Physical-time KMS strip equilibrium condition

Here a complex C${}^{\ast }$-algebra is a complex Banach algebra with associative bilinear multiplication, submultiplicative norm, and a conjugate-linear involution satisfying $\mathrm{\parallel }{A}^{\ast }A\mathrm{\parallel }=\mathrm{\parallel }A{\mathrm{\parallel }}^2$; unital means that it has a nonzero identity $I$ (C star algebra). A $\ast$-automorphism is a bijective complex-linear map preserving multiplication and involution. Let $t\mapsto {\alpha }_t$ be a strongly continuous one-parameter group of such automorphisms, so ${\alpha }_0$ is the identity and ${\alpha }_s{\alpha }_t={\alpha }_{s+t}$. It is strongly continuous when $t\mapsto {\alpha }_t(A)$ is norm-continuous for each fixed $A\in \mathcal{A}$. A state is a bounded complex-linear functional $\omega :\mathcal{A}\to \mathbb{C}$ with $\omega (I)=1$ and $\omega (A^{\ast }A)\ge 0$ for every $A\in \mathcal{A}$; this is the state convention of Infinite algebraic state and finite-region restrictions.

Fix absolute temperature $T>0$ in kelvins and Boltzmann's constant $k_B>0$ in J K${}^{-1}$; the inverse-temperature parameter is $\beta :=(k_{B}T{)}^{-1}>0$, measured in J${}^{-1}$. Let the reduced Planck constant $\mathrm{\hslash }>0$ have units J s. Write $b:=\beta \mathrm{\hslash }$, measured in seconds, and

$S_b:=\{z\in \mathbb{C}:0\le \mathrm{Im}z\le b\},S_b^{\circ }:=\{z\in \mathbb{C}:0<\mathrm{Im}z<b\}\mathrm{.}$

A state $\omega$ is a $\beta$-KMS state for $\alpha$ in physical time when for every $A,B\in \mathcal{A}$ there is a bounded function $F_{A,B}:S_b\to \mathbb{C}$ (that is, ${\sup }_{z\in S_b}\mathrm{\mid }{F}_{A,B}(z)\mathrm{\mid }<\mathrm{\infty }$) that is continuous on $S_b$, complex-differentiable at every point of $S_b^{\circ }$, and satisfies

$F_{A,B}(t)=\omega \text{\hspace{0.17em}⁣}(A{\alpha }_t(B)),F_{A,B}(t+ib)=\omega \text{\hspace{0.17em}⁣}({\alpha }_t(B)A)(t\in \mathbb{R})\mathrm{.}$

For the assigned quantum spin model, $\mathcal{A}$ is the norm-completed quasi-local algebra of Norm-completed quasi-local quantum spin algebra and $\alpha$ is the thermodynamic dynamics of Strongly continuous thermodynamic spin automorphism group, under that theorem's stated hypotheses. The strip width is $\beta \mathrm{\hslash }$ in physical seconds; setting $\mathrm{\hslash }=1$ is a separate natural-unit convention. This is a mathematical equilibrium criterion for the specified algebra and dynamics. It does not assert that a model describes a material, that a system has dynamically relaxed, or that all stationary states are KMS.

This definition takes the C${}^{\ast }$-algebra and strongly continuous dynamics as its inputs. Its quantum-spin application is conditional on the exact algebra-completion and dynamics hypotheses in the linked suppliers; it does not prove those hypotheses or upgrade their draft proof status.