The KMS Condition on a C*-Dynamical System

The KMS condition characterizes equilibrium through analyticity and reversed operator order at the boundaries of a complex-time strip. The canonical KMS treatment gives its Gibbs derivation and physical interpretation. This page supplies the C*-dynamical hypotheses, constructs analytic test elements, and explains invariance and compactness of the equilibrium state set without assuming a global partition function.

Required background. The Algebraic State Space supplies weak-star convergence. Derivations on Operator Algebras distinguishes algebra dynamics from formal unbounded commutators.

The strip condition includes a continuity hypothesis

Let $\mathcal A$ be a unital C*-algebra and $\alpha_t$ a one-parameter group of star automorphisms that is point-norm continuous:

$$\|\alpha_t(a)-a\|\longrightarrow0 \qquad(t\to0,\ a\in\mathcal A).$$

Fix $\beta>0$, with units of inverse energy, and use physical time $t$. The strip height is $\tau=\beta\hbar$. A state $\omega$ is $\beta$-KMS if for every $a,b\in\mathcal A$ there exists a bounded continuous function $F_{a,b}$ on $0\leq\operatorname{Im}z\leq\tau$, analytic in its interior, with

$$\begin{aligned} F_{a,b}(t)&=\omega(a\alpha_t(b)),\\ F_{a,b}(t+i\tau)&=\omega(\alpha_t(b)a). \end{aligned}$$

For general $b$, the definition concerns the scalar function $F_{a,b}$. It does not assert that the operator $\alpha_{i\tau}(b)$ exists. Analytic elements provide a dense class on which that operator notation is legitimate.

For an unbounded Hamiltonian, conjugation on all of $\mathcal B(\mathcal H)$ need not be point-norm continuous. Applying the present C*-version then requires an appropriate invariant algebra on which this continuity holds. A von Neumann-algebra formulation uses its own weak continuity hypotheses and should be stated separately.

Gaussian smearing constructs entire analytic elements

For $\epsilon>0$, with units of time squared, define the norm-convergent Bochner integral

$$b_\epsilon= \frac1{\sqrt{\pi\epsilon}} \int_{\mathbb R}e^{-t^2/\epsilon}\alpha_t(b)\,dt.$$

Automorphisms are isometric, so the integral is bounded by $\|b\|$. Point-norm continuity and the Gaussian approximate identity give $b_\epsilon\to b$ in norm.

For real $s$, translating the integration variable gives a formula that extends to complex $z$:

$$\alpha_z(b_\epsilon)= \frac1{\sqrt{\pi\epsilon}} \int_{\mathbb R}e^{-(t-z)^2/\epsilon}\alpha_t(b)\,dt.$$

The integral is entire in $z$ in norm, justified by locally uniform Gaussian bounds. In particular,

$$\|\alpha_z(b_\epsilon)\| \leq e^{(\operatorname{Im}z)^2/\epsilon}\|b\|.$$

Thus entire analytic elements are norm dense. This construction avoids treating a formal exponential of an unbounded derivation as an everywhere-defined operator on the algebra.

Analytic test elements reduce the condition to an identity

For entire analytic $b$, the KMS condition is equivalent to

$$\omega(a\alpha_{i\tau}(b))=\omega(ba) \qquad(a\in\mathcal A).$$

To see the sufficient direction, use $F_{a,b}(z)=\omega(a\alpha_z(b))$. Applying the identity to $\alpha_t(b)$ gives the required upper boundary. Along real translates, the norm of $\alpha_{t+is}(b)$ equals that of $\alpha_{is}(b)$, so this function is bounded on the strip.

The three-lines bound then gives $|F_{a,b}(z)|\leq\|a\|\|b\|$. Approximate an arbitrary $b$ by entire analytic elements. The same bound for their differences makes the corresponding scalar functions uniformly Cauchy on the closed strip, yielding the required analytic limit. The converse follows from uniqueness of the bounded strip continuation. Bratteli and Robinson give the full equivalent formulations.

This criterion is useful because it tests fixed algebra elements and their defined analytic translates, rather than requiring a separate construction of a function for every pair.

Equilibrium implies invariance, but not conversely

Set $a=1$ in the strip condition. The upper and lower boundary values coincide. Copies of the bounded analytic strip function therefore glue across their boundaries to a bounded entire function. Analyticity across the seams follows from the matching continuous boundary values, for example by Morera’s theorem.

Liouville’s theorem makes the function constant, giving

$$\omega(\alpha_t(b))=\omega(b).$$

Thus a KMS state is stationary. Stationarity alone contains no reversed-order condition and does not identify a temperature.

At fixed $\beta$ and fixed dynamics, the KMS states form a convex weak-star closed subset of the compact state space. The analytic-element identity makes closedness transparent: each identity is an equality of continuous evaluation functionals of $\omega$. Therefore the set is compact, but it may be empty or contain many states. Neither existence nor uniqueness follows from compactness. Multiple equilibrium states and phase questions require analysis of the chosen model.

Exercise: stationary states need not be KMS

Let $\alpha_t=\mathrm{id}$ on $M_2(\mathbb C)$. Which density-operator states satisfy the positive-temperature KMS condition?

Solution

Every element is analytic and $\alpha_{i\tau}(b)=b$, so the condition is $\omega(ab)=\omega(ba)$ for every pair. Taking matrix units forces equal diagonal probabilities and zero off-diagonal entries of the density matrix. Thus only $\rho=I/2$ is KMS, although every state is stationary under this trivial dynamics. On a commutative algebra, by contrast, every state satisfies this condition for trivial dynamics.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States. Models in Quantum Statistical Mechanics, 2nd ed., Springer, 1997, doi:10.1007/978-3-662-03444-6, Section 5.3 — analytic elements and equivalent KMS conditions.
• Naaijkens, Pieter, Quantum Spin Systems on Infinite Lattices, Lecture Notes in Physics 933, Springer, 2017, arXiv:1311.2717 — equilibrium states for infinite-system C*-dynamics.