Trivalent network model for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>d</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> transition metal dichalcogenides in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> structure: Holography from local constraints
Bibliographic record
Abstract
Dimer models are well known as prototypes for locally constrained physics. They describe systems in which every site on a lattice must be attached to one dimer. Loop models are an extension of this idea, with the constraint that two dimers must touch at each site. Here, we present a further generalization where every site must have three dimers attached -- a trivalent network model. As concrete physical realizations, we discuss d$^3$ transition metal dichalcogenides in the 1T structure -- materials with the structural formula MX$_2$ (M = Tc, Re) or AM$'$X$_2$ (A = Li or Na; M$'$ = Mo, W), where X is a chalcogen atom. These materials have a triangular layer of transition metal atoms, each with three valence electrons in $t_{2g}$ orbitals. Each atom forms valence bonds with three of its nearest neighbours. The geometry of the 1T structure imbues each bond with sharp orbital character. We argue that this enforces a ``bending constraint'' so that two dimers attached to the same site cannot be parallel. This leads to a highly structured space of configurations, with alternating bonds along each line of the underlying triangular lattice. There is no dynamics, as constraints forbid local rearrangements of dimers. We construct a phase diagram, identifying configurations that minimize potential energy. We find a rhombus-stripe phase that explains a distortion pattern seen across several materials. Remarkably, the local constraints in this model lead to a simple example of holography. The bonding configuration in the bulk is completely determined by the configuration at the boundary. We recast the model in terms of three Ising chains that are defined on the boundaries of a triangular cluster. As a testable prediction, we propose that a single impurity will generate long-ranged domain walls.
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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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.002 | 0.003 |
| Meta-epidemiology (broad) | 0.002 | 0.006 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".