= Three-site Abelian Higgs chain
{title2=$\mathsf M^2=2g^2v^2R^{\mathsf T}R$}
Two canonically normalized complex <scalar fields> of <vacuum> modulus $v$ and charge rows $R_1=(1,-q,0)$, $R_2=(0,1,-q)$ couple to three canonically normalized Abelian vector <fields>. The scalar <kinetic terms> give $g^2v^2|RB|^2$, so the physical squared-mass <matrix> includes the factor two shown. Its nonzero <eigenvalues> are $2g^2v^2(1+q^2\pm q)$ and its normalized null vector is $(q^2,q,1)/\sqrt{q^4+q^2+1}$. The two scalar phases are absorbed into the massive <vector bosons>, leaving one unbroken local Abelian generator. A compact global gauge-group interpretation additionally requires a compatible charge lattice.
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