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000001 | 
000002 | \begin{theorem}[Parent-Hamiltonian domination with \texorpdfstring{$\eta=0$}{eta=0}]
000003 | \label{lem:norm-stable-gap-parent-path}
000004 | \label{thm:eta-zero-parent-domination}
000005 | Fix finite volume $V$ and let $T_V$ be the primitive blocked admissible transfer operator with
000006 | Perron vector $v_V^\ast$. Let $H_{\mathrm{par}}(V)$ be the blocked admissible parent Hamiltonian of
000007 | \Cref{lem:parent-gap-from-transfer-block}, with
000008 | \[
000009 | m_0(V):=\operatorname{gap}(H_{\mathrm{par}}(V))>0.
000010 | \]
000011 | Let the physical admissible Hamiltonian be written as
000012 | \[
000013 | H_{\mathrm{adm}}(V)
000014 | =
000015 | H_{\mathrm{par}}(V)+K_\Sigma(V)+K_c(V)+K_\Theta(V)+K_g(V)+H_{\mathrm{tr,pos}}(V),
000016 | \]
000017 | where each added term is positive semidefinite and vanishes on the Perron vacuum line generated by
000018 | $v_V^\ast$. Then
000019 | \[
000020 | H_{\mathrm{adm}}(V)\ge H_{\mathrm{par}}(V).
000021 | \]
000022 | Consequently the admissible Hamiltonian has the same vacuum line and satisfies
000023 | \[
000024 | \operatorname{gap}(H_{\mathrm{adm}}(V))
000025 | \ge
000026 | \operatorname{gap}(H_{\mathrm{par}}(V))
000027 | =
000028 | m_0(V).
000029 | \]
000030 | \end{theorem}
000031 | 
000032 | \begin{proof}
000033 | Every summand in
000034 | \[
000035 | H_{\mathrm{adm}}(V)-H_{\mathrm{par}}(V)
000036 | \]
000037 | is positive semidefinite by construction and annihilates the Perron vacuum line, hence the operator
000038 | inequality is immediate and the ground state is unchanged. The gap bound then follows from the
000039 | min-max principle on the orthogonal complement of that line.
000040 | \end{proof}
000041 | 
000042 | \begin{theorem}[Block 4: Stability of the admissible mass gap under the thermodynamic limit]
000043 | \label{thm:gap-stability-thermo-limit}
000044 | Let $(V_n)_{n\ge 1}$ be an admissible exhaustion, let $H_{\mathrm{par}}(V_n)$ be the blocked
000045 | parent Hamiltonians of \Cref{lem:parent-gap-from-transfer-block}, and let the uniform parent-gap
000046 | bound of \Cref{thm:uniform-parent-gap-injective} be
000047 | \[
000048 | \inf_n \operatorname{gap}(H_{\mathrm{par}}(V_n))=:m_0>0.
000049 | \]
000050 | Assume moreover that each finite-volume admissible Hamiltonian $H_{\mathrm{adm}}(V_n)$ satisfies
000051 | the parent-domination hypothesis of \Cref{thm:eta-zero-parent-domination}. Then the unique vacuum
000052 | persists, the spectral gap of the admissible Hamiltonian remains strictly positive in the
000053 | thermodynamic limit, and admissible
000054 | correlators cluster exponentially.
000055 | \end{theorem}
000056 | 
000057 | \begin{proof}
000058 | By \Cref{thm:eta-zero-parent-domination}, every finite-volume admissible Hamiltonian obeys
000059 | \[
000060 | \operatorname{gap}(H_{\mathrm{adm}}(V_n))
000061 | \ge
000062 | \operatorname{gap}(H_{\mathrm{par}}(V_n))
000063 | \ge
000064 | m_0>0.
000065 | \]
000066 | Passing to the thermodynamic limit of \Cref{thm:thermodynamic-limit} preserves the unique vacuum,