The spin Drude weight of the XXZ chain and generalized hydrodynamics
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Bibliographic record
Abstract
Based on a generalized free energy we derive exact thermodynamic Bethe ansatz formulas for the expectation value of the spin current, the spin current-charge, charge-charge correlators, and consequently the Drude weight. These formulas agree with recent conjectures within the generalized hydrodynamics formalism. They follow, however, directly from a proper treatment of the operator expression of the spin current. The result for the Drude weight is identical to the one obtained 20 years ago based on the Kohn formula and TBA. We numerically evaluate the Drude weight for anisotropies \Delta=\cos(\gamma) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>cos</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>γ</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> with \gamma = \pi n/m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> <mml:mi>/</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> , n\leq m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general \gamma=\pi n/m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> <mml:mi>/</mml:mi> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> —obtained by analysis of the quantum transfer matrix eigenvalues—agrees with the bound which has been obtained by the construction of quasi-local charges.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it