Uniformly bounded finite-range quantum spin interaction

Use the lattice $\mathrm{\Gamma }={\mathbb{Z}}^d$, $d\ge 1$, the finite spin dimension $q\ge 1$, the local matrix algebras ${\mathcal{A}}_{\mathrm{\Lambda }}$, and the max-coordinate metric $d_{\mathrm{\infty }}$ of Local spin matrix algebras and their factor inclusions. For finite nonempty $Z\subset \mathrm{\Gamma }$, write ${\mathrm{diam}}_{\mathrm{\infty }}Z={\max }_{x,y\in Z}{d}_{\mathrm{\infty }}(x,y)$; for finite nonempty $X,Y\subset \mathrm{\Gamma }$, write ${\mathrm{dist}}_{\mathrm{\infty }}(X,Y)={\min }_{x\in X,y\in Y}{d}_{\mathrm{\infty }}(x,y)$.

A uniformly bounded finite-range interaction is a family $\mathrm{\Phi }(Z)\in {\mathcal{A}}_Z$, indexed by finite nonempty $Z\subset \mathrm{\Gamma }$, such that $\mathrm{\Phi }(Z{)}^{\ast }=\mathrm{\Phi }(Z)$ and there are an integer $M\ge 1$, a finite $R\ge 0$, and $\kappa <\mathrm{\infty }$ satisfying

$\mathrm{\Phi }(Z)=0\text{if }\mathrm{\mid }Z\mathrm{\mid }>M\text{ or }{\mathrm{diam}}_{\mathrm{\infty }}Z>R,\kappa :={\sup }_{x\in \mathrm{\Gamma }}{\sum }_{Z\ni x}\mathrm{\parallel }\mathrm{\Phi }(Z)\mathrm{\parallel }<\mathrm{\infty }\mathrm{.}$

The norm in this sum is the operator norm in the local matrix algebra ${\mathcal{A}}_Z$. If a model lists several terms with the same support, first add them in that local algebra; the resulting single operator is $\mathrm{\Phi }(Z)$, and $\kappa$ uses its norm. Thus no multiplicity is implicit.

For a finite region $\mathrm{\Lambda }\subset \mathrm{\Gamma }$, define the free-boundary finite-volume Hamiltonian by summing only terms whose complete support lies inside $\mathrm{\Lambda }$; terms meeting $\mathrm{\Lambda }$ but extending outside it are omitted:

$H_{\mathrm{\Lambda }}:={\sum }_{\mathrm{\varnothing }\mathrm{\ne }Z\subseteq \mathrm{\Lambda }}\mathrm{\Phi }(Z)\in {\mathcal{A}}_{\mathrm{\Lambda }}\mathrm{.}$

It is a finite sum of self-adjoint matrices. For a fixed scale $\mathrm{\hslash }>0$ and any finite $\mathrm{\Omega }\supseteq \mathrm{\Lambda }$, embed $H_{\mathrm{\Lambda }}$ in ${\mathcal{A}}_{\mathrm{\Omega }}$ by the canonical factor inclusion and define

${\alpha }_t^{\mathrm{\Lambda }}(A):=e^{it{H}_{\mathrm{\Lambda }}\mathrm{/}\mathrm{\hslash }}A{e}^{-it{H}_{\mathrm{\Lambda }}\mathrm{/}\mathrm{\hslash }}(A\in {\mathcal{A}}_{\mathrm{\Omega }},t\in \mathbb{R})\mathrm{.}$

These actions are compatible as $\mathrm{\Omega }$ increases, because the extra tensor factors carry the identity. On ${\mathcal{A}}_{\mathrm{\Lambda }}$ this is the ordinary finite matrix Heisenberg dynamics.

In physical SI bookkeeping, when $\mathrm{\Phi }(Z)$ is interpreted as an energy, $\mathrm{\Phi }(Z)$, $\kappa$, and $H_{\mathrm{\Lambda }}$ have units J, $t$ has units s, and $\mathrm{\hslash }$ is the adopted reduced Planck constant with units J s. The mathematics below uses only $\mathrm{\hslash }>0$ and asserts no measured parameter value.