Numerics Overview

Quantitative tests of the QHE / BHE analogy

Philosophy

The 2012 paper was conceptually bold but left many quantitative claims as sketches. The numerics program is designed to:

  1. Make the analogy computable — Define precise mappings between LQG and QHE parameters
  2. Test robustness — Check if results hold under finite-size, disorder, and varying approximations
  3. Produce reproducible artifacts — Scripts, figures, and data that can be checked by others

Active Tasks

Task Focus Key Question Status Link
T14 Filling fractions What is \(\nu\) for a given horizon area spectrum? 🟢 Completed View →
T15 Entropy steps Exact entropy via state counting 🟢 Completed View →
T16 Dimer mobility scaling How does the fraction of mobile punctures scale with \(N_p\)? 🟢 Completed View →
T17 Resonance graph spectrum Is the RVB space connected under local moves? 🟢 Completed View →
T18 Chiral RVB & Hall response Does the chiral RVB produce a quantized Hall response? 🟢 Completed View →
T19 Thurston-enhanced numerics Does the geometric structure of Thurston’s moduli space explain the exact Chern numbers? 🟡 In Progress View →

Common Parameters

All numerics use the standard LQG area spectrum with Immirzi parameter \(\gamma = \ln(2)/\sqrt{3} \approx 0.274\) (Domagala–Lewandowski value) or \(\gamma \approx 0.237\) (Meissner value).

The area of a single puncture with spin \(j\) is:

\[ A_j = 8\pi\gamma\ell_P^2 \sqrt{j(j+1)} \]

Total area for \(N\) punctures:

\[ A = \sum_{i=1}^{N} A_{j_i} \]

Filling Fraction (Completed: T14)

For \(j=1/2\) punctures (the ground state), the area per puncture is:

\[ A_{1/2} = 8\pi\gamma\ell_P^2 \sqrt{3/4} = 4\pi\gamma\ell_P^2 \sqrt{3} \]

The filling fraction is defined as:

\[ \nu = \frac{N_p}{k_{IH}} = \frac{N_p}{A / (4\pi\gamma\ell_P^2)} = \frac{N_p}{2 \sum_i \sqrt{j_i(j_i+1)}} \]

Result: For j=1/2 truncation, \(\nu = 1/\sqrt{3} \approx 0.577\) for all \(N_p\).

This is a fixed filling fraction, not tunable like the QHE. To get variable \(\nu\): - Mixed spin configurations give \(\nu \in [0.25, 0.36]\) (continuous, no plateaus) - The charged/dyonic route (v2) gives \(\nu = Q/P\) (tunable by charge ratio)

See full results: T14 — Filling Fractions

Entropy Steps (Completed: T15)

The entropy-area curve has been computed for the standard LQG counting and the RVB restriction:

N_p j_max Total States Entropy S S/N_p RVB S/N_p
6 0.5 20 2.996 0.499 0.299
12 0.5 924 6.829 0.569 0.402
18 0.5 48,620 10.792 0.600 N/A
24 0.5 2,704,156 14.810 0.617 N/A

Key findings: 1. j=1/2 truncation gives entropy at a single area value (no steps) 2. Entropy per puncture approaches ~0.62 for large N_p (vs Bekenstein-Hawking ~2.18) 3. RVB restriction significantly reduces entropy (0.30–0.40 per puncture)

See full results: T15 — Entropy Steps

Code & Reproducibility

  • Language: Python (NumPy, SciPy, Matplotlib) or Julia (Faster for large \(N\))
  • Repo: code/ directory in the GitHub repo
  • Data: Stored as HDF5 or CSV; figures generated automatically

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