3 days without any response (at a time where, in my opinion, good questions on RG are very scarce) indicate that you have not quite hit an audience with your question.

Which seems interesting and worth while to discuss, if you care to explain it quite a lot better. First, I think you should define your concepts more clearly and explicitly, starting with the Lagrangian of QED_(2+1). As of now, I can only guess what you mean. And my guess may very well be wrong.

Hey, thanks for the answer. I had the same impression, so let me reformulate my question. The classic Lagrangian of QED in two space and one time dimensions is defined via

L_classic= -1/4 F_{\mu \nu}^{\mu \nu} + \bar{\psi} (\im \slashed D - m ) \psi

However, if one applies an external magnetic background field perpendicular to the system, one creates relativistic landau levels. The landau level spectrum is assymetric and the zeroth landau level  has a  charge of abnormal parity, while the higher landau levels have charge of normal parity. This is for example shown in Schakels' paper: PhysRevD.43.1428 (see link below).

My question is now: 

Has the abnormal, so-called vacuum charge of the bulk system a partner of different parity at the edge? Meaning, is it mandatory that the full system bulk+edge has a  particle density of even parity?

Remark:

For me this question arises, cause the broken U(1) symmetry on the edge of a Chern-Simons system is cured by the  axial anomaly of the edge theory (anomaly inflow). So in bulk+edge a local U(1) symmetry is given (Callan-Harvey). However, this mechanism does not solve the appearence of a parity abnormal charge. But I'm not even sure if this is necessary.

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.43.1428

I was more interested in the fermion sector. Do you use (an odd number of) 2-component fermions, or a parity symmetric Lagrangian? Do you work with a finite chemical potential for the fermions? For the gauge field, what about a possible Chern-Simons terms? What is your parity anomaly?

I use two-component spinors. My mass term breaks parity in the classical Lagrangian and induces a Chern-Simons (CS) term via one-loop corrections. The quantum corrections generate  an additional anomalous term in the Ward idenentity. 

Yes, usually in the context of parity anomaly (Redlich) the classical classical Lagrangian has parity symmetry  (mass 0). However, one could also observe an anomaloud Ward Id if the mass is non-zero.

Most interesting is that under an external magnetic field the zeroth landau level carries the information of the parity anomaly, cause it carries a charge of anomalous parity.

Is not the parity anomaly discussed by Redlich essentially the same as the Witten anomaly? In any case an anomaly in the presence of SU(N) gauge fields, while you are considering U(1) gauge fields? Or are you talking about something else?

Actually, I am puzzled why a U(1) gauge theory with only one species of 2-component fermions should exists. How can one avoid fermion doubling in a lattice formulation?  The physical realisations I can think of, all start with pairs of 2-component fermions.

Now, if the single-species theory really exist, would not a mass term or external magnetic field break parity explicitly? Nothing anomalous with that.

So, I still feel confused about your meaning of anomaly.

Added: When you refer to existing works you should take the time to be much more explicit about which works you have in mind. "Well known" works are only known to those who knows it well; there may be very few from this set reading your question.  Where is the Callan-Harvey U(1) anomaly cancellation procedure (that you mention in the question) published?

Depends on the boundary conditions.  The Callan-Harvey mechanism is, of course, published in the paper by Callan and Harvey http://inspirehep.net/record/15691 but it simply implies that, since the 2+1 dimensional bulk theory doesn't have well-defined chirality, the chirality, that can be defined  on any two-dimensional boundary, is cancelled by a corresponding flow to or from the bulk. If the anomaly on any such boundary is cancelled by the field content, then the bulk decouples from the boundary and a two-dimensional theory can be defined, independent of any cutoff.

That Callan-Harvey paper discusses "the mathematical relation between non-abelian anomalies in 2 n dimensions, the parity anomaly in 2 n +1 dimensions, and the Dirac index density in 2 n +2"

If the gauge group is non-abelian; if the gauge group is abelian, then everything goes through, as well. And the consequences have been extensively studied on and off the lattice, too. 

To summarize: the 2+1 (or any theory in odd dimensional spacetime) dimensional theory doesn't have a parity anomaly, since chirality isn't well-defined. The Chern-Simons term expresses, precisely,  this fact. Any bulk anomaly is cancelled by surface terms. 

The interesting feature is about the theory defined on the-even dimensional-boundary. It can be anomalous. If it isn't, however, it can decouple from the bulk. 

Stam> It can be anomalous. If it isn't, however, it can decouple from the bulk.

Yes, and that makes it possible to have anomaly cancellations between very different sectors of the model. As is the case for the Standard Model. Unfortunately, this apparently means that lattice gauge theory simulators must run full Standard Model simulations on 5-dimensional lattices (with some possible shortcuts).

By the way, I found a recent (2016) paper by Witten on essentially this topic (on non-orientable manifolds). Since I have not read it yet (it is rather long) I will not link to it. But anyone can find it.

So far I understood the Callan-Harvey formalism, thanks for this great discussion.

Stam: "...simply implies that, since the 2+1 dimensional bulk theory doesn't have well-defined chirality, the chirality, that can be defined on any two-dimensional boundary, is cancelled by a corresponding flow to or from the bulk..."

So in an external background magnetic field the (spin and parity polarized) charge of the zeroth Landau Level

j^0= -eB/(4 \pi) sign(m)

flows in/out from the chiral anomaly of its edge liquid? Thus, overall the charge  is again a scalar in the theory (bulk+edge)?

Yes to both questions. The charge in 2+1 dimensional QED is a Lorentz scalar, as usual. It's just that the magnetic field is a pseudoscalar, rather than a pseudovector. 

Ok, perfect, there is one last point:

Without an external magnetic background field QED_2+1 has an symmetric spectrum, whereas it has assymetric Landau levels if one turns on the magnetic field. It seems that the partner of the zeroth Landau Level is missing. Usually one argues that the corresponding states are non-renormalizable. Are these states also pushed on the edge, if one ramps up the magnetic field?

The non-normalizable (not non-renormalizable!) states depend, of course, also, on the boundary conditions and, indeed, can appear as edge states, whose presence can affect the infinite volume limit. 

This is extremely interesting. Can you explain this a bit more, or do you know some publications, discussing this effect?

The best publication I could find is the paper of Blankenbecler and Boyanowsky (link below):

INDUCED QUANTUM NUMBERS IN 2+1 DIMENSIONAL QED

They discuss the formation of the zeroth Landau Level (LL) as follows:

"... These threshold bound states (states of the zeroth LL) deplete both the positive and negative energy continuum -“borrowing” states from both parts of the spectrum. This causes an asymmetry in the Dirac spectrum ..."

However, they don't discuss the non-normalizable states or rather their role at the edge.

http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-3893.pdf

This: http://web.physics.ucsb.edu/~phys123B/w2015/lecture6.pdf might be useful, for starting out. 

Thanks for the answer. The figure on page 14, which shows the realtivistic LL (here for two copies) is exactly what I'm interested in. In particular, I try to understand the "bumps" in the valence band (of one copy).

Numerically, states on these bumps are edge-states. The bumps get steeper and stepper if one increases the magnetic field and together form a kind of Dirac cone (expansion of the 0-LL edge liquid) in the high magnetic field limit.

These bumps  have the same symmetry under parity [or sign(m)] as the zeroth LL.

However, I don't understand their relation to the Callan Harvey formalism. Are these states of abnormal parity (inverse to the zeroth LL) and pushed towards the edge during an increasement of the magnetic field? 

The relation is that it's possible to describe the edge states quantitatively, by introducing domain walls. In that case one readily finds chiral zeromodes-that can be identified as edge states-on the corresponding boundaries. The bulk magnetic field couples them. 

All right. Are the following statements correct? I assume one single, massive QED_2+1 copy.

The edge liquids of the valance band LLs are caused by Callan-Harvey. These liquids are coupled to the topological Dirac cone (which already exists at B=0) via the magnetic field and thus generate the bumps.

The edge liquids can be described à la Callan-Harvey, i.e. as the response to the presence of domain walls. If they're charged and each carries a net charge,  then, IF there's a bulk magnetic field, THEN the they're coupled. If the net charge of each vanishes, then they are independent. 

The non-trivial issue is, whether the domain walls themselves are stable under fluctuations. And there the interesting answer is that they can become unstable, if the magnetic field isn't just external, but can fluctuate. Cf. for instance, https://arxiv.org/abs/1010.5281 and https://arxiv.org/abs/1001.1885 for the 2n+1 case in general. 

Perfect, thanks a lot.

Then I would guess that  the "bumps" in the valance band LLs are the result of the hybridization of a topological edge liquid with the usual edge liquids from  non-topological LLs?  Is this the right picture?

That would be interesting to understand better; I'm not sure that the statement, as made, is precise enough.