>[!warning] >This content has not been peer reviewed. # Quantum Geometry — RST application **Pillar:** Berry curvature and the quantum metric in Bloch bands are mapped onto the RST framework via conceptual bridges to holonomy ($\theta_W$), the fidelity landscape ($\mu$), and noise projection ($\nu$). This application establishes the **conceptual connection**; calibration against experimental quantum geometry data (Phase 2) will test predictions. --- ## 1. Classical picture (summary) ### Wave function as arrow In quantum mechanics, a particle's state can be represented as an "arrow" on a sphere (or higher-dimensional manifold). For two-state systems, the two eigenstates correspond to poles; superpositions point somewhere on the surface. For many-state systems (e.g. Bloch electrons in a crystal), the arrow occupies a high-dimensional space. ### Parameter-space map As external parameters change (magnetic field $B$, temperature $T$, momentum $\mathbf{k}$), the wave function rotates. Tracking this rotation across the parameter space builds a **map**. Each parameter adds a dimension. The **quantum metric** describes how quickly the arrow rotates as you move across the map—the "rugosity" of the landscape. The **Berry curvature** describes the phase accumulated when the arrow travels a closed loop and returns to the starting point—a holonomy. ### Experimental methods - **ARPES (angle-resolved photoemission spectroscopy):** UV light ejects electrons; energy and momentum are measured. - **Circularly polarized light:** The "ghost field" (Berry curvature) deflects electrons differently by spin; circular dichroism yields Berry curvature. - **Velocity–energy correlation:** The quantum metric relates to how electron velocity changes with energy across the Brillouin zone; measured via upgraded ARPES (Comin, Kang, Yang et al.). ### Torus topology For 2D crystals, the Brillouin zone is topologically a **2-torus** $\mathbb{T}^2$. States at opposite edges are identified. Looping around the torus can accumulate Berry phase; topological materials show nonzero Chern numbers and "points of discontinuity" where the wave function flips. --- ## 2. Conceptual bridges (RST mappings) ### Berry phase ↔ holonomy The electroweak mixing angle $\theta_W$ is interpreted in RST as the **holonomy twist** in the gauge fibre bundle ([[expanded theory/Lexicon and Mapping]]). The $d^* = 9 + \theta_W$ embedding dimension follows from sub-Riemannian gauge geometry: Mitchell's Theorem (Mitchell 1985) states that non-holonomic constraints raise the Hausdorff dimension by the geometric phase excess; Montgomery (2002) provides the framework. **Berry curvature in materials** and **$\theta_W$ in the Standard Model** are the same mathematical structure—geometric phase accumulated on a closed loop. Experimental Berry curvature measurements thus provide an independent probe of the holonomy sector. ### Quantum metric ↔ μ landscape The fidelity function $\mu(\eta, n) = \eta/(1+\eta^n)^{1/n}$ describes how the substrate allocates budget between signal and noise. The **slope** of $\mu$ along the parameter map—how sharply the transition occurs—is controlled by $n$. The quantum metric's "rugosity" (how rapidly the wave function changes with parameters) corresponds conceptually to this slope: - **Steep landscape** (large quantum metric variation) ↔ **high effective $n$** (sharp transition) - **Flat landscape** (slow variation) ↔ **low effective $n$** (gradual transition) So $n_{\text{qgeom}}$ (to be calibrated) may correlate with the integrated quantum metric or its variance across the Brillouin zone. ### Ghost field ↔ noise projection Topological materials host an effective "ghost field"—electrons behave as if experiencing a force with no classical source. This force arises from the geometry (Berry curvature) alone. RST's **noise projection** $\nu = N/W$ gives analogous effective gravity from substrate geometry: in the low-SNR regime, the extra "gravity" attributed to dark matter is the noise-dominated workload, not new matter. Both are **effective forces from geometry** without additional fields or particles. --- ## 3. Proposed Bloch-sector mapping (conjecture) ### η_qgeom (candidate definitions) To apply $\mu(\eta, n)$ to the Bloch sector, $\eta$ must be defined. **Conjectured** options: 1. **Bandwidth / Berry-curvature scale:** $\eta = \Delta E / \Omega_{\text{char}}$, where $\Delta E$ is bandwidth and $\Omega_{\text{char}}$ is a characteristic Berry-curvature scale. 2. **Momentum-space resolution:** $\eta = k_{\text{max}} / k_{\text{noise}}$, where $k_{\text{noise}}$ is the resolution limit. 3. **Integrated metric / gap:** $\eta = \langle g \rangle / \Delta^2$, where $\langle g \rangle$ is the mean quantum metric and $\Delta$ is the gap. These are **conjectures**; Phase 2 calibration will determine which (if any) yields a consistent $n_{\text{qgeom}}$. ### n_qgeom To be **calibrated** in Phase 2 from published Berry curvature and quantum metric data (e.g. kagome, black phosphorus). The prediction-before-fit protocol will compare lattice-class predictions from [[2D Crystal Lattice Classes (RST)]] with fitted values. --- ## 4. Preregistration and falsification *Preregistered before experimental comparison. No post-hoc fitting of $\eta$ or $n$.* ### 4.1 η definition (primary) **Primary** (used in compute script): $\eta(k) = |\mathbf{k} - \mathbf{k}_{\text{Dirac}}| / k_{\text{scale}} + \epsilon$, where $k_{\text{scale}} = |\mathbf{K}|/2$ (half the Gamma–K distance). Small $\eta$ near Dirac (high Berry curvature), large $\eta$ away. **Alternatives** (tested in compute script): - **eta_def_2:** $\eta = |E(\mathbf{k}) - E_{\text{Dirac}}| / \Delta E_{\text{band}}$ (energy distance; best predictor, $r \approx 0.59$) - **eta_def_3:** $\eta = g(\mathbf{k}) / \langle g \rangle$ (metric-relative; $r \approx 0.81$ but circular — uses $g$ to define $\eta$) ### 4.2 n derivation rule **Conjectured derivation** ([[2D Crystal Lattice Classes (RST)]]): For Kagome, $n_{\text{qgeom}} = 5$ from phase-space sum: $n = 2$ (Brillouin zone dimension) + band multiplicity (2–4) → midpoint 5. Explicit: $n = 2 + \lfloor \log_2(n_{\text{bands coupling}}) \rfloor$ or phase-space heuristic. Reproducible: *For Kagome (6-fold, Dirac + flat band): $n \in [4, 6]$, default 5.* ### 4.3 Mapping form **RST prediction:** $g(\mathbf{k}) \propto \mu(\eta(\mathbf{k}), n)$ up to scale. The compute script shows $\mu$ correlates better than $1/\mu - 1$ with quantum metric; we adopt $\mu$ as the primary mapping. **Alternatives tested** (`qg/scan_mapping_forms.py`): $\mu^\alpha$ and $\partial\mu/\partial\eta$; primary remains $\mu$ unless $r > 0.65$ warrants amendment. ### 4.5 Multi-path test (preregistered) **Criterion:** Multi-path test **passes** if mean $r(g, \mu) > 0.5$ across paths Γ→K, Γ→M, K→M. Use `--paths` on compute script; save `multi_path_summary.png`. ### 4.4 Falsification criteria | Test | Criterion | Status | |:---|:---|:---| | **Test 1** (theory-to-theory) | For Kagome TB, correlation $r(g, \mu) > 0.5$ | **Pass:** $r \approx 0.59$ (eta_def_2) | | **Test 2** (prediction-before-fit) | For CoSn, predict $n \in [4, 6]$; $n_{\text{fit}} \in [4, 6]$ → pass | Blocked: calibration used CD-ARPES proxy, not QGT | | **Test 3** (experimental shape) | Experimental $g(\mathbf{k})$, $\Omega(\mathbf{k})$ vs RST prediction; $r > 0.6$ → pass | Blocked: need extracted QGT from Comin/Kang | | **Test 4** (second material) | Black phosphorus (Kang et al. 2025); same protocol; both materials must pass | Pending data | **Blocker:** Experimental quantum geometric tensor. Comin Harvard Dataverse provides raw ARPES, not extracted $g_{ij}$, $\Omega$. Request from authors or replicate extraction. --- ## 5. Links - **Knowledge:** [[expanded theory/knowledge/Berry phase and quantum geometry]] - **Quantization:** [[expanded theory applied/further applications/Quantization/Quantization (RST)]] — $L^n$ geometry, Heisenberg limit - **Electronic Transport:** [[expanded theory applied/further applications/Electronic Transport/Electronic Transport (RST)]] — lattice as substrate, $n_{\text{el}} \approx 5$; $n_{\text{el}}$ does not follow 3D crystal structure - **Substrate Hardware:** [[expanded theory applied/further applications/Substrate Hardware/Substrate Hardware (RST)]] — Sovereign Chain, $d^* = 9 + \theta_W$ - **2D lattice classes:** [[2D Crystal Lattice Classes (RST)]] - **Lexicon:** [[expanded theory/Lexicon and Mapping]] --- ## 6. Epistemological status **Tier 3 (structural):** Conceptual bridge. The Berry–holonomy and ghost-field–noise-projection mappings are structural identifications. The compute script (`rst_quantum_geometry_compute.py`) **derives** $\eta$ and $n$ from lattice/band structure and tests shape correlation with standard quantum geometry—no fitting. Energy-based $\eta$ (eta_def_2) gives $r \approx 0.59$, passing the preregistered cutoff; the mapping $g(\mathbf{k}) \propto \mu(\eta(\mathbf{k}), n)$ has predictive power when $\eta$ is defined from band structure. Calibration on CoSn CD-ARPES and Berry phase (alpha-T3) proxies gives poor fits—those observables are not direct $g$, $\Omega$; the poor fits do not falsify RST. Black P proxy fit falls in predicted $n \in [3, 5]$, supporting framework consistency. --- ## 7. Code and results - **Script (compute):** `rst_quantum_geometry_compute.py` — framework test: compute Berry curvature and quantum metric from Kagome TB model; derive $\eta$, $n$ from structure; compare to RST. **No fitting.** - **Script (calibration):** `rst_quantum_geometry_calibration.py` — fits $n_{\text{qgeom}}$ from proxy or real data. - **Code:** [[Quantum Geometry - Code]] - **Results:** [[Quantum Geometry Results]] — compute test; calibration protocol. --- ## 8. Limitations - **No experimental validation** with real Berry curvature/quantum metric data yet; calibration uses CD-ARPES and Berry-phase current proxies, which have different functional forms than direct $g(\mathbf{k})$, $\Omega(\mathbf{k})$. Kang Dryad Black P QMT data needed for Test 4. - **2D crystals only** in the first instance; 3D bulk materials may require different $\eta$ definitions. - **Electronic Transport null result:** $n_{\text{el}}$ does not correlate with 3D crystal structure (FCC/BCC/HCP). The Bloch-sector $n_{\text{qgeom}}$ may behave differently; no claim that lattice topology universally determines $n$.