Exponential connected clustering from a unique infinite-volume ground-state gap

A cyclic representation of the spin algebra is separable. Indeed, finite tuples of the countable lattice enumerate its finite subsets, and rational real/imaginary entries form a countable dense family in each finite local matrix algebra. Countable Choice permits their countable union. Density follows, for a $D\times D$ matrix, from $\mathrm{\parallel }T\mathrm{\parallel }\le D{\max }_{i,j}\mathrm{\mid }{T}_{ij}\mathrm{\mid }$, obtained by finite Cauchy–Schwarz. The resulting countable family is dense in the completion, and its vectors $\pi (A)\mathrm{\Omega }$ are dense in the cyclic Hilbert space by contractivity. The cyclic representation supplied by [F3] has the same state pairings as any given cyclic representation: the map $[A]\mapsto \pi (A)\mathrm{\Omega }$ preserves $⟨[A],[B]⟩=\omega (B^{\ast }A)$ and extends by density to a unitary. Thus the chosen cyclic ground representation is a separable realization of the actual GNS state, with no extra representation hypothesis.

We verify the advertised finite-range instance of the commutator premise. If $\mathrm{\Phi }$ has range $R$ and per-site norm bound $\kappa$, a pair separated by more than $R$ contributes no interaction term, while for a pair of distance at most $R$ the numerator of the weighted norm is at most $\kappa$. Hence $\mathrm{\parallel }\mathrm{\Phi }{\mathrm{\parallel }}_{F_a}\le \kappa \mathrm{/}{F}_a(R)<\mathrm{\infty }$. The uniform finite-volume bound [F2] passes to the thermodynamic limit for each local $A,B$ and fixed $t$, by norm continuity of commutators. The group is isometric, so $\mathrm{\parallel }[A,{\alpha }_t(B)]\mathrm{\parallel }=\mathrm{\parallel }[{\alpha }_{-t}(A),B]\mathrm{\parallel }$, matching exactly the supplier's time/support convention. This establishes the stated constants without replacing a general infinite-volume gap by a finite-volume gap.

The cyclic restriction in the ground-state definition preserves positive energy. Let $\mathcal{L}=\overline{\pi (\mathcal{A})\mathrm{\Omega }}$ in a ground representation. Covariance and $U_t\mathrm{\Omega }=\mathrm{\Omega }$ imply that $U_t$ and $U_{-t}$ preserve $\mathcal{L}$; its orthogonal projection commutes with $U_t$. For $x\in D(K)$, difference quotients show that this projection preserves $D(K)$ and commutes with $K$. The generator domain inside $\mathcal{L}$ is dense: the vectors $h^{-1}{\int }_0^h{U}_{t}xdt$ lie in it and tend to $x$ for $x\in \mathcal{L}$, as their difference quotients give $iK$ equal to $(U_{h}x-x)\mathrm{/}h$. The restriction is self-adjoint because $K\pm i$ are onto, their bounded inverses preserve $\mathcal{L}$ by the commuting projection, and the restricted symmetric operator has both ranges onto; directly, if $y$ is in its adjoint domain, solve $(K+i)x=(K_{\mathcal{L}}^{\ast }+i)y$ in $\mathcal{L}$, and the zero kernel of $K_{\mathcal{L}}^{\ast }+i$ forces $x=y$. It is positive by restriction. More generally, for every $z$ in the resolvent of $K$, commutation of the projection with $K$ on its domain implies commutation with $(K-z{)}^{-1}$: apply $(K-z)$ and its injectivity to the two candidate images. This inverse therefore restricts to the inverse on $\mathcal{L}$, so $\sigma (K_{\mathcal{L}})\subseteq \sigma (K)$ and the spectral gap is inherited, as is a simple ground kernel. The resolvent surjectivity follows from the self-adjoint closed-range estimate exactly as in the spectral-measure helper. Thus the definition using a possibly noncyclic representation agrees with its cyclic restriction, without a Stone or spectral-existence theorem.

Let $P_0$ be the ground-kernel projection in the separable representation. A unit vector $\psi \in \ker K$ satisfies $U_t\psi =\psi$, since its orbit derivative is $i{U}_{t}K\psi =0$. Its vector state ${\eta }_{\psi }(A)=⟨\pi (A)\psi ,\psi ⟩$ is a positive-energy ground state using the same representation; step 1.3 also supplies its cyclic version if desired. Uniqueness gives ${\eta }_{\psi }=\omega$. Scaling and polarization inside $\ker K$ therefore yield $P_0\pi (A)P_0=\omega (A)P_0$ for every $A$. If $\psi \in \ker K$ is orthogonal to $\mathrm{\Omega }$, this identity implies $⟨\pi (A)\mathrm{\Omega },\psi ⟩=0$ for every $A$. Cyclicity forces $\psi =0$. Since the fixed vector $\mathrm{\Omega }$ has zero orbit derivative, $\mathrm{\Omega }\in D(K)$ and $K\mathrm{\Omega }=0$, so we have $\ker K=\mathbb{C}\mathrm{\Omega }$. The alternative hypothesis stated in the theorem supplies this directly and needs no uniqueness argument.

Put $N=\mathrm{\parallel }A\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$ and $f(t)=\omega ([A,{\alpha }_t(B)])$. The displayed Lieb–Robinson premise, $F_a(s)\le e^{-as}$ and $d_{\mathrm{\infty }}(x,y)\ge r$ give $\mathrm{\mid }f(t)\mathrm{\mid }\le \frac{2N}{C}\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }{e}^{-ar}(e^{v\mathrm{\mid }t\mathrm{\mid }}-1),\mathrm{\mid }f(t)\mathrm{\mid }\le 2N\mathrm{.}$ The first bound is $O(t)$ at zero, including $v=0$. The local observables commute by [F1], so all time-integral hypotheses in [F4] hold. The same estimates apply to the finite-range instance justified in step 1.2.

Apply [F4] using step 2.1. For any $q,\tau >0$, split its absolutely convergent integral at $\mathrm{\mid }t\mathrm{\mid }=\tau$. Step 2.2 gives the short-time bound $\frac{2N\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }}{\pi C}{e}^{-ar}{\int }_0^{\tau }\frac{e^{vt}-1}{t}dt\le \frac{2N\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }}{\pi C}{e}^{-ar+v\tau },$ by [F6], because $(e^{vt}-1)\mathrm{/}t\le v{e}^{vt}$ and $e^{v\tau }-1\le e^{v\tau }$, with zero left side when $v=0$. On the long-time part the trivial bound gives $\frac{2N}{\pi }{\int }_{\tau }^{\mathrm{\infty }}\frac{e^{-q{t}^2}}{t}dt=\frac{N}{\pi }{\int }_{q{\tau }^2}^{\mathrm{\infty }}\frac{e^{-u}}{u}du\le \frac{N}{\pi q{\tau }^2}{e}^{-q{\tau }^2}\mathrm{.}$ Adding the filter error $N{e}^{-g^2\mathrm{/}(4q)}$ bounds the connected correlation. These estimates use only the supplied dynamics and its infinite-volume ground representation.

Choose $\tau =ar\mathrm{/}(v+g\mathrm{/}2)$ and $q=g\mathrm{/}(2\tau )$. They are positive for $r\ge 1$ and $g,a>0$, even when $v=0$. Then $ar-v\tau =q{\tau }^2=\frac{g^2}{4q}=\frac{ag}{2v+g}r=\mu r,\frac{1}{q{\tau }^2}=\frac{2v+g}{agr}\mathrm{.}$ Substitution into step 3.1 proves the displayed bound. Zero observables give $N=0$ and require no division by $N$. The independent finite-volume band multiplicity and transport statements on this page are unchanged; this proof adds no claim that a degenerate gapped state has ordinary connected clustering. The clock conversion $g=\gamma \mathrm{/}\mathrm{\hslash }$ gives exactly the units in the Statement.