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Conserved overlaps bound the finite persistent current correlation

Statement

Take the finite-chain data of the current definition on $H_{L}$ and identify $H : = H_{L}$ . Let $H = H^{*}$ have energy unit joules, let $ℏ > 0$ have unit joule-seconds, and let $ρ$ be a positive-definite trace-one matrix satisfying $[ρ, H] = 0$ . Define $ω (A) : = Tr (ρ A)$ and $α_{t} (A) : = e^{i H t / ℏ} A e^{- i H t / ℏ}$ . Let $J_{L}$ be the total lattice current of the finite-volume current definition and center it with respect to this same $ρ$ , setting $J : = J_{L} - ω (J_{L}) I$ . With no identification of $H$ and $H_{L}$ , the estimate below is an abstract bound for this observable under $α_{t}$ . Its lattice-transport specialization further takes $H = H_{L}$ ; then the Gibbs density $ρ_{β, L}$ in that definition is stationary and is an admissible choice of $ρ$ . More generally, $ρ_{β, L}$ can be used with another $H$ exactly when it also satisfies $[ρ_{β, L}, H] = 0$ . Then $J = J^{*}$ , as established by the local continuity result. For $m \geq 1$ , let $Q_{1}, \dots, Q_{m}$ be linearly independent conserved matrices, $[H, Q_{a}] = 0$ . Put $G_{a b} : = Tr (ρ Q_{a}^{*} Q_{b}), v_{a} : = Tr (ρ Q_{a}^{*} J) .$ Then $G$ is positive definite, the finite Cesàro limit exists, and $\lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} Re ω ⁣ (α_{t} (J) J) d t \geq v^{*} G^{- 1} v \geq 0.$ This is a finite-dimensional persistent-correlation bound. It does not identify a thermodynamic Drude weight or assert that either volume/time order of limits exists.

Statement references

Local charge current and finite-volume correlation conventions

Continuity equation for a local lattice charge current

Finite Cesaro dephasing with degenerate energy blocks

Finite matrix spectral and state calculus

Sirker, §2.1, equations (2.9)–(2.14)

Facts & Assumptions

Given: The finite-dimensional Hilbert space, self-adjoint $H$ , positive-definite stationary density $ρ$ , centered self-adjoint current $J$ , conserved matrices $Q_{a}$ , and $ℏ > 0$ in the Statement.

[A1]

Faithfulness and stationarity: $ρ > 0$ , $Tr ρ = 1$ , and $[ρ, H] = 0$ .

[F1]

Local charge current and finite-volume correlation conventions: the lattice total current is $J_{L} = \sum_{(j, j + 1)} j_{j, j + 1}$ , with $j_{j, j + 1} = (i / ℏ) [h_{j, j + 1}, q_{j + 1}]$ ; the same definition specifies the connected finite-volume correlation convention.

[F2]

Continuity equation for a local lattice charge current: each such bond current is self-adjoint and obeys the local continuity equation under the stated bond-locality and charge-conservation hypotheses.

[F3]

Finite Cesaro dephasing with degenerate energy blocks: for finite-dimensional unitary evolution generated by self-adjoint $H$ , Cesàro averages of every density matrix converge in trace norm to its equal-energy block-diagonal part $\sum_{E} P_{E} ρ P_{E}$ ; blocks inside degenerate eigenspaces remain.

[F4]

Finite matrix spectral and state calculus: finite Hermitian matrices have orthogonal spectral projections, and the finite trace is cyclic.

Proof

technique · orthogonal projection onto the conserved subspace

Given: [A1], [F1]–[F4], a centered self-adjoint $J$ , and linearly independent $Q_{a}$ with $[H, Q_{a}] = 0$ .

1.1givenF1F2algebra

The lattice-current definition fixes $J_{L} = \sum_{j} j_{j, j + 1}$ , and the local-current lemma gives $J_{L} = J_{L}^{*}$ . Subtracting its real mean $ω (J_{L})$ preserves self-adjointness, so $J$ is a matrix observable in $B (H)$ .

1.2givenA1F4algebra

On $B (H)$ define $(A, B)_{ρ} : = Tr (ρ A^{*} B)$ , linear in its second argument. It is positive definite: if $(A, A)_{ρ} = 0$ , then $ρ \geq c I$ for some $c > 0$ in finite dimension, so $0 = Tr (ρ A^{*} A) \geq c Tr (A^{*} A)$ and hence $A = 0$ . The invariance $[ρ, H] = 0$ gives $(α_{t} (A), α_{t} (B))_{ρ} = (A, B)_{ρ}$ by cyclicity of the finite trace, so $α_{t}$ is unitary on this finite-dimensional Hilbert space of matrices.

2.1givenA1F3F4step 1.1algebra

Write $H = \sum_{E} E P_{E}$ over its distinct eigenvalues. Then $α_{t} (J) = \sum_{E, F} e^{i t (E - F) / ℏ} P_{E} J P_{F}$ . For $E \neq F$ , $∣ \frac{1}{T} \int_{0}^{T} e^{i t (E - F) / ℏ} d t ∣ \leq \frac{2 ℏ}{T ∣ E - F ∣} ⟶ 0,$ while for $E = F$ the coefficient is $1$ . The sum is finite, so $M_{T} J : = T^{- 1} \int_{0}^{T} α_{t} (J) d t$ converges in matrix norm to $P J : = \sum_{E} P_{E} J P_{E}$ . This block map is the orthogonal projection onto $N : = {A : [H, A] = 0}$ in $(\cdot, \cdot)_{ρ}$ : $P J \in N$ , and each off-diagonal block of $J - P J$ has zero weighted trace pairing with a block-diagonal matrix because $ρ$ is block diagonal. It is the same energy-block projection as the density-matrix dephasing in [F3]; [F3] states the density-matrix version, while the explicit finite expansion here proves the observable version used.

3.1step 1.2step 2.1algebra

Since $J = J^{*}$ , $(α_{t} (J), J)_{ρ} = ω (α_{t} (J) J)$ . Norm convergence in step 2.1 therefore gives $\lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} Re ω (α_{t} (J) J) d t = Re (P J, J)_{ρ} = ∥ P J ∥_{ρ}^{2},$ because $J - P J$ is orthogonal to $N$ and $P J \in N$ .

4.1givenstep 1.1step 1.2step 3.1algebra∎

Set $S : = span {Q_{1}, \dots, Q_{m}} \subseteq N$ , and let $P_{S}$ be its orthogonal projection. Then $P_{S} J = P_{S} P J$ , so $∥ P J ∥_{ρ}^{2} \geq ∥ P_{S} J ∥_{ρ}^{2}$ . The Gram matrix $G$ is positive definite by linear independence and the positive-definite inner product. Writing $P_{S} J = \sum_{a} c_{a} Q_{a}$ , orthogonality gives $G c = v$ , so $c = G^{- 1} v$ and $∥ P_{S} J ∥_{ρ}^{2} = c^{*} G c = v^{*} G^{- 1} v$ . Together with step 3.1 this proves the bound.

Geometry

The drawing depicts orthogonal projections in a three-dimensional sketch: the line $S$ lies in the conserved subspace $N$ , $P_{S} J$ is the projection onto $S$ , and $P_{N} J$ is the projection onto the full conserved subspace. The dashed segments represent orthogonal residuals. The proof uses the Hilbert-space projection and Gram calculation above, not the sketch.

Depends on

Dependency tree · 2 levels

Sources

J. Sirker, Transport in one-dimensional integrable quantum systems, SciPost Physics Lecture Notes 17 (2020), §2.1, equations (2.9)–(2.14)