Gauss operators, gauge group and nonempty physical projection

The finite-support linear combinations of the joint basis vectors are dense in ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$. On each joint basis vector, $G_x$ multiplies by the real number ${\gamma }_x(\mathbf{n},\mathbf{m})$. The maximal diagonal operator is self-adjoint: if $\varphi \in D(G_x^{\ast })$ and $G_x^{\ast }\varphi =\eta$, testing the adjoint identity against each basis vector gives ${\eta }_{\mathbf{n},\mathbf{m}}={\gamma }_x(\mathbf{n},\mathbf{m}){\varphi }_{\mathbf{n},\mathbf{m}}$. Since $\eta \in {\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$, the defining weighted square sum for $D(G_x)$ is finite, so $D(G_x^{\ast })\subseteq D(G_x)$; conversely Cauchy–Schwarz gives the adjoint identity for every $\varphi \in D(G_x)$.

Every $G_x$ is diagonal in the same basis. Thus the spectral projections of any two $G_x$ are diagonal coordinate projections and commute, which proves strong commutation. On a basis vector, $U(\alpha )$ multiplies by $e^{i{\sum }_x{\alpha }_x{\gamma }_x}$; integer ${\gamma }_x$ makes it $2\pi$-periodic in each coordinate, and the diagonal phases give $U(\alpha +\beta )=U(\alpha )U(\beta )$ and $U(0)=I$.

For finite linear combinations of basis vectors, $\alpha \mapsto U(\alpha )\psi$ is a finite sum of continuous characters. Such vectors are dense and all $U(\alpha )$ have norm one, so for arbitrary $\psi$ approximation by a finite combination proves norm continuity of $\alpha \mapsto U(\alpha )\psi$. This is strong continuity.

From $[E_e,U_f]={\delta }_{ef}{U}_f$ and $[n_y,c_{x\beta }]=-{\delta }_{xy}{c}_{x\beta }$ on the finite occupation core, one has $[G_x,U_e]=({\mathbf{1}}_{x=s(e)}-{\mathbf{1}}_{x=t(e)})U_e$ and $[G_x,c_{y\beta }]={\delta }_{xy}{c}_{y\beta }$. Exponentiating these exact shifts on the joint basis yields the two displayed gauge-transformation laws.

For every $\psi \in {\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$, strong continuity from step 3.1 makes $\alpha \mapsto U(\alpha )\psi$ continuous on the compact torus. Its Hilbert-valued Riemann sums converge in norm: uniform continuity makes sums over partitions of mesh tending to zero a Cauchy family, and completeness supplies their limit. The normalized integral therefore exists, and its operator norm is at most one because each summand is an average of unitary images. For a basis vector with charge labels $\gamma =({\gamma }_x)$, that integral is the vector itself if $\gamma =0$ and zero otherwise, since $(2\pi {)}^{-1}{\int }_0^{2\pi }{e}^{ik\alpha }d\alpha ={\mathbf{1}}_{k=0}$ for each integer $k$. By density of finite-support vectors and the norm bound, $P$ is the diagonal multiplier ${\mathbf{1}}_{\gamma =0}$ on all of ${\mathcal{H}}_{\mathrm{k}\mathrm{i}\mathrm{n}}$; hence $P=P^{\ast }=P^2$ and its range is exactly the joint kernel of the $G_x$.

For each $x$, choose any $b_x$ of the $s$ fermion modes at that site to occupy. The occupation vector so obtained is normalized and has $n_x=b_x$; tensoring it with the zero-link Fourier basis vector gives every ${\gamma }_x=0$. Thus it is a nonzero vector in $\mathrm{ran}P={\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$.