TFPT scientific source excerpt
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000241 | \begin{lemma}[Parent-Hamiltonian gap inherited from the primitive transfer block]
000242 | \label{lem:parent-gap-from-transfer-block}
000243 | Fix finite volume. Let $T_V$ be the primitive blocked transfer operator of
000244 | \Cref{thm:cp-even-vacuum-uniqueness} with unique strictly positive Perron vector $v_V^\ast$, and
000245 | let $H_{\mathrm{par}}$ be the blocked admissible parent Hamiltonian built from the local
000246 | admissibility projectors together with the local Perron vacuum projector onto $v_V^\ast$. Then
000247 | $H_{\mathrm{par}}$ has the same unique local vacuum sector and a strictly positive parent gap
000248 | \[
000249 | m_0:=\operatorname{gap}(H_{\mathrm{par}})>0.
000250 | \]
000251 | \end{lemma}
000252 | 
000253 | \begin{proof}
000254 | Every summand of $H_{\mathrm{par}}(V)$ is positive semidefinite by construction. By design, all
000255 | local admissibility penalties annihilate the admissible Perron sector, and the Perron vacuum
000256 | projector annihilates the distinguished Perron line generated by $v_V^*$. Hence
000257 | \[
000258 | H_{\mathrm{par}}(V)v_V^*=0,
000259 | \]
000260 | so $\mathbb C v_V^*$ is contained in the kernel.
000261 | 
000262 | Conversely, let $\psi\in\ker H_{\mathrm{par}}(V)$. Then
000263 | \[
000264 | 0=\langle \psi,H_{\mathrm{par}}(V)\psi\rangle
000265 | \]
000266 | and since $H_{\mathrm{par}}(V)$ is a finite sum of positive semidefinite terms, each summand must
000267 | have vanishing expectation on $\psi$. Therefore $\psi$ lies in the kernel of every local penalty
000268 | term and in the kernel of the Perron projector term. By the definition of the parent Hamiltonian,
000269 | the common zero set of these penalties is exactly the Perron vacuum line. Hence
000270 | \[
000271 | \ker H_{\mathrm{par}}(V)=\mathbb C v_V^*.
000272 | \]
000273 | 
000274 | Because the blocked finite-volume Hilbert space is finite dimensional, the spectrum of
000275 | $H_{\mathrm{par}}(V)$ is discrete and nonnegative. The zero eigenvalue is simple, so the smallest
000276 | nonzero eigenvalue exists and is strictly positive. This number is precisely
000277 | \[
000278 | m_0(V):=\min\bigl(\operatorname{spec}(H_{\mathrm{par}}(V))\setminus\{0\}\bigr)>0.
000279 | \]
000280 | \end{proof}
000281 | 
000282 | \begin{lemma}[Diagonal admissibility weights separate the retained basis]
000283 | \label{lem:diagonal-weights-separate-basis}
000284 | Let $\mathcal D_V$ be the commuting family of diagonal admissibility weights entering the blocked