TFPT scientific source excerpt
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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
000285 | transfer operator on the retained basis of $\mathcal K_{+,\mathrm{adm}}^{(V)}$. If
000286 | \[
000287 | u\neq v
000288 | \]
000289 | are two retained basis words, then there exists $D\in\mathcal D_V$ such that
000290 | \[
000291 | D_{uu}\neq D_{vv}.
000292 | \]
000293 | \end{lemma}
000294 | 
000295 | \begin{proof}
000296 | Two distinct retained basis words differ in at least one blocked admissibility datum: local
000297 | transport count, carrier label, or retained selector weight. The blocked diagonal admissibility
000298 | weights were constructed precisely from these discrete data, so some diagonal weight separates the
000299 | two words. Hence the family $\mathcal D_V$ separates the retained basis.
000300 | \end{proof}
000301 | 
000302 | \begin{theorem}[Injective blocked transfer and uniform parent gap]
000303 | \label{thm:uniform-parent-gap-injective}
000304 | Let $P_V$ be the blocked positive transport operator on the retained auxiliary space, and let
000305 | $\mathcal D_V$ be the family of diagonal admissibility weights. Under
000306 | \TFPTxref{prop:strict-positive-c6-green,lem:diagonal-weights-separate-basis}, after finite blocking the
000307 | algebra generated by $P_V$ and $\mathcal D_V$ is the full matrix algebra on the retained auxiliary
000308 | space. Equivalently, the blocked admissible transfer tensor is injective. Therefore the
000309 | corresponding blocked parent Hamiltonians satisfy a system-size-independent gap bound
000310 | \[
000311 | \inf_n \operatorname{gap}(H_{\mathrm{par}}(V_n))\ge m_0>0.
000312 | \]
000313 | Consequently no separate uniform parent-gap assumption remains in
000314 | \Cref{thm:gap-stability-thermo-limit}.
000315 | \end{theorem}
000316 | 
000317 | \begin{proof}
000318 | By \TFPTxref{prop:strict-positive-c6-green}, every local admissible $C_6$ kernel has strictly positive
000319 | matrix entries. After finite blocking this yields a blocked operator $P_V$ whose matrix entries are
000320 | strictly positive on the retained basis. By
000321 | \Cref{lem:diagonal-weights-separate-basis}, the diagonal family $\mathcal D_V$ separates the
000322 | retained basis words. Hence every subspace invariant under the algebra
000323 | \[
000324 | \mathrm{Alg}(P_V,\mathcal D_V)
000325 | \]
000326 | must first be invariant under all separating diagonal matrices, and is therefore a coordinate
000327 | subspace spanned by a subset of basis vectors. But a strictly positive matrix sends every basis
000328 | vector to a vector with nonzero component in every retained basis direction, so no proper
000329 | coordinate subspace can remain invariant under $P_V$. Thus the generated algebra acts irreducibly.
000330 | 
000331 | Burnside's theorem now gives
000332 | \[
000333 | \mathrm{Alg}(P_V,\mathcal D_V)=M_n(\mathbb C)
000334 | \]
000335 | on the finite blocked auxiliary space. This is exactly the injectivity criterion after finite
000336 | blocking for the corresponding MPS / finitely-correlated transfer tensor. The standard
000337 | Fannes--Nachtergaele--Werner parent-Hamiltonian theorem therefore yields a uniform spectral gap
000338 | independent of system size. Hence
000339 | \[
000340 | \inf_n \operatorname{gap}(H_{\mathrm{par}}(V_n))>0.
000341 | \]
000342 | Then \Cref{thm:eta-zero-parent-domination} gives
000343 | \[
000344 | \operatorname{gap}(H_{\mathrm{adm}}(V_n))
000345 | \ge
000346 | \operatorname{gap}(H_{\mathrm{par}}(V_n))
000347 | \ge
000348 | m_0>0,
000349 | \]
000350 | and the thermodynamic-limit argument of \Cref{thm:thermodynamic-limit} yields a unique gapped vacuum
000351 | with exponential clustering.
000352 | \end{proof}
000353 |