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$b_1, b_2$ and $b_3$ are reciprocal primitive vectors. $G$ is the set of all vectors that are in the reciprocal lattice and, as you said, is given by the linear combination of the reciprocal primitive vectors. The set of points, $G$, just define the lattice vectors or the locations of the origin of each Brillouin zone. Now we need to look within each ...


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deriving the desired expression for $\boldsymbol k$ and figuring out what $\boldsymbol k$ really is we will begin with the definition of bloch's theorem. $$\psi_{\boldsymbol k}(\boldsymbol r + \boldsymbol t_n) = e^{i \boldsymbol k \cdot \boldsymbol t_n} \psi_{\boldsymbol k} (\boldsymbol r)$$ here $\boldsymbol k$ is a wave vector defined in the primitive ...


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The bare coupling vanishes in the continuum limit, because the renormalized quantities are associated with measurements over very large scales (compared to the lattice spacing). One way to think about this is that since QCD is confining, the coupling at the scale of the lattice spacing must vanish if the coupling at the continuum scale is to remain finite. ...



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