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To my understanding, they both refer to circular symmetry. I know U(1) is the complex plane, and I think O(2) is real numbers. I have seen them used more or less interchangably, but is there actually some subtle difference or are they just two ways to say the same thing?

I'm specifically interested about this question as it applies to phase transitions in materials, specifically with BKT-like transitions. It seems people use U(1) and O(2) interchangeably to describe a BKT transition.

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    $\begingroup$ Related, but not exactly the same, since they're talking about SO(2): math.stackexchange.com/questions/308418/… $\endgroup$
    – Nike Dattani
    May 15, 2020 at 2:15
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    $\begingroup$ You might like to state explicitly what is the relevance of this question to materials modeling. $\endgroup$
    – Peter Morgan
    May 16, 2020 at 13:35

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The 1-dimensional unitary group U(1) simply corresponds to all complex numbers with a modulus of 1. This is isomorphic to the special orthogonal group SO(2), which corresponds to all real 2x2 rotation matrices. This is the case because because any element of U(1) is uniquely defined by its complex phase (going from 0 to 2pi) and this can be mapped uniquely onto the rotation angle in SO(2).

SO(2) is a subgroup of the orthogonal group O(2). O(2) also contains the improper rotations. As a consequence O(2) should be seen as a "larger" group than U(1), I would say.

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    $\begingroup$ What do you mean by a "larger" group? $\endgroup$
    – taciteloquence
    May 19, 2020 at 5:37
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    $\begingroup$ I guess the technical definition here is that it is possible to create a surjective homomorphism from O(2) to U(1) but it is not possible to create an injective homomorphism from O(2) to U(1). In particular this also means that there is no isomorphism. $\endgroup$
    – Felix
    May 20, 2020 at 9:18
  • $\begingroup$ @Felix That went way over my head, haha. Is it correct to say that O(2) and U(1) refer to the same symmetry even though they are not the same group? $\endgroup$
    – taciteloquence
    May 21, 2020 at 2:52

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