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Continuity equation for a local lattice charge current

Statement

Use the finite open or periodic chain and bond-current definition in Local charge current and finite-volume correlation conventions. Each $j_{j, j + 1}$ is self-adjoint, and with ${\dot{q}}_{j} : = (i / ℏ) [H_{L}, q_{j}]$ the exact local identity ${\dot{q}}_{j} = j_{j - 1, j} - j_{j, j + 1}$ holds at every site. Consequently the total charge $Q_{L} : = \sum_{j \in Λ_{L}} q_{j}$ , where $Λ_{L}$ is the open-chain site set or periodic site set from the definition, is conserved: $[H_{L}, Q_{L}] = 0$ . The current orientation is from $j$ to $j + 1$ , so positive outgoing current lowers the charge at site $j$ .

Statement references

Local charge current and finite-volume correlation conventions

Sirker, §2, equations (2.1)–(2.2)

Facts & Assumptions

Given: The finite open chain or periodic ring, self-adjoint nearest-neighbor bond terms $h_{j, j + 1}$ , local charges $q_{j}$ , and $ℏ > 0$ in the Statement.

[A1]

Locality: $[h_{k, k + 1}, q_{j}] = 0$ whenever $j \notin {k, k + 1}$ ; absent open-chain boundary currents are zero.

[A2]

Bond charge conservation: $[h_{j, j + 1}, q_{j} + q_{j + 1}] = 0$ for every oriented bond.

[F1]

Local charge current and finite-volume correlation conventions: $H_{L}$ is the sum over open-chain or periodic-ring bonds, $j_{j, j + 1} = (i / ℏ) [h_{j, j + 1}, q_{j + 1}]$ , and absent open-chain boundary currents are zero.

Proof

technique · direct commutator calculation

Given: [A1]–[A2], [F1], and the open-chain or periodic-ring site and bond indices.

1.1givenF1algebra

Since $h_{j, j + 1}$ and $q_{j + 1}$ are self-adjoint, $[h_{j, j + 1}, q_{j + 1}]^{*} = - [h_{j, j + 1}, q_{j + 1}]$ . Thus $j_{j, j + 1} = (i / ℏ) [h_{j, j + 1}, q_{j + 1}]$ is self-adjoint.

1.2givenA1F1algebra

The Heisenberg derivative is ${\dot{q}}_{j} = (i / ℏ) \sum_{k} [h_{k, k + 1}, q_{j}]$ . By [A1], only the existing bonds immediately to the left and right of $j$ can contribute. The left contribution is $(i / ℏ) [h_{j - 1, j}, q_{j}] = j_{j - 1, j}$ when that bond exists, and is zero by the open-boundary convention otherwise.

1.3givenA1A2F1algebra

For the right contribution, [A2] gives $[h_{j, j + 1}, q_{j}] = - [h_{j, j + 1}, q_{j + 1}]$ , so $(i / ℏ) [h_{j, j + 1}, q_{j}] = - j_{j, j + 1}$ when that bond exists. If it is absent at the open right endpoint, both terms are zero by convention.

2.1step 1.2step 1.3algebra∎

Combining steps 1.2 and 1.3 proves ${\dot{q}}_{j} = j_{j - 1, j} - j_{j, j + 1}$ . Summing over $j \in Λ_{L}$ cancels every existing open-chain or periodic bond once with each sign, so ${\dot{Q}}_{L} = 0$ , equivalently $[H_{L}, Q_{L}] = 0$ .

Diagram

The arrows show the declared positive orientation of the bond currents. At site $j$ , the incoming arrow contributes positively to ${\dot{q}}_{j}$ and the outgoing arrow negatively. This picture explains the sign convention only; the commutator proof above is independent of it.

Depends on

Local charge current and finite-volume correlation conventions

Dependency tree · 2 levels

Sources

J. Sirker, Transport in one-dimensional integrable quantum systems, SciPost Physics Lecture Notes 17 (2020), §2, equations (2.1)–(2.2)