Modular KMS condition - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra, let $\varphi$ be a faithful normal state, and let $\sigma^\varphi$ be its modular automorphism group. The modular KMS condition is the boundary identity

\varphi(xy)=\varphi\!\left(y\,\sigma_{-i}^\varphi(x)\right)

for elements $x,y$ that are entire analytic for $\sigma^\varphi$ . Equivalently, the real-time correlation functions admit bounded analytic continuations to a strip whose two boundary values are related by cyclically interchanging $x$ and $y$ . With the displayed sign convention, this is the KMS condition at inverse temperature $1$ .

Characterization of modular dynamics

The state is invariant under its modular group: $\varphi\circ\sigma_t^\varphi=\varphi$ . More strongly, among suitably continuous one-parameter automorphism groups, the KMS boundary relation determines the modular group of a faithful normal state. This is why modular flow can be described intrinsically, without choosing a particular GNS realization Takesaki, vol. II, Chapter VIII, §1.

For a normal semifinite faithful weight, the analogous identity is imposed on the analytic elements lying in the weight's finite left and right domains. That domain qualification is essential because a weight may take the value $+\infty$ .

Tracial and finite-dimensional cases

If $\varphi$ is tracial, its modular group is trivial, and the boundary identity reduces to $\varphi(xy)=\varphi(yx)$ . On $M_n(\mathbb C)$ , let $\varphi(x)=\operatorname{Tr}(\rho x)$ for an invertible density matrix $\rho$ . Then

\sigma_t^\varphi(x)=\rho^{it}x\rho^{-it},

and the modular KMS identity follows by ordinary trace cyclicity. This example also shows that KMS cyclicity is twisted by modular evolution when $\rho$ is not scalar.

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter VIII, §1 on modular automorphism groups and their KMS characterization.
Ola Bratteli and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States, Models in Quantum Statistical Mechanics, 2nd ed., Springer, 1997. DOI record. Relevant: §5.3 on KMS states and analytic boundary conditions.