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Here is a partial answer that depends on a particular choice of local gauge constraint. In a U(1) gauge theory, the usual gauge constraint is just Gauss' Law, $$\nabla \cdot \mathbf{E} = \rho.$$ This in turn implies Coulomb's Law $\mathbf{E} \sim 1/r$ for the electric field surrounding a deconfined point charge. Such a long-range interaction ought to be ...

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Is the existence of deconfined gauge charges a sufficient condition to ensure gaplessness? I think the answer is NO, such as the $Z_2$ gauge theory in 2+1D and 3+1D. I believe that the existence of deconfined gauge charges of a continuous gauge group is a sufficient condition to ensure gaplessness? Hastings and I have a paper ...

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In the continuum limit the lattice spacing $a$ goes to zero, therefore the Brillouin zone grows to infinity. If the Fermi velocity shall remain constant, the hopping parameter has to be rescaled as $t \propto 1/a$ (remember that the bandwidth is on the scale of $t$ and $v_F = \nabla_k E(\vec k)$), therefore only the features close to the Dirac points remain ...

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What is the fundamental reason of the fermion doubling? Answer: there is no fundamental reason of the fermion doubling. Adding proper lattice interaction can always get rid of the doublers, and it works for both abelian and non-abelian gauge groups, as long as the resulting chiral theory is free of all anomalies. (See Xiao-Gang Wen arXiv:1305.1045 and ...

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