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For a discrete $\mathbb{Z}_2$ symmetry for instance, in some patches of the Universe, it can spontaneously breaks $\mathbb{Z}_2$ and goes in the "right" minimum, while in some others patches it can go in the "left" minimum. This leads to domains walls. Do we have the equivalent of domain walls for continuous symmetries, let's say $U(1)$, because it can go to the minimum $v$ as well as any of the other minima $v\times \exp(i\theta)$?

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    $\begingroup$ is this classical field theory or quantum field theory? $\endgroup$ Feb 3, 2022 at 15:35
  • $\begingroup$ I strongly suspect this is all about classical fields, from the tag « cosmology ». Personally, I’m more interested in the classical definition. $\endgroup$
    – Cham
    Feb 3, 2022 at 15:44
  • $\begingroup$ Classical field theory $\endgroup$
    – ketherok
    Feb 3, 2022 at 18:58

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Yes, when $U(1)$ is spontaneously broken we have cosmic strings (aka vortices), which can be global or local depending on whether the $U(1)$ is global or local symmetry. Then there are magnetic monopoles which arise when a non-abelian symmetry $G$ is spontaneously broken to a smaller symmetry group $H$, such that the quotient space $G/H$ (which is the vacuum manifold of the theory) has non-trivial second homotopy group $\pi_2(G/H)$. For example, when $G=SU(2)$ and $H=U(1)$ the quotient $G/H$ is isomorphic to two-sphere ($S^2$), and $$\pi_2(SU(2)/U(1))=Z~~~({\rm integers})$$ If $\pi_2(G/H)$ is trivial -- no monopoles.

Similarly, the existence of cosmic strings can be inferred from non-trivial first homotopy group $\pi_1$ of $U(1)$ which is also given by integers. This classification of topological defects can be generalized as $$ \pi_n(G/H) $$ which basically represents how many times (if at all) $S^n$ wraps around the vacuum manifold $G/H$. $\pi_1$ tells us about cosmic strings, $\pi_2$ about point defects, or monopoles, $\pi_3$ classifies so-called cosmic textures. There are also higher-dimensional generalizations of these defects.

In addition, another way to think about topological defects, is by looking for non-contractible $S^n$ spheres. In the presence of a cosmic string you cannot continuously contract a circle to a point (if the circle surrounds the string), in the presence of a monopole you cannot contract a two-sphere around the monopole, and so on.

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    $\begingroup$ @AccidentalFourierTransform "Domain walls are not topological solitons" - I disagree with this statement $\endgroup$
    – Kosm
    Feb 3, 2022 at 14:27
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    $\begingroup$ @AccidentalFourierTransform in any case, I'm not familiar with that definition, where is it from? In my mind topological defects are just solitons. $\endgroup$
    – Kosm
    Feb 3, 2022 at 14:38
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    $\begingroup$ @Cham or unless there is a lower-energy vacuum. Then DW is metastable and can collapse by tunnelling. I guess this kind of DW can be classified into its own category $\endgroup$
    – Kosm
    Feb 3, 2022 at 14:46
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    $\begingroup$ @AccidentalFourierTransform I understood the question in terms of (classical) field theories, as asking about the generalization of DW for continuous $G$, as stable solutions to field equations $\endgroup$
    – Kosm
    Feb 3, 2022 at 15:28
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    $\begingroup$ @Kosm fair enough, I was thinking about QM, in fact the tag that I added may be wrong. I will remove it and ask OP whether they care about classical field theory or QFT. Cheers! $\endgroup$ Feb 3, 2022 at 15:35

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