TITLE: Vorticial ground states for the O(2) rotor model
QUESTION [5 upvotes]: Is there a sensible notion of a ground state for the classical $O(2)$ rotor model "frustrated at infinity by a single unit of counterclockwise vorticity"? Here is a picture of the kind of thing I mean, taken from Brian Skinner's blog:

I'm not sure the phrase in quotation marks is standard terminology, so let me clarify the question. The model I have in mind assigns to each site $i$ in the lattice $\Lambda = {\bf Z}^2$ a a unit-length vector ${\bf s}_i \in {\bf R}^2$ that we call a spin. The interaction energy between two adjacent sites $i,j$ is $- {\bf s}_i \cdot {\bf s}_j$. We naively define the energy of a spin-configuration as the sum of these interaction energies (summed over all pairs of adjacent sites in $\Lambda$), with the understanding that since this sum almost always diverges, we need to regularize it in some principled way. More on regularization later.
If we didn't bring some frustration into the situation, it's pretty clear what the ground-states of the model would have to be, regardless of the regularization scheme one chooses: spin-configurations in which all the spins line up with each other. However, I want to impose a couple of conditions. The first is that the configuration should be "slowly varying near infinity": for all $\epsilon > 0$, there should be a radius $R$ such that if $i$ and $j$ are adjacent sites at distance $>R$ from the origin, the angle between ${\bf s}_i$ and ${\bf s}_j$ is less than $\epsilon$. (Note that I stipulated adjacency between $i$ and $j$ in this condition; if $i$ and $j$ are far apart, I don't require ${\bf s}_i$ and ${\bf s}_j$ to be close.) The second is that the configuration should be "vorticial": if one looks at the spins along any closed lattice path sufficiently far away from the origin, the (slowly-changing) spins should make a single counterclockwise turn.
I would like to know whether within this class of "vorticial" spin-configurations there are ones that minimize energy, and if so, what those configurations look like. Note that the class of vorticial configurations (unlike the product space it sits inside) is not compact, so it's not guaranteed that an actual minimum exists (even leaving aside the issue of what "energy-minimizing" means when the naively-defined energy diverges). 
One way to regularize the sum is to change the global guage so that two vectors being parallel counts as energy 0 rather than energy $-1$. However, I think that even with this change, all the candidates for "vortex ground-states" have energy that diverges like the harmonic series.
One could sidestep divergence by replacing $\Lambda$ by a large finite patch of $\Lambda$ and imposing "vorticial" conditions along the boundary, and then look for limiting behavior of the resulting energy-minimizing configuration as the patch grows to exhaust $\Lambda$.
One might work directly with $\Lambda$ but mitigate divergence issues by looking not at energies of configurations but energy-differences between two configurations. Indeed, in this vein one might replace the search for globally energy-minimizing configurations by a search for locally energy-minimizing configurations, having the property that changing the spins at any finite number of sites results in higher energy (as measured by an increase in the sum of the finitely-many interaction energies that involve the affected sites).
Another way to regularize would be to use a summable multiplier as described in Regularizing divergent sums over lattices.
Assuming that a minimum does exist, or a family of minima, one might seek an exact solution.
If this is old hat for condensed matter theorists, where can I learn about it?
(Condensed matter is not my field, so if I am misusing terminology or laboring under misconceptions I hope readers of this forum will set me straight.)

REPLY [3 votes]: I think your question is interesting and reasonable. But I don't know the answer, so I'll say something else.
Kosterlitz and Thouless won the physics Nobel prize this week for their work on this subject.  According to them, the 'vortex' configuration in Brian Skinner's picture has infinite energy - at least for an ideal thin film of magnetic material that's infinite in spatial extent.  So, they didn't try to find an energy-minimizing configuration of this sort: indeed they used the infinite energy to argue that we'd never see a single vortex by itself.   Instead, they noticed that a 'vortex-antivortex' pair can have finite energy - and they showed that as you heat up a thin film starting from absolute zero, a 'phase transition' occurs when the concentration of vortex-antivortex pairs suddenly increases.
I wrote a blog article about this today:

John Baez, Kosterlitz-Thouless transition, Azimuth, 7 October 2016.