In what condensed matter systems (without chiral anomaly) do we need two $U(1)$ gauge fields? In condensed matter systems, we use a $U(1)$ gauge field to describe the electric current by charge carriers. If there is a chiral anomaly, there will be a vector $U(1)$ and an axial $U(1)$. Suppose there is no chiral anomaly, is there any physical meaning for a Lagrangian with two $U(1)$ gauge fields?
For example,
$$\mathcal{L}=-\frac{1}{4}F_{ab}F^{ab}-\frac{1}{4}H_{ab}H^{ab}+\text{(interactions among $A_a$, $B_a$, and other fields)},$$
where $F=dA$ and $H=dB$. What condensed matter systems in the real world can be described by such a Lagrangian, and what is the meaning of the second gauge field?
 A: Yes, for fractional quantum Hall effect, we DO have two (with one being effective) $U(1)$ gauge fields.
In low dimensional systems, a fermion can be represented by an effective boson coupled to a Chern-Simons gauge field. A quintessential example is the 2D fractional quantum Hall effect, where electrons can be transformed into effective bosons coupled simultaneously to the statistic-changing Chern-Simons $U(1)$ gauge field and the usual $U(1)$ magnetic field. The fractional quantum Hall effect is resulted from Boson condensation which is allowed when the two $U(1)$ fields cancel out each other on average.
For more details, see here.
A: Often when one writes down a Lagrangian it tends to have symmetries that we did not intend to put in there. I think this is happening here as well. There seems to be a symmetry between $A_a$ and $B_a$. One can transform them into mixtures of each other and still end up with the same Lagrangian. So in the end one can combine $A_a$ and $B_a$ into one gauge field with a larger symmetry group. (Basically, this means you started with two Abelian gauge fields and then end up with one non-Abelian gauge field.) Thus one would end up with the usual situation where there is just one gauge field.
One can break this larger symmetry explicitly by having them couple to different fermion fields. Then you would end up with two different Abelian gauge fields. I think there are models in literature where people have tries this.
