Central Charge from Entanglement Entropy: Calabrese-Cardy Confirmed, With a Real Lesson About "Critical"¶
In plain terms: every quantum critical point has a number attached to it, called the central charge, that tells you what "universality class" it belongs to -- the Ising model's is exactly 1/2. This experiment checks whether Dense-Evolution can actually recover that known number from a real simulated ground state, by fitting how entanglement grows with subsystem size. It also caught a real trap: fitting at a point that looks critical (a very clean straight-line fit) but isn't the true critical point gives an extremely convincing wrong answer -- the plot below shows exactly how deceptive that can be.
![Entanglement entropy vs. ln[(2N/pi)*sin(pi*L/N)] at three values of g -- the true critical point (g=1.0), a finite-size point that looks equally clean but is wrong (g=0.86), and an off-critical negative control (g=1.8)](../assets/central_charge_calabrese_cardy/central_charge_fit.svg)
All three fits look like clean straight lines by eye -- but only the blue one (g=1.0) is fit at the actual critical point. The red line (g=0.86) has an even higher R² (0.999997 vs 0.999924) while extracting a central charge nearly double the correct value. High fit quality alone does not mean the extracted physics is right.
Tests whether the critical transverse-field Ising model's ground-state entanglement entropy really follows the open-chain CFT prediction (Calabrese & Cardy, J. Stat. Mech. 2004, P06002; Tong SS4.4.3 "c is for Cardy"):
S(L) = (c/6) * ln[ (2N/pi) * sin(pi*L/N) ] + const
and whether fitting it recovers the known Ising CFT central charge c=1/2.
Formula verified directly against the paper's own text (now indexed in quantumrag's quantum_info collection, calabrese_cardy_2004_entanglement_entropy_qft.pdf): eq. (4) confirms the single-boundary-point asymptotic form \(S_A \sim (c/6)\log(\ell/a)\), and eq. (19) gives the finite periodic-system form \((c/3)\log[(L/\pi a)\sin(\pi\ell/L)]\) this experiment's open-chain formula is derived from via the standard doubling trick (an open chain of length \(N\) unfolds to a periodic chain of length \(2N\) with a reflection symmetry; a boundary-touching subsystem of length \(L\) maps to half of a periodic interval of length \(2L\), giving \((c/6)\log[(2N/\pi)\sin(\pi L/N)]\)).
Two independent validations¶
1. Computation check: dense_evolution.partial_trace/von_neumann_entropy on the many-body Lanczos ground state is cross-checked against a completely independent method -- free-fermion (Jordan-Wigner + Bogoliubov-de Gennes) exact diagonalization via Peschel's formula (J. Phys. A 36, L205, 2003) for the entanglement entropy of a Gaussian fermionic state through its Majorana covariance matrix. The Majorana-correlator algebra was self-tested against brute-force many-body ED at N=6 (diff ~1e-15) before being trusted at N=12. At N=12, the two independent methods agree to ~1e-10 across every L tested.
2. Critical-point check -- a real methodological pitfall found and fixed: ising_exact_verification.py's g*=0.8600 is the finite-size susceptibility-peak location -- the right point for that script's <ZZ> ansatz-tracking purpose, but not the same as the textbook self-dual CFT critical point g=1.0 (H = -sum ZZ - g*sum X is Kramers-Wannier self-dual at g=1 in the thermodynamic limit). At finite N these two different notions of "critical" do not coincide.
Result¶
| g | meaning | extracted c | R² | vs theory (0.5) |
|---|---|---|---|---|
| 1.0 | self-dual CFT point | 0.565 | 0.999924 | Δ=0.065 |
| 0.86 | susceptibility peak (finite-size) | 0.983 | 0.999997 | Δ=0.483 |
| 1.8 | off-critical (negative control) | 0.014 | 0.912692 | -- (should not fit) |
Fitting at g*=0.86 gives an almost perfectly clean fit (R²=0.999997) to a wrong answer -- extracted c≈0.98, roughly double the true 0.5. It looks even more convincing than the correct point's fit, which is exactly why this is worth documenting: a high R² alone does not mean the extracted physics is right if the wrong reference point was used. At the true self-dual point g=1.0, extracted c=0.565 -- much closer to 0.5, with the residual ~13% gap a plausible finite-size correction at this modest N=12 (the CFT formula is an asymptotic large-N/L result). The off-critical negative control behaves correctly: near-zero extracted c and a visibly worse fit (R²=0.91), consistent with area-law saturation instead of CFT log-scaling.
Status¶
Confirmed, not a confound (unlike Experiment 35) -- the CFT prediction genuinely holds at the correct critical point, and the methodological lesson (susceptibility-peak != CFT point at finite size) is itself a real, reusable finding for any future work in this repo that needs "the critical point" of this specific N=12 open TFIM chain.
Not yet done: validating dense_evolution.MPSSimulator's JSD-budget bond-truncation against this benchmark (does truncation bias the extracted central charge?) -- the original motivation for building this. Left as a follow-up: doing it properly requires isolating truncation error from state-preparation error (MPS is circuit-based, so reaching the ground state needs a real preparation circuit, e.g. VQE or Trotterized adiabatic evolution, not just loading the exact statevector), which is a separate, real piece of work.