How can one define vortex on a lattice

[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 don't 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?

 

[sirchasm]

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

 

[jensa]

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 don't 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?

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 $\varphi_{12}+\varphi_{23}+\varphi_{34}+\varphi_{41}=2\pi n$.

 

[wdlang]

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 $\varphi_{12}+\varphi_{23}+\varphi_{34}+\varphi_{41}=2\pi n$.

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

 

[wdlang]

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 $\varphi_{12}+\varphi_{23}+\varphi_{34}+\varphi_{41}=2\pi n$.

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

 

[jensa]

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

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 $\varphi_i'=\varphi_i+2\pi n$

 

[wdlang]

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 $\varphi_i'=\varphi_i+2\pi n$

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!

That make your n arbitrary

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

 

[jensa]

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!

That make your n arbitrary

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

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.

 

[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?

 

[wdlang]

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?

Thanks for your interests.

It baffles me for a long time. I post this thread because recently i noticed a paper. originally it was posted in the quantum physics section, but then it was moved here.

'uniformly frustrated bosonic josephson-junction arrays', PRA 79, 021604(R) (2009)

In the caption of fig.1, the author indicates how he identify a vortex. he resorts to the current between two sites, not the phase difference between them. The former is gauge invariant but the later is not. But i still can not see the goodness of his criterion.

I sent an email to him. He told me that if the sum is larger than 1, then he identify the presence of a vortex on that plaquett.