# Can we tell apart $A$ and $B$? -- from a non-relativistic system $A$ to an emergent relativistic system $B$

We know that the non-relativistic system like Graphene with a particular filling of electrons give rise to relativistic Dirac equations at low energy, with multi-flavors of Dirac particles.

In terms of free theory specified by the $$k$$-spaces, the different flavors of Dirac particles may live on the different points (say $$K,K'$$ points) in the $$k$$-space Brillouin zone.

At the low energy, we may live in the vacuum with several flavors of Dirac particles (quarks, leptons, etc) similar to the one in a certain non-relativistic system like Graphene (but we are in the 3+1 dimensional Dirac semi-metal instead of 2+1 dimensional Graphene).

Let us call a non-relativistic Dirac semi-metal-like/Graphene-like system $$A$$, and the relativistic system with Dirac particles as the system $$B$$.

My question is that can we tell apart $$A$$ from $$B$$? If God or some physicists told us that we actually live in a relativistic system $$B$$, could we determine at low energy that we are living in a non-relativistic system $$A$$ where the relativistic phenomenon of $$B$$ is just a low-energy emergent phenomenon?

And the other way around, suppose some physicists told us that the ground state of non-relativistic Dirac semi-metal-like/Graphene-like system (for instance at half-filling of electrons) are the same and can-not be distinguished from the relativistic systems of Dirac fermions in QFT. Could we justify or falsify the statements? For example, the two-flavor free Dirac fermions in relativistic QFT (system $$A$$) in the sub-atomic particle physics, do not require the labels of special $$K,K'$$ points in the $$k$$-space Brillouin zone for the system $$B$$. So do those special points imply some subtle difference between the systems $$A$$ and $$B$$?

A picture illustrates 2+1 dimensional Graphene and its Dirac cone:

When we say a phenomenon is emergent, we mean that the phenomenon only appears in the low-energy limit. So to check if a phenomenon is emergent or not, we just need to go to high enough energy and see if the phenomenon is still there. Consider a tight-binding model of graphene, where the electron dispersion relation is given by $$E(\boldsymbol{k})=\pm t\big|e^{\mathrm{i}k_y}+2e^{-\mathrm{i}k_y/2}\cos(\sqrt{3}k_x/2)\big|,$$ where $\boldsymbol{k}=(k_x,k_y)$ denotes the electron quasimomentum. Expand the dispersion relation around either $K$ or $K'$ point and let $\boldsymbol{p}=\boldsymbol{k}\pm(4\pi/3\sqrt{3},0)$ be the momentum deviation, we obtain $$E^2=\frac{9t^2}{4}\boldsymbol{p}^2\pm\frac{9t^2}{8}(p_x^3-3p_xp_y^2)-\frac{27t^2}{64}\boldsymbol{p}^4+\mathcal{O}(p^5),$$ which is definitely distinguishable from the relativistic dispersion relation $E^2=c^2\boldsymbol{p}^2$ by the presence of all the higher order terms in momentum (which are also the high-energy corrections that break the emergent Lorentz symmetry). These Lorentz breaking terms can be observed (most directly) from ARPES spectrums for condensed matter systems.
Finally, it worth mention that the subtlety of $K$ and $K'$ point is not essential. Because a single Dirac cone can emerge as the surface state of 3D topological insulators right at the $\Gamma$ point, which does not have the subtlety of special momentum points and fermion doubling. There are papers[1] proposing that the Standard Model of particle physics could emerge as the 3D surface of a 4D interacting topological insulator.