Thermal Density Operators and the KMS Condition

The Kubo–Martin–Schwinger condition is equilibrium stated directly in terms of time-translated observables. In a finite system with a trace-class Gibbs operator it follows from trace cyclicity; in an infinite system it remains meaningful when no global density matrix exists. Stationarity alone is weaker: any density operator commuting with the Hamiltonian is stationary, but only a thermal choice has the KMS relation at one inverse temperature for the full observable algebra.

The operator-ordering form of thermal equilibrium used below goes back to Martin and Schwinger 1959, §§ II–III, pp. 1346–1354.

Required background. Hilbert Positivity and Unitary Evolution fixes the state and adjoint structure. Retarded, Advanced, and Spectral Correlators fixes causal correlators. Helpful background. Thermal OPE and KMS Crossing develops the conformal specialization.

Gibbs cyclicity gives KMS

Let

$$\rho_\beta=\frac{e^{-\beta H}}{Z}, \qquad A(t)=e^{iHt}Ae^{-iHt}, \qquad Z=\operatorname{Tr}e^{-\beta H}.$$

For bounded operators, or for unbounded operators on a common domain where the products and traces exist, define

$$F_{AB}(z)=\langle A(z)B\rangle_\beta.$$

The Gibbs factors move an operator through the trace according to

$$e^{-\beta H}A(t)e^{\beta H}=A(t+i\beta).$$

Trace cyclicity therefore gives the KMS boundary relation

$$\boxed{ \langle A(t)B\rangle_\beta =\langle B A(t+i\beta)\rangle_\beta }.$$

The full statement includes analyticity of $F_{AB}(z)$ in a strip, conventionally $0<\operatorname{Im}z<\beta$, and continuous boundary values. The boundary equality without its domain and growth conditions is not the complete KMS property.

For a grand-canonical state $\rho\propto e^{-\beta K}$ with $K=H-\mu_aQ_a$, one must say whether time evolution is generated by $K$ or by the physical $H$. With $K$-evolution the formula above is unchanged. With $H$-evolution a charged operator acquires a chemical-potential twist; Chapter 3 develops that translation.

Bosonic and fermionic thermal relations

Trace cyclicity itself does not insert a minus sign. The familiar fermionic sign enters when odd operators are assembled into an imaginary-time-ordered Green function. For a field of fermion parity $\lvert A\rvert\in\{0,1\}$,

$$G_A(\tau)=-\langle\mathrm T_\tau A(\tau)A^\dagger(0)\rangle$$

is periodic for even fields and antiperiodic for odd fields:

$$G_A(\tau+\beta)=(-1)^{\lvert A\rvert}G_A(\tau)$$

up to the overall Green-function sign convention displayed above. Thermal Boundary Conditions and Graded Traces separates this thermal antiperiodicity from a supertrace insertion.

KMS also produces detailed balance for spectral weights. For Wightman functions

$$G^>(t)=\langle A(t)B(0)\rangle, \qquad G^<(t)=\langle B(0)A(t)\rangle,$$

the global Fourier convention gives

$$G^>(\omega)=e^{\beta\omega}G^<(\omega)$$

for neutral bosonic operators in the displayed convention. This is a frequency-domain equilibrium test, not a generic property of stationary correlators.

Harmonic-oscillator test

For $H=\omega(a^\dagger a+\frac12)$,

$$a(t)=e^{-i\omega t}a, \qquad n_B(\omega)=\frac1{e^{\beta\omega}-1}.$$

Then

$$\begin{aligned} \langle a(t)a^\dagger\rangle_\beta &=(1+n_B)e^{-i\omega t},\\ \langle a^\dagger a(t+i\beta)\rangle_\beta &=n_Be^{-i\omega t}e^{\beta\omega} =(1+n_B)e^{-i\omega t}. \end{aligned}$$

This checks the direction of the imaginary shift. It also shows why a stationary diagonal state need not be thermal. If $\rho=\sum_np_n|n\rangle\langle n|$, KMS for $a,a^\dagger$ requires

$$\frac{p_{n+1}}{p_n}=e^{-\beta\omega}$$

for every $n$. An arbitrary diagonal distribution commutes with $H$ but fails this ratio.

Equilibrium, stationarity, and detailed balance

• Stationarity means $\langle A(t+s)\rangle=\langle A(t)\rangle$ and, in finite dimensions, $[\rho,H]=0$.
• KMS equilibrium adds analytic and boundary relations for all eligible pairs of observables at one $\beta$.
• Detailed balance is a related symmetry of transition rates or correlation spectra whose exact form depends on the dynamical framework.
• Passivity means no cyclic operation extracts net work; complete passivity connects KMS to thermodynamic equilibrium in the infinite-system setting.

Confusing these notions can make a generalized Gibbs ensemble, a diagonal post-quench state, or a driven steady state look thermal merely because its one-point functions are time independent.

The shared equilibrium convention table records the distinction between a finite Gibbs trace and an intrinsic KMS state.

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

Show that a two-level density matrix $\rho=\operatorname{diag}(p_0,p_1)$ is KMS at inverse temperature $\beta$ for the full matrix algebra precisely when $p_1/p_0=e^{-\beta\Delta}$, where $H=\operatorname{diag}(0,\Delta)$.

Solution

Apply KMS to $A=|0\rangle\langle1|$ and $B=A^\dagger$. The two boundary values are $p_0e^{-i\Delta t}$ and $p_1e^{-i\Delta t}e^{\beta\Delta}$, so equality requires $p_1/p_0=e^{-\beta\Delta}$. This ratio normalizes to the Gibbs state and is then sufficient by trace cyclicity.

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.
• Kubo, Ryogo. “Statistical-Mechanical Theory of Irreversible Processes. I.” Journal of the Physical Society of Japan 12, no. 6 (1957): 570–586. doi:10.1143/JPSJ.12.570.
• Martin, Paul C., and Julian Schwinger. “Theory of Many-Particle Systems. I.” Physical Review 115, no. 6 (1959): 1342–1373. doi:10.1103/PhysRev.115.1342.