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### The role of Symmetry Breaking in the Formation of Topological Defects

Symmetry breaking is a collective behavior of the physical system. Although one can read-off that information locally from the Lagrangian or Halmintonian of the system, but the phenomenon itself can only be realized through a collection of degrees of freedom. The simplest explanation can be seen from the following example: consider a material in superconductor phase (with the familiar gauge symmetry U(1) is broken), and take a look at a single electron. The electron itself is an gauge invariant object, and it couldn't tell whether the phase of the material breaks symmetry or not. Therefore, by definition, symmetry breaking is a collective phenomenon.

Topological defects, in a conventional sense, is a quantized object (therefore, macroscopic objects like ocean vortices or smoke rings aren't topological defects, although there are many similarities between them). It's usually the case that topological defects are a lot more stable compare to a composite object in the theory since these quantized defects are protected by the system itself (unpatchable defects), at least up to a certain energy scale - the symmetry broken scale. Symmetry breaking supports topological defects, that can be seen, but also note that most of the time, topological defects are formed only if the system has a symmetry breaking. That's not always true, since there are cases when topological defects are formed without symmetry breaking: if the spatial region is non-trial, there are topological properties arised from the geometry and stable topological objects can appear, for example a compact space can supports Kaluza-Klein type topological defect which corresponds to the winding number - the topological label for fields when one examines a loop around the compacted direction. Another example is the field value space itself is non-trivial, for example in the Ising model (as the field can only receive 2 different values)

Without symmetry breaking, all field configurations in trivial spatial region are are topologically equivalent since each field value can locally change to a new value without constrain. With symmetry breaking and the result order parameter space compacted, then it's straight-forward see that quantization can naturally arise from that compactness (for example, if the order parameter space is topologically equivalent to a circle then one can has quantized winding number, which is an integrer). This quantization structure can be read-off from homotopy and homotopy group, associated possible topological defects with distinct labels.