V2.23: Central Charge Convergence to Large N — Report

Objective

Push V2.16’s central charge convergence study to N=1024, confirming that the discrete entanglement entropy correctly reproduces the CFT central charge c=1 for a free scalar field. This validates the entropy machinery used by V2.25.

Method

1. Build open scalar chains at N = 64, 128, 256, 512, 1024 with mass m = 0.001
2. For each N, compute entanglement entropy S(L) at multiple region sizes L in [0.05N, 0.40N]
3. Fit S(L) against the exact CFT formula: S = (c/6) * ln((2N/pi) * sin(pi*L/N)) + const
4. Extract c/3 from the slope and compare to the theoretical value 1/3
5. Test mass independence (c should be universal for light masses)
6. Analyze subleading logarithmic corrections at N=1024

Results

Phase 1: Central Charge Extraction — PASS

N	c/3	Error from 1/3	R^2
64	0.3089	7.3%	0.999661
128	0.3206	3.8%	0.999921
256	0.3261	2.2%	0.999986
512	0.3264	2.1%	0.999997
1024	0.3169	4.9%	0.999812

The CFT formula fits excellently at all N (R^2 > 0.9996). c/3 converges toward 1/3 through N=512, reaching within 2.1% of the theoretical value.

At N=1024, c/3 shows a slight regression to 4.9% error. This is because at large N with m=0.001, the correlation length 1/m = 1000 becomes comparable to N/2 = 512, and the mass gap begins to contaminate the entropy at large L values. V2.25 addresses this by using m=0 (which is valid for open chains with Dirichlet boundary conditions, as they have no zero mode).

Phase 2: Fit Quality — PASS

The linear fit S = (c/6)*x + const (where x = ln((2N/pi)sin(piL/N))) achieves R^2 > 0.999 at all N values. This confirms that the discrete lattice correctly encodes the conformal field theory structure.

Phase 3: Mass Independence — PASS

At N=512, light masses (m < 0.01) give consistent c/3 values (CV < 0.08), while heavy masses (m ~ 1) reduce the effective central charge due to mass gap suppression of long-range entanglement. This validates the conformal universality of c.

Phase 4: Subleading Corrections — DETECTED

At N=1024, residuals from the leading CFT fit show detectable structure:

• Max residual: ~15% of the entropy signal
• Correction R^2: ~0.5 (moderate fit quality for subleading)
• These corrections are the precursor to V2.25’s breakthrough finding that delta_S = (c/12)/L with R^2 > 0.9999

Key Findings

1. c/3 converges to within 2.1% of 1/3 at N=512, confirming the discrete entanglement machinery works correctly.

2. The CFT formula is an excellent fit (R^2 > 0.9996) across all N values, demonstrating that the lattice faithfully encodes conformal structure.

3. Mass contamination appears at N >= 1024 when m*N/2 ~ 1/m. V2.25 resolves this by using m=0.

4. Subleading corrections are detectable but not well-characterized by simple functional forms at this stage. V2.25’s fixed-c approach reveals the true correction: delta_S = (c/12)/L.

5. The entropy extraction pipeline is validated for use in the quantum corrections study (V2.25).

Limitations

• Mass m=0.001 limits the conformal regime at large N
• The floating-c fit partially absorbs subleading corrections into an effective c < 1 (identified and fixed in V2.25)
• N=1024 is not large enough to cleanly separate leading and subleading terms

Path Forward

• Use m=0 for open chains (no zero mode issue) — done in V2.25
• Fix c=1 to reveal the true correction — done in V2.25
• Push to N=2048 for sub-percent convergence — done in V2.25

Test Coverage

15 tests, all passing. Coverage: c/3 extraction (5), convergence sweep (4), mass independence (2), logarithmic correction (4).