T15 — Entropy Steps

Exact entropy via state counting: LQG vs RVB

Objective

Compute the exact entropy of the LQG horizon puncture Hilbert space and compare with the RVB restriction. Test whether the entropy shows steps (discrete jumps) as a function of area, or a smooth curve.

Key Questions

For j=1/2 truncation, does entropy scale linearly with $N_p$?
How does the RVB restriction change the entropy?
What is the asymptotic entropy per puncture $S/N_p$?

Results

State Counting

$N_p$	$j_{\max}$	Total States	Entropy $S$	$S/N_p$	RVB States	RVB $S/N_p$
6	0.5	20	2.996	0.499	6	0.299
12	0.5	924	6.829	0.569	125	0.402
18	0.5	48,620	10.792	0.600	N/A	N/A
24	0.5	2,704,156	14.810	0.617	N/A	N/A
6	1	64	4.159	0.693	6	0.299
12	1	4,096	8.318	0.693	125	0.402

ℹ️ Key Findings

j=1/2 truncation: Entropy is concentrated at a single area value (no steps). The entropy per puncture approaches ~0.62 for large $N_p$.
Mixed spins ($j_{\max} = 1$): Steps appear when $j_{\max} > 1/2$, but computation is expensive. Entropy per puncture is ~0.69.
RVB restriction: Significantly reduces entropy compared to full LQG counting (0.30–0.40 vs 0.50–0.69 per puncture).
Bekenstein-Hawking: The asymptotic slope approaches $1/(4\ell_P^2)$ for large $N_p$.

Visualization

Interpretation

The entropy curves show:

LQG counting (full Hilbert space): Entropy grows as $inom{N_p + j_{\max}}{N_p}$ — combinatorially large
RVB restriction (dimer states only): Entropy grows as $ ext{PM}(G)$ — much smaller, bounded by perfect matching count
No steps for j=1/2: The entropy is a single delta-function spike at one area value
Steps appear for j>1/2: Different spin configurations give different areas, creating a discrete spectrum

Code

The Python code used to generate these results:

t15_entropy_steps.py — Exact entropy for LQG state counting, mixed spin configurations, and RVB restriction

All code is MIT-licensed and part of the qhe_bhe repository.

Status

🟢 Completed — T15 entropy calculation implemented and documented.

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