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To support this question I'm going to use a theoretical example.

Suppose we have some Hamiltonian (H) which is a function of a continuous parameter, say $\alpha$. Also suppose that as a function of alpha this Hamiltonian moves from a space of trivial topology to a space of non-trivial topology (i.e sphere($S^2$) -> torus($T$)).

Clearly the Hamiltonian is continuous as I have stipulated, but the change in topology across the phase transition is certainly discontinuous in the sense that topology has been generated by something (usually the breaking of some continuous symmetry).

So my question more robustly stated is: Is there a good model for describing this change mathematically, smoothly, in terms of the underlying manifolds? And how well understood is this process?

Super
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  • Did you try to go through some simple example? – Norbert Schuch Aug 26 '15 at 11:46
  • I don't know of a simple example, otherwise I wouldn't be asking the question. I'm not sure a simple example exists, since you're changing the physical topology of the system. – Super Aug 26 '15 at 12:42
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    You could for instance consider the mapping from the torus to the sphere which maps $(k_x,k_y)$ to the normalized vector in $\mathbb R^3$ proportional to $(\sin(k_x),\sin(k_y),2-cos(k_x)-cos(k_y)-m)$, see e.g. http://arxiv.org/abs/cond-mat/0505308. – Norbert Schuch Aug 26 '15 at 13:26
  • Perhaps I should qualify that by saying algebraically topological instead of geometrically topological since it has been shown that the topological space doesn't really matter too much in that case. I.e. see the comments in this discussion by Heidar: http://physics.stackexchange.com/q/70057/ – Super Aug 26 '15 at 16:26

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