Macroscopic quantum spin tunneling with two interacting spins
Bibliographic record
Abstract
We study the simple Hamiltonian, $H=\ensuremath{-}K({S}_{1z}^{2}+{S}_{2z}^{2})+\ensuremath{\lambda}{\stackrel{P\vec}{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{P\vec}{S}}_{2}$, of two large, coupled spins which are taken equal, each of total spin $s$ with $\ensuremath{\lambda}$ the exchange coupling constant. The exact ground state of this simple Hamiltonian is not known for an antiferromagnetic coupling corresponding to the $\ensuremath{\lambda}>0$. In the absence of the exchange interaction, the ground state is fourfold degenerate, corresponding to the states where the individual spins are in their highest weight or lowest weight states, $|\ensuremath{\uparrow},\ensuremath{\uparrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\uparrow}\ensuremath{\rangle}$, in obvious notation. The first two remain exact eigenstates of the full Hamiltonian. However, we show that the two states $|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle},|\ensuremath{\downarrow},\ensuremath{\uparrow}\ensuremath{\rangle}$ organize themselves into the combinations $|\ifmmode\pm\else\textpm\fi{}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|\ensuremath{\uparrow},\ensuremath{\downarrow}\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle})$, up to perturbative corrections. For the antiferromagnetic case, we show that the ground state is nondegenerate, and we find the interesting result that for integer spins the ground state is $|+\ensuremath{\rangle}$ and the first excited state is the antisymmetric combination $|\ensuremath{-}\ensuremath{\rangle}$ while for half odd integer spin, these roles are exactly reversed. The energy splitting, however, is proportional to ${\ensuremath{\lambda}}^{2s}$, as expected by perturbation theory to the $2s$th order. We obtain these results through the spin coherent state path integral.
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How this classification was reachedexpand
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) | 0.003 | 0.003 |
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; both teacher heads agree on what is shown here.
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".