# Is the artificial gauge field a gauge field?

The so-called artificial gauge fields are actually the Berry connection. They could be $U(1)$ or $SU(N)$ which depends on the level degeneracy.

For simplicity, let's focus on $U(1)$ artificial gauge field, or artificial electromagnetic field. One can show that it is indeed gauge invariant.

My criterion for a gauge field is that it should satisfy Maxwell's equations (or Yang-Mills' equations in the non-Abelian case).

Then I doubt whether the artificial gauge field is really a gauge field.

How to write down the $F_{\mu\nu}^2$? What's the conserved current?

And can we make gapped Goldstone modes using artificial gauge fields?

• I always thought the Berry connection is not a well defined one. I would be happy to learn more about your progress on this question. Below I've tried to be of help. – FraSchelle Dec 10 '12 at 22:50
• Thank you for your answer. Why do you "feel" Berry connections are ill-defined? You may post your feelings as another interesting question. – Machine Dec 11 '12 at 13:41
• Thanks for your interest. I'm working on :-p It's hard, since I firs have to learn more about connection in fiber bundle, and more about dynamics of two-levels system (since I consider only the simplest case). I'll post something as soon as I have better idea about that. For the moment it is pure speculation, and my feeling may be just wrong. The annoying point is of course the time dependent vs. adiabatic construction of quantum mechanics. – FraSchelle Dec 11 '12 at 13:46

The Berry connection/Curvature can be formulated as the connection/curvature of a principal bundle over the parameter space, in this sense it can be thought of as a "gauge theory". It can also contain topological information, such as the first Chern number (measuring "magnetic charge"), second Chern number (measuring "instanton charge") etc., so geometrically/topologically it looks like a gauge theory.

But it is not a usual gauge field for various reasons. First of all, the Berry connection is a "gauge field" on the parameter space and not the space-time. Secondly, the Berry phase is a purely geometrical/topological object and does not have any dynamics. So I don't think it makes sense to have a Maxwell/Yang-Mills term in the (spacetime) Lagrangian, since these terms contain the dynamical features of a gauge field. Also for this reason, I don't think it makes any sense to talk about conserved currents, etc. for the Berry connection.

It is however possible to have emergent dynamical gauge fields, but I only know about these examples in strongly interacting theories (search for example for quantum spin-liquids).

• Yes, currently, the artificial gauge fields are background fields. But you can prepare a position dependent Zeeman field, then your Berry connection depends on position. I think Prof. Xiao-Gang Wen proposed many models for emergent gauge fields of different kinds. – Machine Dec 11 '12 at 13:37

I believe part of your question has been answered by the paper

Simon, Barry. Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase. Phys. Rev. Lett. 51 no. 24, pp. 2167–2170 (1983). doi:10.1103/PhysRevLett.51.2167.

Also, the gauge field only verifies the Maxwell equations without source I think, so you have $dA = F$ and that's it. Simon shows that explicitly.

Obviously, time is badly implemented in the Berry connection structure, so there must be some trouble when trying to apply relativistic arguments. I believe there is no need for conserved current : in Berry's idea what you ('re suppose to) conserve is the ground state subspace.

• Hi Oaoa, welcome to physics.stackexchange. I don't think I understand what you mean by "it verifies the Maxwell equations without a source". The equation $F = \text dA$ (and its non-abelian generalization $F = \text dA + A\wedge A$) just gives the curvature ("fields strength") from the connection. Isn't the Maxwell equation you are thinking about this one: $\text d\star F =0$? (The Bianchi identity $\text d F=0$ is purely geometrical constraint.) – Heidar Dec 11 '12 at 0:00
• @Heidar Note it as you want, the Maxwell equation without source define the gauge fields. So they are the Bianchi identity if you wish. For me, they're still $\nabla\times E+\partial_{t}B=0$ and $\nabla\cdot B=0$ (or $dF = 0$ if you wish). I believe (but I've no clue about that up to now), that the equations with sources $d\star F=0$ can not be established for the Berry phase. Moreover, I believe (but I've no clue about that neither) that, due to the absence of time evolution in the Berry calculation, $\partial_t B = 0$ and the fields $E$ and $B$ do not talk to each other. – FraSchelle Dec 11 '12 at 9:05
• @Heidar And sorry also, because I always prefer to note $dA = F$ than $dF = 0$, which is just a consequence of the first (as you said, it's Bianchi identity after all), because I was thinking of the Maxwell equations. But you're right I should note $dA+A\wedge A=F$ in general, thanks for pointing this out. – FraSchelle Dec 11 '12 at 9:09
• I think I was inaccurate, you did not say anything wrong. You are of course right that two of the Maxwell equations are satisfied, written in fancy form as $\text d F=0$. My point was however that this is just a purely geometrical constraint (Bianchi identity), and not the dynamical equations. It is therefore "trivial" that these are satisfied. The $F_{\mu\nu}F^{\mu\nu}$ term in the Lagrangian give rise to the equations $\partial_\mu F^{\mu\nu}=0$ (or $\text d\star F=0$ in fancy notation), the other equation $\text d F=0$ has to be imposed by hand (which is done by putting $F=\text dA$). – Heidar Dec 11 '12 at 9:53
• Thanks for your comments. I think there is a point of confusion. $\text d A=F$ is satisfied for the Berry phase, but you can only write it as $\nabla\times A=B$ if the parameter space is 3+1 dimensional (and then $\nabla$ is derivatives wrt. the parameters, not space). A usual space-time gauge field is of the form $A(\mathbf x, t)$, where $(\mathbf x,t)$ are space-time points. But the Berry connection is of the form $A(\lambda_i)$, where $\lambda_i$, ($i=1,\dots N$) are the parameters you are changing adiabatically. So there are no time, and the parameter space can be any dimensions. – Heidar Dec 11 '12 at 14:57