# How can one define vortex on a lattice

1. Feb 17, 2009

### wdlang

vortex in superfluidity is well known

in a continuum system, a vortex is well defined

along a closed (continuous) path, the phase change is well defined.

However, on a lattice, i donot think the phase change from site to site can be well defined.

but people are talking about vortex in a josephson junction array.

Could anyone explain this to me?

2. Feb 18, 2009

### sirchasm

It's to do with electron fluid-dynamics, and lattice algebras.

3. Feb 18, 2009

### jensa

I am not sure I understand what you mean. Are you saying that the phase-difference across a josephson junction is not well defined?

Anyway, I think of a Josephson junction array as a grid where the vertices represent superconducting islands. The phase difference between islands on vertex i and j is $\varphi_{ij}$. Going around in a loop the total phase-difference must add up to a multiple, n, of 2pi, this is a vortex with a trapped flux of n times the flux quantum. For example if we go around vertices 1,2,3,4 and then back to 1 ( I hope it is clear what I mean with this). We have $\varphi_{12}+\varphi_{23}+\varphi_{34}+\varphi_{41}=2\pi n$.

4. Feb 19, 2009

### wdlang

Thanks a lot

but consider the case: \varphi_i=i*\pi/2

then intuitively, we have a vortex

but according to your rule, n=0

5. Feb 19, 2009

### wdlang

the problem is that how do you define \varphi_{ij}

if you define it as \varphi_{ij}=\varphi_i-\varphi_j

then you will always get n=0

in the continuum case, you do not have this problem

6. Feb 19, 2009

### jensa

Why then, would you say, are things any different in the continuous case? In the array you could think of the phase as a piece-wise continuous function, being constant in the islands and jumping across the junctions. Now if you make the jumps sufficiently smooth you have exactly the same situation as in the continuous case. You can then write the closed loop integral as:

$$\Phi=\oint d\vec{l}\cdot\vec{A}\propto\oint d\vec{l}\cdot\vec{\nabla}\varphi=\varphi_{12}+\varphi_{23}+\varphi_{34}+\varphi_{41}=2\pi n$$

The reason why you can have non-zero n is because of the following:

Let us parametrize the loop by $s\in[0,1]$. The order parameter has the form $\Psi(s)=|\Psi(s)|e^{i\varphi(s)}$ and must be single valued, i.e. $\Psi(1)=\Psi(0)$, but this does not mean $\varphi(1)=\varphi(0)$ but rather $\varphi(1)=\varphi(0)+2\pi n$ the n being determined by the flux through the loop.

So the accumulated phase difference is 2pi n.

The point is, for the continuous case as well as the jj-array case, is that after the loop you end up with a different phase than you started with (even-though you get back to the same point). In the jj-array case this means if you start your loop at vertex i with phase $\varphi_i$ and then go around the loop you will end up again at vertex i but with the phase $\varphi_i'=\varphi_i+2\pi n$

Last edited: Feb 19, 2009
7. Feb 20, 2009

### wdlang

Yes, you need to take a continuous loop to link the sites!

But in doing so, there is a pitfall

on the segment from site i to site j, with the two ends fixed, the phase can wind with an arbitrary number of loops on the unit circle!

anyway, i am happy that someone is interested in my question, which always baffles me.

8. Feb 21, 2009

### jensa

Yes, I too find it quite interesting.
I see now what you are saying, and I agree that it is a little dicey. Although I would like to add that there is a certain class of josephson junctions, usually called weak links, where the order parameter in the junctions is small but finite (the order parameter may be suppressed due to geometrical constriction or the intermediate material may be such that superconductivity is induced within the junction). I suppose you would agree that for such junctions there is no problem as the loop can be taken such that the orderparameter is non-vanishing along the whole loop and the phase is everywhere defined and continuous.

However, the junctions that are used in jj-arrays are most likely tunnel junctions with insulating material so that the order-parameter really vanishes inside the junction. I remember that Legget briefly discusses the difference between these two types of junctions in his book on quantum liquids, but I do not remember if he discusses this issue in particular.

I would say that your concern is not just an issue specific to vortices in jj-arrays but regards almost all jj-devices involving tunnel junctions. Let us think of a simpler example, like a rf-SQUID (i.e. superconducting loop interrupted by one tunnel junction). The flux through the loop determines the phase difference across the junction. In this case the arbitrariness you are talking about (i.e. the arbitrary number,n, of windings on the unit circle) seems to be fixed by the number of whole flux quanta through the loop.

$$\Phi=\tilde{\Phi}+n\Phi_0 \rightarrow \varphi=\tilde{\varphi}+2\pi n$$

If we add another junction to the loop (dc-SQUID) we have two phase-differences that add up to a total accumulated phase-difference, $\phi=\phi_1+\phi_2$ which is fixed by the flux through the loop. While we don't know the different "winding numbers" $n_1,n_2$ we do know that $n_1+n_2=n$ where n is the number of whole flux quanta through the loop.

What I am trying to say with this is that I don't really see a problem with having an arbitrary number of windings of the phase difference as long as they all add up to account for the number of flux quanta through the loop/vortex.

9. Feb 22, 2009

### jensa

On second thought I think I have been focusing too much on josephson junctions in this thread. I think Josephson junction arrays are described by the same model as spins on a 2D lattice - the XY model. Here it seems you can also have topological defects such as vortices in terms of the spin-alignments. Obviously the same question arises here - how do you define vortices on something which is not continuous. As far as I can tell people often treat these systems in the continuum limit where vortices can be defined in a rigorous way. Probably the same thing with josephson junction arrays.

I am surprised that this thread has not received many replies. Perhaps it would attract more attention in the quantum physics section?

10. Feb 22, 2009