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000001 | \section{Existence of equilibrium states}
000002 | 
000003 | \begin{theorem}[Ground and KMS states]
000004 | \label{thm:states}
000005 | The infinite-volume system \((\A,\tau)\), and its invariant subalgebra
000006 | \((\A^G,\tau|_{\A^G})\), have at least one ground state.  For every
000007 | \(\beta\in(0,\infty)\), they have at least one \(\beta\)-KMS state.
000008 | \end{theorem}
000009 | 
000010 | \begin{proof}
000011 | The state space \(S(\A)\) is weak-* compact.  Choose an exhaustion
000012 | \(\Lambda_n\uparrow\Gamma\), a normalized ground vector of \(H_{\Lambda_n}\),
000013 | and extend its vector state from \(\A_{\Lambda_n}\) to \(\A\).  Gauge-average
000014 | the extension if necessary.  A subnet converges weak-* to a state
000015 | \(\omega_\infty\).  If \(A\) is local, then for all sufficiently large
000016 | \(n\), the finite-volume ground-state variational inequality gives
000017 | \[
000018 |  -i\,\omega_n(A^*\delta_{\Lambda_n}(A))
000019 |  =\omega_n\!\left(A^*[H_{\Lambda_n},A]\right)\geq0.
000020 | \tag{5.1}\label{eq:groundineq}
000021 | \]
000022 | For local \(A\), the commutator stabilizes to
000023 | \(\delta(A)=i\sum_{X:X\cap\supp A\ne\varnothing}[\Phi(X),A]\).
000024 | Passing to the weak-* limit proves the ground-state condition
000025 | \(-i\omega_\infty(A^*\delta(A))\geq0\) on the local core, hence
000026 | \(\omega_\infty\) is a ground state.
000027 | 
000028 | For finite \(\beta\), take the finite Gibbs state
000029 | \[
000030 |  \omega_{n,\beta}(A)=
000031 |  \frac{\operatorname{Tr}_{\Lambda_n}
000032 |  (e^{-\beta H_{\Lambda_n}}A)}
000033 |  {\operatorname{Tr}_{\Lambda_n}(e^{-\beta H_{\Lambda_n}})}
000034 | \tag{5.2}\label{eq:gibbs}
000035 | \]
000036 | and any extension to \(\A\), again gauge-averaged if desired.  Weak-*
000037 | compactness supplies a convergent subnet \(\omega_\beta\).
000038 | To pass the KMS identity without assuming complex-time volume convergence,
000039 | smear a local \(A\) with a Gaussian \(f\):
000040 | \[
000041 |  A_f=\int_{\mathbb R}f(t)\tau_t(A)\,dt,\qquad
000042 |  A_{f,n}=\int_{\mathbb R}f(t)\tau_t^{\Lambda_n}(A)\,dt.
000043 | \tag{5.3}\label{eq:smear}
000044 | \]
000045 | Theorem~\ref{thm:dynamics}, isometry, and the Gaussian tail imply
000046 | \(A_{f,n}\to A_f\) in norm; the same holds after the imaginary translation
000047 | of the entire Gaussian smear.  The finite Gibbs KMS identity
000048 | \[
000049 |  \omega_{n,\beta}(A_{f,n}B)
000050 |  =\omega_{n,\beta}\!\left(B\tau_{i\beta}^{\Lambda_n}(A_{f,n})\right)
000051 | \tag{5.4}\label{eq:finiteKMS}
000052 | \]
000053 | therefore passes to the subnet limit.  Gaussian-smeared local elements are
000054 | a norm-dense entire analytic subalgebra.  Equation
000055 | \eqref{eq:finiteKMS} on this core is equivalent to the full
000056 | \(\beta\)-KMS condition, so \(\omega_\beta\) is a KMS state.
000057 | Restriction gives states on \(\A^G\); alternatively, gauge averaging
000058 | preserves both ground and KMS conditions because the gauge action commutes
000059 | with \(\tau\).
000060 | \end{proof}
000061 | 
000062 | \section{Numeric twin}