Peierls–Bogoliubov inequality

A variational upper bound on free energy: F(H) ≤ F(H0) + ⟨H−H0⟩_{H0}, equivalently a tangent-line bound for log Tr e^{A}.

Peierls–Bogoliubov inequality

Definitions and notation

Quantum Gibbs state and quantum partition function .
(Helmholtz) free energy in statistical mechanics .
Golden–Thompson lemma (often used to establish convexity properties).

Statement

Let $A$ and $B$ be self-adjoint operators on a finite-dimensional Hilbert space (or more generally assume $e^{A}$ and $e^{A+B}$ are trace-class so the traces below are finite). Define

Z(A) := \operatorname{Tr}(e^{A}), \qquad \langle B\rangle_{A} := \frac{\operatorname{Tr}(e^{A}B)}{\operatorname{Tr}(e^{A})}.

Then the Peierls–Bogoliubov inequality states:

\log Z(A+B) \;\ge\; \log Z(A) + \langle B\rangle_{A}.

Thermodynamic form. Let $H$ and $H_0$ be (finite-volume) Hamiltonians and fix $\beta>0$ . Set

Z := \operatorname{Tr}(e^{-\beta H}), \qquad Z_0 := \operatorname{Tr}(e^{-\beta H_0}), \qquad F(H) := -\frac{1}{\beta}\log Z.

Let $\langle\cdot\rangle_{0}$ denote expectation in the Gibbs state $\rho_0 = e^{-\beta H_0}/Z_0$ . Then

F(H) \le F(H_0) + \langle H - H_0\rangle_{0}.

Key hypotheses and conclusions

Hypotheses

$A,B$ are self-adjoint and $Z(A),Z(A+B)$ are finite.
In the Hamiltonian form: $Z$ and $Z_0$ are finite at the chosen inverse temperature $\beta$ .

Conclusions

$\log Z(\cdot)$ is bounded below by its tangent plane: $\log Z(A+B)\ge \log Z(A)+\langle B\rangle_A$ .
Free energy admits a variational upper bound in terms of any trial Hamiltonian $H_0$ and its Gibbs expectation.

Proof idea / significance

Consider the function $\phi(t)=\log \operatorname{Tr}(e^{A+tB})$ . Under mild conditions, $\phi$ is convex in $t$ . Convexity implies the supporting line inequality $\phi(1)\ge \phi(0)+\phi'(0)$ , and a direct differentiation yields $\phi'(0)=\operatorname{Tr}(e^{A}B)/\operatorname{Tr}(e^{A})=\langle B\rangle_A$ .

This inequality is the basis of Bogoliubov’s variational method: choose a tractable $H_0$ (e.g. mean-field) and optimize the upper bound $F(H) \le F(H_0)+\langle H-H_0\rangle_{0}$ over parameters in $H_0$ .