Weighted interactions and gapped finite-volume paths

Fix $d\ge 1$, the site set $\mathrm{\Gamma }={\mathbb{Z}}^d$, its max-coordinate metric $d_{\mathrm{\infty }}$, and the local spin algebras ${\mathcal{A}}_{\mathrm{\Lambda }}$ of the uniformly finite-dimensional spin system. A finite interaction is a family $\mathrm{\Phi }(Z)=\mathrm{\Phi }(Z{)}^{\ast }\in {\mathcal{A}}_Z$ indexed by finite nonempty $Z\subset \mathrm{\Gamma }$. For a finite $\mathrm{\Lambda }\subset \mathrm{\Gamma }$, define

$H_{\mathrm{\Lambda }}^{\mathrm{\Phi }}:={\sum }_{\mathrm{\varnothing }\mathrm{\ne }Z\subseteq \mathrm{\Lambda }}\mathrm{\Phi }(Z),{\tau }_t^{\mathrm{\Lambda },\mathrm{\Phi }}(A):=e^{it{H}_{\mathrm{\Lambda }}^{\mathrm{\Phi }}\mathrm{/}\mathrm{\hslash }}A{e}^{-it{H}_{\mathrm{\Lambda }}^{\mathrm{\Phi }}\mathrm{/}\mathrm{\hslash }},$

where $\mathrm{\hslash }>0$, $A$ is embedded in a common finite local algebra, and the finite sum is a self-adjoint matrix. When a path ${\mathrm{\Phi }}_s$, $s\in [0,1]$, is used, each ${\mathrm{\Phi }}_s(Z)$ is $C^1$ in matrix norm and the derivative is taken term by term.

For $r\ge 0$ put $F(r)=(1+r{)}^{-d-1}$. The admissible weights used here are

$F_a(r):=e^{-ar}F(r)(a>0),F_{\rho }(r):=(1+r{)}^{-\rho }(\rho >d)\mathrm{.}$

For an admissible weight $G$, define the weighted interaction norm

$\mathrm{\parallel }\mathrm{\Phi }{\mathrm{\parallel }}_G:={\sup }_{x,y\in \mathrm{\Gamma }}\frac{1}{G(d_{\mathrm{\infty }}(x,y))}{\sum }_{\begin{array}{c}Z\subset \mathrm{\Gamma }\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e},\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{t}\mathrm{y}\\ x,y\in Z\end{array}}\mathrm{\parallel }\mathrm{\Phi }(Z)\mathrm{\parallel }\mathrm{.}$

The derivative norm of a path is

$\mathrm{\parallel }\mathrm{\partial }\mathrm{\Phi }{\mathrm{\parallel }}_G:={\sup }_{s\in [0,1]}{\sup }_{x,y\in \mathrm{\Gamma }}\frac{1}{G(d_{\mathrm{\infty }}(x,y))}{\sum }_{Z\ni x,y}\mathrm{\mid }Z\mathrm{\mid }\mathrm{\parallel }{\mathrm{\partial }}_s{\mathrm{\Phi }}_s(Z)\mathrm{\parallel }\mathrm{.}$

A finite-volume spectral band is a nonempty selected subset ${\mathrm{\Sigma }}_{\mathrm{\Lambda }}(s)\subseteq \sigma (H_{\mathrm{\Lambda }}^{{\mathrm{\Phi }}_s})$ together with the orthogonal projection onto the sum of its eigenspaces. It may be specified by an interval $I_{\mathrm{\Lambda }}(s)$ with smoothly varying endpoints, in which case ${\mathrm{\Sigma }}_{\mathrm{\Lambda }}(s)=\sigma (H_{\mathrm{\Lambda }}^{{\mathrm{\Phi }}_s})\cap I_{\mathrm{\Lambda }}(s)$. The band is uniformly isolated with gap $\gamma >0$ along a specified finite-volume exhaustion when

$\mathrm{dist}\text{\hspace{0.17em}⁣}({\mathrm{\Sigma }}_{\mathrm{\Lambda }}(s),\sigma (H_{\mathrm{\Lambda }}^{{\mathrm{\Phi }}_s})\setminus {\mathrm{\Sigma }}_{\mathrm{\Lambda }}(s))\ge \gamma$

for every volume in that exhaustion and every $s\in [0,1]$. For spectral-flow arguments, the selected eigenvalues must also form a locally fixed spectral cluster: for each $s_0$, a contour in the resolvent set encloses exactly the selected eigenvalues for all $s$ in a neighborhood of $s_0$. In finite dimension this implies that its projection is $C^1$ when the Hamiltonian path is $C^1$. This cluster condition holds, for example, for the first $r_{\mathrm{\Lambda }}$ eigenvalues counted with multiplicity when the integer $r_{\mathrm{\Lambda }}$ is fixed along the path and the gap after them stays open. Uniform distance between the selected and unselected spectra alone does not prevent an interval from changing membership discontinuously. A ground band is such a band containing the lowest eigenvalue or a finite-volume cluster of lowest eigenvalues; no uniqueness is implicit.

For physical units, take interaction terms and Hamiltonians in joules, time in seconds, and $\mathrm{\hslash }$ in joule-seconds. The path parameter $s$, lattice distance, and admissible weights are dimensionless; the interaction norm therefore has energy units and the derivative norm has energy units per dimensionless path parameter.

Local spin matrix algebras and their factor inclusions

Uniformly bounded finite-range quantum spin interaction