Full-algebra quantum Ising KMS–DLR correspondence

For each finite cylinder assignment $\sigma$ on $\mathrm{\Lambda }$, put ${\mu }_{\omega }(\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }})=\omega (P_{\sigma })$, where $P_{\sigma }$ is the product-basis rank-one projection. Positivity of $\omega$ gives nonnegative weights, ${\sum }_{\sigma }{P}_{\sigma }=I_{\mathrm{\Lambda }}$ gives normalization, and for $\mathrm{\Lambda }\subset {\mathrm{\Lambda }}^{\mathrm{\prime }}$ the identity $P_{\sigma }={\sum }_{\tau {\mathrm{\mid }}_{\mathrm{\Lambda }}=\sigma }{P}_{\tau }$ gives consistency. By [A1] and [F9], these cylinder weights extend uniquely to a Borel probability measure ${\mu }_{\omega }$.

Fix a finite $\mathrm{\Delta }$, configurations $\sigma ,\tau$ on $\mathrm{\Delta }$, an assignment $\eta$ on $C_{\mathrm{\Delta }}$, and a cylinder projection $Q_{\zeta }$ on finitely many sites outside $\mathrm{\Delta }\cup C_{\mathrm{\Delta }}$. Let $Q_{\eta }$ be the product projection for $\eta$ and let $E_{\sigma ,\tau }$ be the local matrix unit sending $e_{\tau }$ to $e_{\sigma }$. Since every interaction term that meets $\mathrm{\Delta }$ has all its exterior sites in $C_{\mathrm{\Delta }}$, changing the $\mathrm{\Delta }$ configuration from $\tau$ to $\sigma$ while keeping $\eta$ fixed changes the finite diagonal Hamiltonian by exactly $\mathrm{\Delta }E:=E_{\mathrm{\Delta }}(\sigma \mid \eta )-E_{\mathrm{\Delta }}(\tau \mid \eta )$. Terms not meeting $\mathrm{\Delta }$ commute with $E_{\sigma ,\tau }{Q}_{\eta }$, so [F3] yields ${\alpha }_t(E_{\sigma ,\tau }{Q}_{\eta })=e^{it\mathrm{\Delta }E\mathrm{/}\mathrm{\hslash }}{E}_{\sigma ,\tau }{Q}_{\eta }$.

For a complex-valued cylinder function, define its integral as the real integral of its real part plus $i$ times the real integral of its imaginary part. For a DLR measure $\mu$, define ${\omega }_{\mu }(A)=\int ⟨A{e}_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}},e_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}⟩d\mu (\sigma )$ for $A\in {\mathcal{A}}_{\mathrm{\Lambda }}$, using the inner product linear in its first argument from [F13]. The value is independent of enlarging $\mathrm{\Lambda }$ because the added factors are identities. It is linear, equals $1$ on the identity, and ${\omega }_{\mu }(A^{\ast }A)=\int \mathrm{\parallel }A{e}_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}{\mathrm{\parallel }}^{2}d\mu (\sigma )\ge 0$; Cauchy–Schwarz gives $\mathrm{\mid }{\omega }_{\mu }(A)\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }$. Hence it defines a bounded positive normalized functional on ${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}$. If local $A_n\to A$ in the completion, continuity of product and adjoint from [F12] gives $A_n^{\ast }{A}_n\to A^{\ast }A$, while each ${\omega }_{\mu }(A_n^{\ast }{A}_n)\ge 0$; the continuous extension is therefore positive and normalized and is a state.

Apply [F5] to $A=E_{\tau ,\sigma }{Q}_{\eta }{Q}_{\zeta }$ and $B=E_{\sigma ,\tau }{Q}_{\eta }$. The lower edge is $e^{it\mathrm{\Delta }E\mathrm{/}\mathrm{\hslash }}\omega (P_{\tau }{Q}_{\eta }{Q}_{\zeta })$. The function $g(z)=e^{iz\mathrm{\Delta }E\mathrm{/}\mathrm{\hslash }}\omega (P_{\tau }{Q}_{\eta }{Q}_{\zeta })$ is entire and bounded on the closed strip because $\mathrm{\Delta }E$ is real and $0\le \mathrm{Im}z\le \beta \mathrm{\hslash }$; it has the same lower edge. By [F6], $g$ is the KMS function, and its upper edge equals the reversed product boundary from [F5]. At $t=0$ this gives ${\mu }_{\omega }(P_{\sigma }{Q}_{\eta }{Q}_{\zeta })=e^{-\beta \mathrm{\Delta }E}{\mu }_{\omega }(P_{\tau }{Q}_{\eta }{Q}_{\zeta })\mathrm{.}$

Let $A,B$ be local and choose finite $\mathrm{\Delta }$ containing their supports and every interaction support that meets the support of $B$. For each boundary assignment $\eta$ on $C_{\mathrm{\Delta }}$, let $H_{\mathrm{\Delta }}^{\eta }$ be the diagonal matrix with eigenvalues $E_{\mathrm{\Delta }}(\sigma \mid \eta )$ and let ${\rho }_{\mathrm{\Delta }}^{\eta }=e^{-\beta H_{\mathrm{\Delta }}^{\eta }}\mathrm{/}{Z}_{\mathrm{\Delta }}^{\eta }$. For bounded real measurable $f$, the DLR fixed-point equation [F14] gives $\mu (f)=\int {\gamma }_{\mathrm{\Delta }}(f\mid \xi )d\mu (\xi )$; extend this identity to complex $f$ by real and imaginary parts. If $D$ is exterior-measurable, properness gives ${\gamma }_{\mathrm{\Delta }}({\mathbf{1}}_{D}f\mid \xi )={\mathbf{1}}_D(\xi ){\gamma }_{\mathrm{\Delta }}(f\mid \xi )$. For a bounded cylinder function $f$ depending only on spins in $\mathrm{\Delta }\cup C_{\mathrm{\Delta }}$, the kernel depends on the remaining exterior only through $C_{\mathrm{\Delta }}$, so for every finite-boundary assignment $\eta$, ${\int }_{Q_{\eta }}fd\mu =\mu (Q_{\eta }){\sum }_{{\sigma }_{\mathrm{\Delta }}}{\gamma }_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )f({\sigma }_{\mathrm{\Delta }}\eta )\mathrm{.}$ Apply this identity to the diagonal cylinder function $f=\mathrm{diag}(A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }})$. For each fixed $\eta$, this diagonal cylinder function depends only on the spins in $\mathrm{\Delta }$ and the specified boundary assignment on $C_{\mathrm{\Delta }}$: $A,B$ are supported in $\mathrm{\Delta }$, and $H_{\mathrm{\Delta }}^{\eta }$ is a matrix on that finite factor with coefficients determined by $\eta$. Thus it meets the support condition of the preceding finite-mixture identity. The finite sum of diagonal entries is the trace against ${\rho }_{\mathrm{\Delta }}^{\eta }$, so ${\int }_{Q_{\eta }}\mathrm{diag}\text{\hspace{0.17em}⁣}\left(A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}\right)d\mu =\mu (Q_{\eta }){\mathrm{Tr}}_{\mathrm{\Delta }}\text{\hspace{0.17em}⁣}\left({\rho }_{\mathrm{\Delta }}^{\eta }A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}\right)\mathrm{.}$ Here the trace identity follows by summing ${\gamma }_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )=e^{-\beta E_{\mathrm{\Delta }}({\sigma }_{\mathrm{\Delta }}\mid \eta )}\mathrm{/}{Z}_{\mathrm{\Delta }}^{\eta }$ against the diagonal matrix coefficients. By [F3], on $Q_{\eta }$ the global local dynamics of $B$ equals conjugation by $H_{\mathrm{\Delta }}^{\eta }$: terms in the global generator not meeting $\mathrm{\Delta }$ commute with $B{Q}_{\eta }$, and the terms in $H_{\mathrm{\Delta }}^{\eta }$ disjoint from the support of $B$ commute with $B$. Summing the displayed identities over the finitely many boundary assignments gives, for real $t$, ${\omega }_{\mu }(A{\alpha }_t(B))={\sum }_{\eta \in \{-1,+1{\}}^{C_{\mathrm{\Delta }}}}\mu (Q_{\eta }){\mathrm{Tr}}_{\mathrm{\Delta }}\text{\hspace{0.17em}⁣}\left({\rho }_{\mathrm{\Delta }}^{\eta }A{e}^{it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-it{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}\right)\mathrm{.}$

For fixed $\mathrm{\Delta },\eta ,\zeta$, step 2.1 implies $e^{\beta E_{\mathrm{\Delta }}(\sigma \mid \eta )}{\mu }_{\omega }(P_{\sigma }{Q}_{\eta }{Q}_{\zeta })$ is constant as $\sigma$ varies. Summing over $\sigma$ and using ${\sum }_{\sigma }{P}_{\sigma }=I_{\mathrm{\Delta }}$ shows that the constant is ${\mu }_{\omega }(Q_{\eta }{Q}_{\zeta })\mathrm{/}{Z}_{\mathrm{\Delta }}^{\eta }$, including when this numerator is zero. Therefore ${\mu }_{\omega }(P_{\sigma }{Q}_{\eta }{Q}_{\zeta })={\gamma }_{\mathrm{\Delta }}(\sigma \mid \eta ){\mu }_{\omega }(Q_{\eta }{Q}_{\zeta })\mathrm{.}$

For each $\eta$, define $F_{A,B}^{\eta }(z)={\mathrm{Tr}}_{\mathrm{\Delta }}({\rho }_{\mathrm{\Delta }}^{\eta }A{e}^{iz{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }}B{e}^{-iz{H}_{\mathrm{\Delta }}^{\eta }\mathrm{/}\mathrm{\hslash }})$. By [F7] under [A2]–[A3], this is the finite Gibbs KMS strip function for $H_{\mathrm{\Delta }}^{\eta }$ and the pair $A,B$; it is bounded and continuous on the closed strip, holomorphic inside, and has lower and upper boundaries given by the finite Gibbs traces in their indicated orders. The finite sum ${\sum }_{\eta }\mu (Q_{\eta })F_{A,B}^{\eta }(z)$ has the same regularity and is bounded because the weights are nonnegative and sum to one. Step 2.2 identifies its boundaries with ${\omega }_{\mu }(A{\alpha }_t(B))$ and ${\omega }_{\mu }({\alpha }_t(B)A)$, proving the KMS condition for local pairs.

Every exterior cylinder event is a finite disjoint union of events $Q_{\eta }{Q}_{\zeta }$ after refining it to specify all spins in $C_{\mathrm{\Delta }}$. Summing step 3.1 gives the conditional identity for each interior assignment and every exterior cylinder. Every full cylinder is a finite disjoint union of events $P_{\sigma }{D}_{\sigma }$ with $D_{\sigma }$ an exterior cylinder; properness in [F2] then gives $({\mu }_{\omega }{\gamma }_{\mathrm{\Delta }})(P_{\sigma }{D}_{\sigma })={\mu }_{\omega }(P_{\sigma }{D}_{\sigma })$, so the two probability measures agree on all full cylinders. The class of Borel sets on which they agree contains the whole space, is closed under relative complements of nested members, and under disjoint countable unions, so it is a lambda-system. Full cylinders together with $\mathrm{\varnothing }$ and $X$ form a pi-system generating the Borel sigma-algebra. [F10] states that every lambda-system containing a pi-system contains its generated sigma-algebra, hence the two measures agree on that sigma-algebra. Thus ${\mu }_{\omega }{\gamma }_{\mathrm{\Delta }}={\mu }_{\omega }$ and ${\mu }_{\omega }\in {\mathcal{D}}_{\beta }$.

For arbitrary $A,B\in \mathcal{A}$, choose local $A_n\to A$ and $B_n\to B$ using [F12], and let $F_n$ be their local-pair KMS functions from step 3.2. On each strip boundary, the difference $F_n-F_m$ is bounded by the state norm and the isometric automorphism property [F15] by quantities tending to zero, for example on the lower edge by $\mathrm{\parallel }{A}_n-A_m\mathrm{\parallel }\mathrm{\parallel }{B}_n\mathrm{\parallel }+\mathrm{\parallel }{A}_m\mathrm{\parallel }\mathrm{\parallel }{B}_n-B_m\mathrm{\parallel }$; the analogous upper-edge bound has the factors reversed. The three-lines bound [F6] makes $F_n$ uniformly Cauchy on the entire closed strip. Its uniform limit is bounded and continuous there, has the required two boundary values by norm continuity, and is holomorphic inside by [F11]. Thus ${\omega }_{\mu }\in {\mathcal{K}}_{\beta }$.

For a KMS state $\omega$, [F4] and [A2] make every local density matrix diagonal, so for local $A$ its value is ${\sum }_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}\omega (P_{\sigma })⟨A{e}_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}},e_{\sigma {\mathrm{\mid }}_{\mathrm{\Lambda }}}⟩={\omega }_{{\mu }_{\omega }}(A)$; density gives equality on $\mathcal{A}$. Conversely, ${\omega }_{\mu }(P_{\sigma })=\mu (P_{\sigma })$ for every cylinder, so uniqueness in [F9] makes $\mu \mapsto {\omega }_{\mu }$ injective. Steps 4.1 and 4.2 prove that the two maps take values in ${\mathcal{D}}_{\beta }$ and ${\mathcal{K}}_{\beta }$, and these identities show they are inverse affine maps.