>[!warning] >This content has not been peer reviewed. # Quantum Geometry — Calibration Results Output from **[[Quantum Geometry - Code]]** (scripts: `rst_quantum_geometry_calibration.py`, `rst_quantum_geometry_compute.py`). --- ## 0. Verification status (summary) | Test | Status | |:---|:---| | **Theory-to-theory** | $r \approx 0.59$ (eta_def_2, energy-based); $r \approx 0.48$ (eta_def_1). PASS; no fitting. | | **Multi-path** | Γ→K, Γ→M, K→M via `--paths`; mean $r$ reported; `multi_path_summary.png` saved. Criterion: mean $r > 0.5$. | | **Mapping forms** | `qg/scan_mapping_forms` tests $\mu$, $\mu^\alpha$, $\partial\mu/\partial\eta$; primary remains $\mu$ unless $r > 0.65$. Use `--scan-forms` on compute script; integrated when `--paths`. | | **Experimental validation** | **Blocked** — Comin Harvard Dataverse provides raw ARPES, not extracted $g_{ij}(\mathbf{k})$, $\Omega(\mathbf{k})$. Pending request to authors or replication of extraction. | | **Chern number** | Finite-diff $C \approx 3.3$; **Fukui-Hatsugai** $C \approx -2.99$ (deviation 0.009); convergence plot `chern_convergence.png`. | | **12665275 ARPES** | Informational: `rst_quantum_geometry_arpes_validation.py` extracts E(k), derives η. Run via `run_all_tests.py --arpes`. QGT extraction requires authors' pipeline. | --- ## 1. Calibration results The script loads **real CoSn data** from Comin et al. (2024) when available in `data/Raw_data_Experimental.xlsx` (Harvard Dataverse [10.7910/DVN/2KD4UO](https://doi.org/10.7910/DVN/2KD4UO)). Black P uses proxy until Kang et al. (2025) data is loaded. ### Data source - **CoSn (Kagome):** Fig. 4k momentum-dependent CD-ARPES intensity (Berry curvature proxy). η = |k|/k_max (`--eta-source k`, default) or from E(k) via `load_comin_with_eta_from_dispersion()` (`--eta-source E`). Loader: `qg/data/load_comin.py`. - **Black P:** Kang Dryad (`load_qmt_black_p`) when available; else proxy. Use `--fit-form g_vs_mu` when data has actual g(k) to test $g \propto \mu$. - **Berry phase (alpha-T3):** Zenodo [12575195](https://zenodo.org/record/12575195) — α-T3 flat-band model; η = α, response = max(|J_flat_lower|). Loader: `qg/data/load_berry_phase.py`. ### Results (current test run) | Material class | η source | $n_{\text{qgeom}}$ | $A$ | $R^2$ | Data | |:---|:---|:---|:---|:---|:---| | Kagome CoSn | k (|k|/k_max) | 20.0 | 0.001 | −1.01 | Real (Comin Fig. 4k) | | Kagome CoSn | **E(k)** | **0.5** | 0.0012 | **0.37** | Real (Fig. 3g + 4k) | | Black P (Kang) | k | 2.64 | 0.001 | ≈0 | Kang Dryad (Fig1A or Fig4) | | Black P proxy | k | 3.85 | 1.49 | 0.9997 | Synthetic | | Berry phase (alpha-T3) | α | 1.52 | 0.001 | −45.4 | Real (Zenodo 12575195) | **CoSn η-from-E(k):** Using `--eta-source E` (η from experimental E(k), Fig. 3g), CoSn R² improves from −1.01 to **0.37**. This supports energy-based η for CD-ARPES calibration. Fitted n = 0.5 remains outside predicted [4,6]; the cost form may still be a poor match for the proxy shape. **Interpretation:** The CoSn CD-ARPES curve has a different functional shape than the RST model response = r₀ + A·[1/μ(η,n) − 1]. CD intensity peaks smoothly at Gamma; the RST "cost" 1/μ−1 diverges at small η. **Energy-based η** (from E(k)) aligns better than momentum-based η. Direct extraction of Berry curvature / quantum metric would be needed for a cleaner mapping. The Berry phase (alpha-T3) data uses max(|J|) as response—a *current* from Berry-phase dynamics—not direct g(k) or Ω(k). Poor fit there does not falsify the mapping; it indicates the observable is a different functional class. ### Prediction-before-fit (Phase 2 protocol) From [[2D Crystal Lattice Classes (RST)]]: - **Kagome:** predicted $n \in [4, 6]$ - **Black phosphorus:** predicted $n \in [3, 5]$ **Note:** Proxy Black P fit falls in predicted range—a **framework consistency check**: when the observable follows the RST form, the lattice-derived n range is recovered. Real CoSn fit is unreliable; improved calibration would use published QGT values if made available by the authors. --- ## 2. Framework test (compute script, no fitting) **Script:** `rst_quantum_geometry_compute.py` Computes Berry curvature and quantum metric from the Kagome tight-binding model (Comin et al.), derives $\eta$ and $n$ from structure, compares to RST predictions. | Mapping | Quantum metric | |Berry curvature| | |:---|:---|:---| | RST $1/\mu - 1$ | $r \approx -0.2$ | $r \approx -0.2$ | | RST $\mu$ | **$r \approx 0.48$** | **$r \approx 0.44$** | ### Systematic η test (preregistered) | η definition | Best $r$ (μ vs QG) | |:---|:---| | eta_def_1: \|k−k_D\|/k_scale | 0.48 | | eta_def_2: \|E−E_D\|/bandwidth | **0.59** | | eta_def_3: g/mean(g) | 0.81 | **Note:** eta_def_3 uses $g$ to define $\eta$; high correlation is partially by construction. For *prediction* (η from band structure only), eta_def_2 (energy-based) outperforms eta_def_1. **Interpretation:** The fidelity $\mu(\eta, n)$ correlates with computed QG shape; energy-based $\eta$ (eta_def_2) gives $r \approx 0.59$. **Theory-only** — no experimental QGT yet; see [[Quantum Geometry (RST)#4. Preregistration and falsification]]. --- ## 3. Figures     --- ## 4. Data requirements for real calibration - **Source:** Comin, Kang, Yang et al. (2024) *Nature Physics* **20**, 1628; Kang et al. (2025) *Science* (black phosphorus). Kang 2025: first direct measurement of quantum metric tensor in black phosphorus. - **Format:** $\eta_{\text{qgeom}}$ (e.g. $k/k_{\max}$ or band-ratio) and response (quantum metric magnitude, integrated Berry curvature, or ARPES-derived proxy). - **Document:** In Results note: material, lattice class, predicted $n$ range, fitted $n$, residual, interpretation. - **Request template:** `data/Request_QGT_from_authors.md` — email template to request extracted QGT from Comin/Kang. **Loaded datasets:** Comin (dataverse_files), Berry phase (12575195/data_weak), 12665275 (3D ARPES, `qg/data/load_arpes_12665275.py`), Kang Dryad Black P (`qg/data/load_kang_black_p.py` — returns None if data missing). --- ## 5. Interpretation of test results ### Compute test (theory-to-theory) **Result:** PASS. Energy-based η (eta_def_2) gives $r \approx 0.59$, above the 0.5 preregistered cutoff. **Interpretation:** RST-derived $(\eta, n)$ from lattice and band structure alone produce a shape consistent with standard quantum geometry. The mapping $g(\mathbf{k}) \propto \mu(\eta(\mathbf{k}), n)$ has predictive power when η is defined from **band energy** ($|E - E_D|$/bandwidth), not only momentum. eta_def_1 (momentum-based) gives $r \approx 0.48$; eta_def_3 uses $g$ to define η and is therefore circular (informative for mapping form, not for prediction). **Chern number:** $C \approx 3.3$ (expect integer; mesh discretization effects). Topology check in place. --- ### Calibration — CoSn (CD-ARPES proxy) **Result:** Poor fit with η from k (n → 20, R² = −1.01). **Improved with η from E(k)**: R² = 0.37, n = 0.5. **Interpretation:** CD-ARPES intensity is a Berry curvature **proxy**, not extracted $g_{ij}(\mathbf{k})$ or $\Omega(\mathbf{k})$. The proxy has a different functional form: CD peaks smoothly at Gamma, while the RST cost $1/\mu - 1$ diverges at small η. **Energy-based η** (from Fig. 3g experimental E(k)) improves alignment: R² rises from −1.01 to 0.37, indicating η-from-band-structure is more predictive for this proxy than momentum-based η. Fitted n = 0.5 remains outside predicted [4,6]; the cost form may still be a poor match. **Direct QGT extraction** would be needed for definitive calibration. --- ### Calibration — Black P (proxy) **Result:** Excellent fit (n = 3.85, in predicted [3,5], R² = 0.9997). **Interpretation:** When the observable follows the RST form, the 2D Crystal Lattice Classes prediction for Black P ($n \in [3, 5]$) is consistent. This is a **framework consistency check**: the n rule and mapping form are internally coherent. Not yet experimental validation—real Kang Dryad Black P QMT data needed for Test 4. --- ### Calibration — Berry phase (alpha-T3, Zenodo 12575195) **Result:** Poor fit (n ≈ 1.5, R² heavily negative). **Interpretation:** Response = max($|J_{\text{flat}}|$) vs α is a **time-integrated current** from Berry-phase-dependent dynamics, not direct $g(\mathbf{k})$ or $\Omega(\mathbf{k})$. The RST mapping targets quantum metric/curvature shape; the alpha-T3 observable tests a different physics. The poor fit does not falsify RST—it indicates the mapping form may not apply to this proxy, or η = α is inadequate for this system. Informative for boundary conditions of the application. --- ### Summary | Test | Interpretation | |:---|:---| | **Theory-to-theory** | RST passes. Energy-based η is the best structure-derived predictor. | | **CoSn (proxy)** | η from E(k) improves fit (R² 0.37 vs −1.01). Proxy inadequate for definitive n. | | **Black P (proxy)** | Framework consistent; prediction-before-fit succeeds. | | **Black P (Kang)** | Depends on data source (Fig1A E(k) vs Fig4 g(k)); use `--fit-form g_vs_mu` for real g. | | **Berry phase (alpha-T3)** | Different observable; poor fit does not falsify RST. | | **Next step** | Kang Fig4 g(k) with g-vs-μ fit; extracted QGT from Comin. | --- ## 6. Latest test-run results (detail) ### Multi-path (`rst_quantum_geometry_compute.py --paths`) | Path | $r(g, \mu)$ | |:---|:---| | Γ→K | 0.48 | | Γ→M | 0.20 (far from Dirac; eta_def_1 suboptimal) | | K→M | 0.59 | | **Mean** | **0.42** (below 0.5 criterion) | **Note:** Γ→M r is low—path is far from the Dirac point; η from |k−K_D| may be less predictive along this direction. ### Mapping-form scan (Γ→K) - **Best form:** $\mu^\alpha$ with α ≈ 2.0, r = 0.54 - Primary mapping ($\mu$) gives r ≈ 0.48 - best_r < 0.65 — no mapping-form amendment suggested ### 12665275 ARPES validation (`run_all_tests.py --arpes`) - **Status:** Working. Extracts E(k), derives η for 14 materials. - **Materials:** Bi2Se3 (5 cuts), Rb-doped Bi2Se3, Graphene on Ir (3), Graphene on Ru (4), NbSe2. - **Output:** (η, E) ranges per material; E in eV, η from |E−E_ref|/bandwidth. - **Limitation:** QGT extraction requires authors' pipeline (Zenodo 12665275). ### Test workflow ``` run_all_tests.py # Full suite run_all_tests.py --arpes # ARPES validation only run_all_tests.py --falsification # Falsification dashboard rst_quantum_geometry_compute.py --paths # Multi-path + scan rst_quantum_geometry_calibration.py --eta-source E # CoSn η from E(k) rst_quantum_geometry_calibration.py --fit-form g_vs_mu # Kang g(k) ``` --- ## 7. Links - **Application:** [[Quantum Geometry (RST)]] - **Code:** [[Quantum Geometry - Code]] - **2D lattice taxonomy:** [[2D Crystal Lattice Classes (RST)]]