Infinite-Volume KMS States, Passivity, and Phase Multiplicity

In an infinite system the formal operator $e^{-\beta H}$ usually has infinite trace, so equilibrium cannot be defined by a global Gibbs density matrix. Instead a state is a positive normalized functional on a chosen algebra of local observables and is thermal when its time correlations satisfy the KMS analytic boundary condition. Different extremal KMS states at the same temperature encode distinct thermodynamic phases; their mixtures are equilibrium states but need not cluster.

The trace-free infinite-volume equilibrium framework and phase multiplicity are developed by Haag, Hugenholtz, and Winnink 1967, pp. 215–236.

Required background. Thermal Density Operators and the KMS Condition derives finite-volume KMS. Thermodynamic Limits, Phases, and Ensemble Equivalence fixes phase selection. Helpful background. Vacua, States, and Representations explains why inequivalent representations appear in infinite systems.

Local KMS without a global trace

Let $\mathfrak A$ be an algebra of quasilocal observables and $\alpha_t$ the physical time-evolution automorphism. A state $\omega$ is a positive normalized linear functional,

$$\omega(A^*A)\ge0, \qquad \omega(\mathbf1)=1.$$

It is a $\beta$-KMS state if, for a dense set of eligible $A,B\in\mathfrak A$, there is a function $F_{A,B}(z)$ analytic in $0<\operatorname{Im}z<\beta$ with boundary values

$$F_{A,B}(t)=\omega(A\alpha_t(B)), \qquad F_{A,B}(t+i\beta)=\omega(\alpha_t(B)A).$$

The placement of $A$ and $B$ differs from the previous page only by which operator is chosen as the time-translated one. The invariant content is the same cyclic imaginary-time boundary relation. The algebra, dynamics, strip, and eligible domain are part of the statement.

Finite-volume Gibbs states restricted to any fixed local region can converge along subsequences as $V\to\infty$. A limiting state may satisfy KMS even though no Hilbert-space trace represents it globally. Boundary conditions or an infinitesimal source can select different limits.

Passivity as an operational equilibrium test

Let $\delta(A)=\left.\mathrm d\alpha_t(A)/\mathrm dt\right|_{t=0}$ generate the dynamics. The work supplied to the system by a cyclic unitary operation $U$ is

$$W_{\mathrm{on}}=-i\,\omega\!\left(U^*\delta(U)\right).$$

For finite-system dynamics $\delta(A)=i[H,A]$, this is $W_{\mathrm{on}}=\omega(U^*HU-H)$. A state is passive when $W_{\mathrm{on}}\ge0$ for every allowed cycle, equivalently when the extracted work $W_{\mathrm{ext}}=-W_{\mathrm{on}}$ is never positive. It is completely passive if every finite tensor power is passive. Under the hypotheses of Pusz and Woronowicz, completely passive states are ground states or KMS states Pusz and Woronowicz 1978.

Passivity is stronger than stationarity and weaker than a casual assertion that every nonequilibrium steady state is thermal. The theorem’s $C^*$-dynamical-system setting, allowed operations, and tensor-product construction must be retained when exporting the result.

Extremal phases, mixtures, and clustering

The convex set of $\beta$-KMS states may contain several extremal points. An extremal KMS state cannot be written as a nontrivial convex mixture of other KMS states at the same dynamics and temperature. In short-range systems, extremality is closely related to decay of connected correlations for suitable translated local observables,

$$\omega(A\,\tau_{\mathbf x}(B)) -\omega(A)\omega(B)\longrightarrow0 \qquad (\lvert\mathbf x\rvert\to\infty),$$

with hypotheses that depend on the algebra and translation action.

At coexistence, let $\omega_+$ and $\omega_-$ be symmetry-related extremal KMS states. Their mixture

$$\omega_{\mathrm{mix}}=p\,\omega_++(1-p)\omega_-$$

is also KMS because the condition is linear in the state. If an order parameter has values $\pm m$, however, the mixture has a nondecaying connected contribution $4p(1-p)m^2$. Thermal equilibrium therefore does not imply extremality or clustering.

The GNS representations of distinct phases can become disjoint: no finite-energy local operation connects them in the infinite system. This is the representation-theoretic version of the superselection created by the thermodynamic limit.

A controlled phase-selection statement

For finite regions $\Lambda$ with boundary condition $b$, write $\omega_{\Lambda,b}$. A responsible infinite-volume claim has the form

$$\omega_b(A)=\lim_{\Lambda\nearrow\mathbb R^{d-1}} \omega_{\Lambda,b}(A)$$

for every local $A$ in a specified convergence class. It then verifies KMS under the limiting dynamics, identifies whether $\omega_b$ is extremal, and tests clustering or another phase criterion. Writing $\rho=e^{-\beta H}/Z$ after $Z$ has diverged skips every one of these steps.

Scope and theorem boundary

This page supplies the physics-facing definitions and consequences. Proofs of existence, uniqueness, decomposition, and equivalence between clustering and extremality depend on interaction range, locality, asymptotic abelianness, and topology; theorem-first treatment belongs to Mathematical QFT. The strongest safe conclusion here is conditional: a declared limiting state satisfying the KMS boundary condition is an equilibrium state for the declared dynamics, and multiple extremal such states represent phase multiplicity.

The schematic below organizes the relationships used on this page. Inspect it with this question in mind: How do Gibbs states, infinite-volume KMS states, thermal boundary conditions, Matsubara modes, and graded traces fit together?

Finite Gibbs traces imply KMS analyticity; the KMS condition survives without a trace in infinite volume, while thermal-circle periodicity and grading follow only with the operator statistics and insertion specified. Solid connections show the primary relation; dashed outlines or arrows mark qualifications and failure boundaries. The diagram is schematic and not to scale.

The surrounding discussion supplies the relevant equations and checks in text form; the figure is a navigational summary.

Exercise

Let $\omega_\pm(M)=\pm m$ and suppose each phase clusters. Show that the equal mixture does not cluster for $M$.

Solution

For widely separated $M_x,M_0$, each phase gives $\omega_\pm(M_xM_0)\to m^2$. The equal mixture has $\omega(M)=0$ but $\omega(M_xM_0)\to m^2$, so the connected correlator tends to $m^2$ rather than zero. Linearity preserves KMS while extremality and clustering are lost.

References

• Haag, Rudolf, Nico M. Hugenholtz, and Marius Winnink. “On the Equilibrium States in Quantum Statistical Mechanics.” Communications in Mathematical Physics 5 (1967): 215–236. doi:10.1007/BF01646342.
• Pusz, W., and S. L. Woronowicz. “Passive States and KMS States for General Quantum Systems.” Communications in Mathematical Physics 58 (1978): 273–290. doi:10.1007/BF01614224.
• Ruelle, David. Statistical Mechanics: Rigorous Results. Singapore: World Scientific, 1999. doi:10.1142/4090.