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I am trying to follow this paper and track the dynamics of vortex motion on a discrete (square) lattice. The idea is to simulate the time evolution of the Gross-Pitaevskii (GP) equation, which reads (in rescaled units)

$$i \partial_t \psi = - \nabla^2 \psi + |\psi|^2 \psi.$$

As initial condition one takes a wavefunction of the form of Eq. (10) in the above-mentioned paper,

$$\psi(z) = \prod_{z_+ \in [Z_+]} \frac{(z-z_+)}{|z-z_+|} \prod_{z_- \in [Z_-]} \frac{(z^*-z_-^*)}{|z-z_-|}$$

where $z=x+iy$ are complex coordinates. This wavefunction describes a set of vortices (V) with coordinates $z_+$ in $Z_+$ and a set of anti-vortices (AV) with coordinates $z_-$ in $Z_-$. This is equivalent to using a wavefunction of the type $e^{i \phi}$ (polar coords.) for vortices and $e^{-i \phi}$ for anti-vortices, with $\phi$ the polar angle. One can calculate the velocity as gradient of the phase of the complex wavefunction. An example of the associated velocity field, for a V at (x,y)=(0,2.5) and AV at (0,-2.5) is shown below.

velocity-lines

This should correspond to Fig. 1 in the paper. However, there are several things I do not understand.

First, why do the authors say that they cannot look at the dynamics of just one vortex, but must include an anti-vortex in the initial conditions to satisfy the periodic boundary conditions?

Second, apart from the initial V-AV pair, they also seem to include "a few mirrow images with respect to the boundary". However, they do not specify how many nor where they are positioned. (I assume that the mirrow image of a V is and AV and vice-versa, please correct me if I'm wrong).

Lastly, they mention "evolving the system in imaginary time" in order to "cool it down" (although no temperature is included in the model!) and then "turning on a superflow" and continuing the evolution in real-time. I though all was needed was the discretization of the GP equation and following its evolution given the initial wavefunction and periodic boundary conditions ($\psi(\text{left margin})=\psi(\text{right margin})$ and $\psi(\text{top margin})=\psi(\text{bottom margin})$).

Edit

Following the suggestion of @BebopButUnsteady, I repeated the 'unit' of my system, the V-AV pair, until obtaining the following tiling pattern (vortices are circles, AVs are squares)

tiles

Let us first look at the real (left) and imaginary (right) parts of the wavefunctions for the initial V-AV pair:

initial-boundaries

One can see that, especially in the imaginary part, the values on the boundaries are not equal. Now we start adding copies of the "unit cell" along the x axis, and plot a cut though the imaginary part of the wf:

boundaries-tiled

It appears that, as we increase the number of copies, the values at the ends get closer and closer to 0, which would be the ideal periodic case. Of course this is just numerics, I have no formal proof yet that this procedure actually converges.

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Minor comment to the question (v1): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/cond-mat/0503717 –  Qmechanic Jul 10 '13 at 17:49
    
Firstly, because you are annealing you only need it to be roughly periodic. Second, I think you missed the point of the mirror images - you should generate the velocity field with all the mirror images, but then "crop" the result so that only a central pair of vortices are left in your lattice. Also one direction is already symmetric - you do not need mirror images along that direction. There may be issues with the whole thing because I don't think it converges in the limit of images going to infinity, but in that case you should ask the authors what they intended. –  BebopButUnsteady Jul 10 '13 at 21:39
    
@BebopButUnsteady, assuming however that it does somehow converge to perfect periodic BC, it is unclear to me why I have to do the annealing before the actual time evolution, what do you mean by "shedding waves and other junk"? Also, how can the superflow (which is just the gradient of the phase of $\psi$) be turned on/off? –  Andrei Jul 11 '13 at 7:18
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1 Answer

up vote 3 down vote accepted

You cannot have a total vorticity with periodic boundary conditions, since if you take a path around all of your vortices, it will have a non-zero circulation. But you have periodic bc, so you can continuously deform that path to a point, and a point has zero circulation.

Mirror images are not quite the same as in electrostatics. We want periodic boundary conditions. To get periodic boundary conditions you imagine tiling the plane with your system. This will trivially be periodic and so you can just take a tile, and work with that. So you want the mirror image of a vortex to be a vortex.

(In electrostatics you usually use mirror images to enforce not periodic boundary conditions, but to enforce constant voltage. That's why you flip the sign of the mirror charge. Here we want periodic b.c. If you flipped the sign of the vortices I believe you would get *anti*periodic boundary conditions, but don't quote me on that.)

This evolving in imaginary time is presumably an "annealing" type of operation. You are free to run the GP equation on any initial condition you want. However, to cleanly see the interaction of the vortices, we want to the vortices to be in their ground state. Otherwise when we turn on the time evolution they will get rid of their excess energy by shedding waves and other junk.

One way to get to the ground state is to evolve you equation in "imaginary time". Your usual time evolution is $\exp(i\hat{H}t)$. If plug in $t = i\tau$ you get $\exp(-\hat{H}\tau)$. Applying this to a state exponentially suppresses the higher-energy components, so you get rid of the high energy stuff. This is related to finite-temperature (just replace $\tau$ with $\beta$ and you have the partition function), but for your purposes you can just consider it a convenient mathematical trick.

Note that since you're annealing anyway, the specific details of what state you start might not be so important, since you will end in the same place anyway (hopefully).

Finally, they are a little thin on the details, so if you plan to use this work, you should just send the authors an email asking for details.

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Excellent answer. Just a remark: the trick $t=i\tau$ is called Wick's rotation / Matsubara technique in the literature. The other way round, when you want to discuss temperature effect plus a bit of (time-dependent) interaction is called analytic continuation. This details may help Andrei. Thanks again for your answer. –  FraSchelle Jul 11 '13 at 7:43
    
Thanks, knowing what terms are used in the literature does help indeed :) @Oaoa, you wouldn't happen to know any good references for a novice on this topic? Wikipedia is very mathematical, I would need something more connected to my application. –  Andrei Jul 11 '13 at 10:38
    
@Andrei Unfortunately, I'm not so much interested in vortex. But you may find some things about the Mastubara formalism and the analytic continuation from the temperature formalism in a (rather unclear and badly written) book by Kopnin Non-equilibrium superconductivity at the Oxford publications. The Matsubara details are well reviewed in the famous book by Abrikosov, Gor'kov and Dzialoshinski Methods of quantum field theory in statistical physics, Dover. The book by Volovik The universe in a Helium droplet may contains some interesting details regarding vortex matter... –  FraSchelle Jul 11 '13 at 11:22
    
... at least you can find good references there. Volovik spends all his time in superfluids theory. His book is freely available from his homepage ltl.tkk.fi/wiki/Grigori_Volovik at Aalto University. You may also find (non-mathematical) details in the book by Leggett Quantum liquids, Oxford. Finally, modern textbooks are Altland and Simons, Xiao-Gang Wen, and Nagaosa books (these 4 books (there are 2 Nagaosa books) contain condensed matter or many-body or quantum field somewhere in their titles), and may contain vortex staff also. –  FraSchelle Jul 11 '13 at 11:28
    
Thanks a lot @Oaoa, I will definately have a look! On a last note, do you understand why annealing is really necessary in my case -- and this funny business with turning "on" the superflow? –  Andrei Jul 11 '13 at 11:55
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