Symmetries and anomalies of Kitaev spin-S models: Identifying symmetry-enforced exotic quantum matter
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Bibliographic record
Abstract
We analyze the internal symmetries and their anomalies in the Kitaev spin- S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> </mml:math> models. Importantly, these models have a lattice version of a \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> 1-form symmetry, denoted by \mathbb{Z}_2^{[1]} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true" form="postfix">]</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> . There is also an ordinary 0-form \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> symmetry, where \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> are \pi <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>π</mml:mi> </mml:math> spin rotations around two orthogonal axes, and \mathbb{Z}_2^T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:math> is the time reversal symmetry. The anomalies associated with the full \mathbb{Z}_2^{(x)}×\mathbb{Z}_2^{(y)}×\mathbb{Z}_2^T×\mathbb{Z}_2^{[1]} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo stretchy="true" form="prefix">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true" form="postfix">]</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> symmetry are classified by \mathbb{Z}_2^{17} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>17</mml:mn> </mml:msubsup> </mml:math> . We find that for S∈\mathbb{Z} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℤ</mml:mi>
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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