Addendum: Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators
Bibliographic record
Abstract
In this Article, we describe that a PT-symmetric Hamiltonian H (PT) and anti-PT-symmetric Hamiltonian H (APT) are related by a similarity transformation with a unitary operator U given in Eq. (7). This statement may cause confusion or incorrect arguments due to its mathematical validity limitation. Therefore, here we provide an additional explanation. This relation is strictly valid only if H (PT) and H (APT) are traceless and off-diagonal elements in H (PT) are purely real. In case of non-traceless cases, one can employ an energy-shifting gauge transformation that renders H (PT) and H (APT) traceless, in order to make valid use of the relation. In addition, a similarity relation between traceless H (PT) and H (APT) can be extended for H (PT) with complex-valued off-diagonal elements by a generalized unitary-transformation operator $$U\prime = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{*{20}{l}} 1 \hfill & {{\mathrm{{e}}}^{i\theta }} \hfill \\ {{\mathrm{{e}}}^{ - i\theta }} \hfill & { - 1} \hfill \end{array}} \right],$$ where θ is phase angle of an off-diagonal element of a traceless H (PT) .
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.064 | 0.013 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".