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Static finite quantum response and covariance

Physical scope

Nonzero finite-dimensional static canonical model with a bounded Hermitian control observable; noncommuting response uses Kubo–Mori covariance, while ordinary variance is asserted only in the commuting case.

Statement

Adopt the finite canonical state rule of Canonical quantum thermal preparation. Let $H (h) = H_{0} - h B$ on a finite-dimensional complex Hilbert space of dimension $d \geq 1$ , where $H_{0}, B$ are Hermitian, $h$ is a real static control, $h B$ has units J, and $β > 0$ is fixed in J $^{- 1}$ . Put $ρ_{h} = e^{- β H (h)} / Z (h)$ . With $δ B = B - Tr (ρ_{h} B) I$ , define ${Cov}_{K M, ρ_{h}} (B, B) = \int_{0}^{1} Tr (ρ_{h}^{1 - u} δ B ρ_{h}^{u} δ B) d u .$ Then the static susceptibility is $\frac{d}{d h} Tr (ρ_{h} B) = β {Cov}_{K M, ρ_{h}} (B, B) \geq 0.$ Here ${Var}_{ρ_{h}} (B) = Tr (ρ_{h} (B - Tr (ρ_{h} B) I)^{2})$ is the ordinary variance. The Kubo–Mori covariance equals this variance exactly when $[H (h), B] = 0$ ; for noncommuting matrices the Kubo–Mori expression is the applicable covariance. This is a static model deduction, not a dynamical response or observation.

Facts & Assumptions

Given: A finite-dimensional complex Hilbert space of dimension $d \geq 1$ , Hermitian $H_{0}, B$ , real differentiable control $h$ , fixed $β > 0$ , $h B$ in energy units, and canonical preparation at each control value.

[F1]

Canonical quantum thermal preparation: at each $h$ , the adopted canonical state is $e^{- β H (h)} / Z (h)$ .

[F2]

Duhamel derivative and finite Kubo–Mori response: for finite Hermitian matrix paths, the response Hessian is the Kubo–Mori covariance; its diagonal quadratic form is nonnegative and bounded above by ordinary variance, with equality exactly when the observable commutes with the Gibbs state.

Proof

1.1givenF1F2algebra

The parameter matrix is $M (h) = - β H (h) = - β H_{0} + β h B$ , so $M^{'} (h) = β B$ . The finite Duhamel/quotient calculation in [F2] therefore gives $\frac{d}{d h} Tr (ρ_{h} B) = β {Cov}_{K M, ρ_{h}} (B, B)$ , with $ρ_{h}$ given by [F1].

2.1givenstep 1.1F1F2algebra∎

The eigenbasis expression in [F2] writes the Kubo–Mori quadratic form as a sum of $∣ (δ B)_{i j} ∣^{2} L (p_{i}, p_{j})$ , where $L (p_{i}, p_{j}) > 0$ . Every summand is nonnegative, so the susceptibility is nonnegative. Since $E \mapsto e^{- β E} / Z (h)$ is injective on the real eigenvalues of $H (h)$ , those operators have the same eigenspaces and $[H (h), B] = 0$ iff $[ρ_{h}, B] = 0$ . The equality characterization in [F2] therefore gives the ordinary variance formula exactly in the commuting case and strict inequality otherwise.

Static finite quantum response and covariance

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources