$O(N)$ symmetry in three dimensions

Question

Recently, In a research article on magnetism, I came across the term

"$O(N)$ symmetry for three dimensions with the limit $N->infinity$".

What does it mean? When I tried to search about it, could not find much in any Physics text/reference book. Although some Mathematics references on group theory have a discussion about it. But, I am more interested in getting the physical meaning of it. Will appreciate any explanation/reference about it. Is it a old convention symbol of any symmetry element or point/space group?

Deschele Schilder · Answer

It's very simple. $O(N)$ is the group distance preserving transformations in a Euclidean space of $N$ dimensions. It's called the orthogonal group. Look for example here to find all relevant information. I'm not sure how the article on magnetism fits in here, but I think you know best yourself.

Answered by Deschele Schilder on June 23, 2021

Cosmas Zachos · Answer

Go to the references your source is expressly sending you to, namely Rao, M., Krishnamurthy, H. R., & Pandit, R. (1990): "Magnetic hysteresis in two model spin systems", Physical Review  B42 (1) 856.
The three dimensions are spacetime dimensions, and N refers to the numbers of components in the vector representation of O(N). So their fields $Phi(x)^a$ live in a 3dim space indexed by x, and an internal ("spin") Ndim space indexed by the indices a, vectors rotating by the N(N-1)/2 rotations of the O(N) group. In the large N limit, the structure of these models simplifies, as you probably know, and their results follow.

$O(N)$ symmetry in three dimensions

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