects paints a certain picture of their formation but there are some limitations. I will describe both the picture and the limitations. I will start out by describing phase transitions and the “effective potential” way of treating them in field theory. Then I will describe why topological defects come about and finally get to the work on their formation in a phase transition. It is impossible to describe every aspect of this subject within these lectures especially since the subject branches out into a large variety of different areas of research. The references provided in the bibliography should help the reader dive deeper into whichever area he/she chooses to pursue. 2 Phase Transitions 2.1 Effective Potential: Formalism In statistical mechanics, the basic quantity one tries to find is the partition function Z βH Z = e− X where β = 1/T and the sum is over all states with energy H. From the partition function one can derive the Helmholtz free energy A by A Z = e− and the derivatives of A then lead to the thermodynamic functions. To discuss phase transitions, one performs a Legendre transform on the Helmholtz free energy to get the Gibbs free energy G. For a gas, we have G(P,T )= A + P V while for an ensemble of spins, G(M,T )= A + HM where H is the external magnetic field and M is the magnetization of the system. In the magnetic system, we have ∂G = H ∂M 3 and so, in the absence of an external magnetic field, the minima of G describe the various phases of the system. In field theory, these concepts carry over almost word for word 6. For the time being we restrict ourselves to the zero temperature case. Then, in analogy with the partition function, one defines the generating functional in the presence of an external current J(x) (analogous to the external magnetic field) T Z[J]= Dφexp[i dt d3x( [φ]+ J(x)φ(x))] (1) Z Z0 Z L where, the time integration is over a large but finite interval and is a suitable Lagrangian density for the system. In other words, L iH Z[J]=< Ω e− T Ω > exp[ E[J]] (2) | | ≡ − where, Ω > is the vacuum state and H is the Hamiltonian. The energy functional| E[J] is the analog of the Helmholtz free energy. (Note however that E is not really an energy - for example, it does not have the dimensions of energy.) Now, we find, i ( + φ) δE[J] Dφe L J φ(x) = R δJ(x) −R i ( + φ) Dφe L J R = < ΩRφ(x) Ω > | | J φ (x) ≡ − cl which is (minus) the order parameter. Next we get the effective action, which is the analog of the Gibbs free energy, by Legendre transforming the energy functional: 4 Γ[φcl]= E[J] d yJ(y)φcl(y) . − − Z To obtain the value of the order parameter for any given external current, we need to solve: δΓ[φ ] cl = J(x) δφcl(x) − and, in particular, when the external current vanishes, the phases are described by the extrema of the effective action. So far we have taken the external current to be a function of space but, in practice, the current is taken to be a constant and φcl is also constant. Then, in this restricted case, only non-derivative terms can be present in the effective 4 action and the effective action leads to an effective potential with quantum corrections: 1 V (φ ) Γ[φ ] eff,q cl ≡− cl VT where, is the spacetime volume. VT These issues and, in particular, the correspondence between statistical mechanics and field theory, are described very clearly in the textbooks by Peskin and Schroeder 6 and Rivers 7. 2.2 Effective Potential: Quantum Corrections The actual calculation of the quantum effective potential is done by using perturbation theory 8,9. The general result is as follows. For the model: 1 1 = D φ Dµφ V (φ ) F a F aµν + L 2 µ i i − i − 4 µν LF where ∂ φ = (∂ ieAa T a)φ µ µ − µ F a = ∂ Aa ∂ Aa + ef Ab Ac µν µ ν − ν µ abc µ ν = iψγ¯ µD ψ ψ¯Γ ψφ LF µ − i i in conventional notation, the one loop correction to the potential is the Coleman- Weinberg correction: 1 µ2 M 2 m2 V (φ)= [Tr(µ4ln ) + 3Tr(M 4ln ) 4Tr(m4ln )] .
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