Translation-invariant finite-range spin interaction

Use the lattice $\mathrm{\Gamma }={\mathbb{Z}}^d$, the finite-site Hilbert spaces, local matrix algebras, factor inclusions, and norm-completed local algebra $\mathcal{A}$ of Local spin matrix algebras and their factor inclusions and Norm-completed quasi-local quantum spin algebra. Write ${\tau }_x$ for spatial translation by $x\in \mathrm{\Gamma }$. For a finite region $\mathrm{\Lambda }$ and its configuration basis $(e_s{)}_{s\in \{1,\dots ,q{\}}^{\mathrm{\Lambda }}}$, the bijection $s\mapsto s_x$ defined by $s_x(y+x)=s(y)$ gives a unitary $U_{x,\mathrm{\Lambda }}:{\mathcal{H}}_{\mathrm{\Lambda }}\to {\mathcal{H}}_{\mathrm{\Lambda }+x}$ by $U_{x,\mathrm{\Lambda }}{e}_s=e_{s_x}$. On $A\in {\mathcal{A}}_{\mathrm{\Lambda }}$ put ${\tau }_x(A)=U_{x,\mathrm{\Lambda }}A{U}_{x,\mathrm{\Lambda }}^{\ast }$.

For $x,y\in \mathrm{\Gamma }$, the basis relabellings satisfy $U_{x,\mathrm{\Lambda }+y}{U}_{y,\mathrm{\Lambda }}=U_{x+y,\mathrm{\Lambda }}$, since both sides send each configuration vector $e_s$ to $e_{s_{x+y}}$. Consequently ${\tau }_0$ is the identity and ${\tau }_{x+y}={\tau }_x\circ {\tau }_y$ on local observables; continuity extends the action law to $\mathcal{A}$.

This local map is independent of the chosen localization region: enlarging $\mathrm{\Lambda }$ tensors both representatives with identities, and the same site relabelling gives the corresponding enlarged unitary. On each finite region it is a unitary conjugation, hence isometric and preserves products, adjoints, and identities. It therefore extends uniquely to an isometric -automorphism of the norm completion $\mathcal{A}$: if $A_n$ is local and converges to $A$, define ${\tau }_x(A)={\lim }_n{\tau }_x(A_n)$; isometry makes this limit independent of the approximating sequence, and ${\tau }_{-x}$ is its inverse. This construction uses the norm-completion property and the isometric local maps; it does not use the separate finite-matrix C-identity condition.

Let $\mathrm{\Phi }$ be a uniformly bounded finite-range interaction in the sense of Uniformly bounded finite-range quantum spin interaction, with parameters $M,R,\kappa$. It is translation invariant when ${\tau }_x(\mathrm{\Phi }(Z))=\mathrm{\Phi }(Z+x)$ for every finite nonempty support $Z$ and every $x\in \mathrm{\Gamma }$. Let ${\mathcal{S}}_{\mathrm{\Phi }}$ contain one representative of each nonzero translation orbit, anchored so that the lexicographically least site of each representative is $0$, using the standard coordinate lexicographic order on ${\mathbb{Z}}^d$. The least site exists because each support is finite and nonempty; every representative contains $0$. The list ${\mathcal{S}}_{\mathrm{\Phi }}$ is finite: every such support has at most $M$ sites and lies in the finite cube $[-\lceil R\rceil ,\lceil R\rceil {]}^d$. For finite $\mathrm{\Lambda }$, the free-boundary Hamiltonian is $H_{\mathrm{\Lambda }}={\sum }_{\mathrm{\varnothing }\mathrm{\ne }Z\subseteq \mathrm{\Lambda }}\mathrm{\Phi }(Z)$, as in the interaction definition.

When $\mathrm{\Phi }(Z)$ represents an energy, its norm, $H_{\mathrm{\Lambda }}$, and the constant $J_{\mathrm{\Phi }}:={\sum }_{S\in {\mathcal{S}}_{\mathrm{\Phi }}}\mathrm{\parallel }\mathrm{\Phi }(S)\mathrm{\parallel }$ have units joules. The definition is mathematical and makes no empirical identification of an interaction with a material.