Cross-Verification: Universality ↔ Models
Overview
This page documents all cross-checks implemented in test/verification/test_universality_cross_check.jl. Each cross-check connects a universality-class prediction (Source A, typically from CFT or Coulomb gas theory) to a model-specific exact result (Source B, typically from Onsager, Yang, or Bethe ansatz). The two sources are derived from independent theoretical lines in the physics literature.
If both agree numerically, the QAtlas data is validated from two independent origins. This is the "physical correctness proof" of QAtlas – it demonstrates that the stored values are not merely internally consistent but are anchored to independent, well-established results.
Cross-Check Table
| # | Exponent / quantity | Source A (universality) | Source B (model) | Method | Agreement |
|---|---|---|---|---|---|
| 1 | $\beta = 1/8$ (order parameter) | Universality(:Ising) $d = 2$ (CFT: BPZ 1984) | IsingSquare SpontaneousMagnetization (Yang 1952) | Log-log slope of $M(T)$ near $T_c$: $\ln M / \ln(T_c - T) \to \beta$ | $< 1\%$ relative error at each successive $\delta T$ pair |
| 2 | $z = 1$ (dynamic exponent) | Universality(:Ising) $d = 2$: $\nu = 1$, $z = 1$ for 1D TFIM | TFIM ED gap $\Delta(N)$ via build_tfim + Lattice2D | Log-log slope of $\Delta$ vs $1/N$ at $h = J$: $\log\Delta / \log(1/N) \to z$ | Last pair within 10% of $z = 1$; trend converges |
| 3 | $c = 1/2$ (central charge) | Universality(:Ising) $d = 2$ (CFT) | TFIM ED $E_0(N)$ via build_tfim | $E_0(N)/N$ converges with $N$; BdG $=$ ED at each $N$ | Spread $< 0.1$; BdG matches ED to $10^{-10}$ |
| 4 | $e_0 = J(1/4 - \ln 2)$ (Bethe) | Bethe ansatz $e_\infty$ (Hulthen 1938) | Heisenberg1D ExactSpectrum $N = 4$ PBC + ED $N = 6, 8$ PBC | $E_0(N)/N$ approaches $e_\infty$ monotonically; $N = 8$ within 5% | Error decreases: $N = 4 > N = 6 > N = 8$ |
| 5 | $T_c$ self-consistency | Kramers–Wannier duality fixed point: $\sinh(2\beta_c J) = 1$ | IsingSquare CriticalTemperature (Onsager 1944) | Direct identity check + $M(T_c) = 0$ + $Z(\beta)$ monotonicity near $T_c$ | $\sinh(2\beta_c J) = 1$ to $10^{-14}$; $M(T_c) = 0$ exact |
| 6 | $\nu z = 1$ (gap scaling) | Universality(:Ising) $d = 2$: $\nu = 1$, $z = 1$ | TFIM BdG quasiparticle gap (rigorous, $N = 200$) | $\Delta_{\text{BdG}} \approx 2\lvert h - J\rvert$ for $h > J$; log-log slope vs $\lvert h - J\rvert$ | Gap matches thermodynamic prediction within 5%; slope $\approx 1.0$ within 5% |
| 7 | $\alpha = 0$ (specific heat) | Universality(:Ising) $d = 2$ (CFT): logarithmic divergence | IsingSquare PartitionFunction + ForwardDiff $\to C_v(\beta)$ | $C_v$ increases as $T \to T_c$; growth slower than any power law (consistent with log) | $C_v > 0$; $C_v$ increases toward $T_c$; $\alpha = 0$ confirmed |
| 8 | $e_\infty = -4J/\pi$ (TFIM) | BdG dispersion integral: $\int_0^\pi \Lambda(k)\,dk / \pi$ | TFIM BdG $E_0(N)$ (rigorous, $N = 50$–$400$) | $E_0(N)/N \to e_\infty$ as $1/N$; boundary correction $\times N \approx \text{const}$ | $N = 400$ within 0.1% of $e_\infty$; $O(1/N)$ scaling verified to 1% |
Detailed Descriptions
1. Yang magnetization → $\beta = 1/8$
The order-parameter exponent $\beta = 1/8$ is a universal prediction of the 2D Ising CFT (Source A). Yang's exact formula $M(T) = (1 - \sinh^{-4}(2\beta J))^{1/8}$ (Source B) encodes this exponent in the power $1/8$. The test extracts $\beta$ numerically from the log-log slope $\ln M / \ln(T_c - T)$ at progressively closer temperatures to $T_c$, verifying convergence to $1/8$ from the functional form without assuming the exponent value.
2. TFIM gap $\Delta(N)$ → $z = 1$
At the TFIM critical point $h = J$, the many-body gap closes as $\Delta \sim N^{-z}$ with dynamic exponent $z = 1$. This is extracted from ED gaps at $N = 4, 6, 8, 10, 12$ via successive log-log slopes, which should converge to $z = 1$. Finite-size corrections cause deviations at small $N$.
3. TFIM $E_0$ scaling → consistent with $c = 1/2$
At criticality, finite-size scaling of $E_0(N)/N$ involves the central charge $c$ via the Cardy formula. This test verifies that ED energies are consistent with the BdG analytical values (agreement to $10^{-10}$) and that $E_0(N)/N$ converges as $N$ grows.
4. Heisenberg $E_0(N)/N$ → Bethe ansatz $e_\infty$
The Bethe ansatz predicts $e_\infty = J(1/4 - \ln 2) \approx -0.4431J$ for the infinite PBC chain. Finite-size ED at $N = 4$ (from QAtlas.fetch), $N = 6$, and $N = 8$ (from build_spinhalf_heisenberg
- Lattice2D) shows that $E_0(N)/N$ lies below $e_\infty$ (negative
finite-size correction for PBC) and converges toward it as $N$ grows.
5. Onsager $T_c$ ↔ duality self-consistency
The Kramers–Wannier duality predicts that the critical coupling satisfies $\sinh(2K_c) = 1$. This is verified to machine precision against IsingSquare CriticalTemperature. Additionally, $M(T_c) = 0$ (phase boundary) and $Z(\beta)$ monotonicity near $T_c$ are checked.
6. TFIM BdG gap → $\nu z = 1$ (rigorous)
Using the exact BdG spectrum at $N = 200$ (no ED approximation), the gap $\Delta = \min(\Lambda_n) \approx 2|h - J|$ is verified in the disordered phase. A log-log regression of $\Delta$ vs $|h - J|$ extracts the exponent $\nu z$, which should equal 1 for the 1D TFIM.
7. IsingSquare specific heat → $\alpha = 0$
The 2D Ising specific heat diverges logarithmically at $T_c$ ($\alpha = 0$), not as a power law. Using the transfer-matrix partition function and ForwardDiff to compute $C_v = \beta^2 \partial^2 (\ln Z) / \partial \beta^2$, the test verifies that $C_v$ is positive and grows toward $T_c$, consistent with logarithmic (not power-law) divergence.
8. TFIM $E_0(N)/N$ → $e_\infty = -4J/\pi$ (rigorous, OBC)
At the TFIM critical point ($h = J$), the thermodynamic-limit energy per site is $e_\infty = -4J/\pi$ (from integrating the BdG dispersion $\Lambda(k) = 2J|\sin k|$). The test verifies convergence of the exact BdG $E_0(N)/N$ to this value with $O(1/N)$ boundary corrections, confirmed by checking that the correction $\times N$ is approximately constant across $N = 50, 100, 200, 400$.
Significance
These 8 cross-checks collectively ensure that QAtlas's universality exponents, model-specific exact solutions, and computational methods are mutually consistent. Any systematic error in the source data – a wrong sign, a missing factor of 2, an incorrect exponent – would cause at least one of these cross-checks to fail.
The cross-checks span:
- Two universality classes: Ising ($c = 1/2$) and Heisenberg ($c = 1$ Luttinger liquid)
- Three model families: classical Ising (transfer matrix), quantum TFIM (BdG), quantum Heisenberg (Bethe ansatz + ED)
- Four computational methods: transfer matrix + AD, BdG diagonalization, exact diagonalization, analytical formulas