Fisher Curvature Scaling at Critical Points: An Exact Information-Geometric Exponent from Periodic Boundary Conditions

Abstract

We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a $d$-dimensional lattice with periodic boundary conditions and $n = L^d$ sites, the Fisher manifold has $m = d \cdot n$ dimensions (one per bond), and we find $|\mathcal{R}(J_c)| \sim n^{d_R}$ with $d_R = (dν+ 2η)/(dν+ η)$, where $ν$ and $η$ are the correlation-length and anomalous-dimension critical exponents. For 2D Ising ($ν= 1$, $η= 1/4$), this predicts $d_R = 10/9$, confirmed by exact transfer-matrix computations ($L = 6$--$9$: $d_R = 1.1115 \pm 0.0002$) and multi-seed MCMC through $L = 24$. For 3D Ising ($ν= 0.630$, $η= 0.0363$), the prediction $d_R = 1.019$ is consistent with MCMC on $L^3$ tori up to $L = 10$ (power-law fit: $d_R = 1.040$). For 2D Potts $q = 3$ (predicted $33/29 \approx 1.138$), FFT-MCMC through $L = 40$ shows $d_\mathrm{eff}$ oscillating non-monotonically around $\sim 1.20$, consistent with $O(1/(\ln L)^2)$ logarithmic corrections. For $q = 4$ (predicted $22/19$), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity $R_3 = -R_1/2$, $R_4 = -R_2/2$ holds to 5--6 digits for all models. This exponent is distinct from Ruppeiner thermodynamic curvature and reflects the collective geometry of the growing Fisher manifold. We provide falsification criteria and predictions for additional universality classes.

Figures (3)

• Figure 1: Log-log plot of $|R(J_c)|$ vs $n=L^2$ for 2D Ising on $L\times L$ tori. Filled circles: exact TM ($L=3$--$9$). Open circles with error bars: multi-seed MCMC ($L=10$--$24$, 3 seeds each). Solid line: $d_{R} = 10/9$ prediction. Inset: effective exponent $d_\mathrm{eff}(L,L+1)$ vs $L$, converging to the prediction (horizontal dashed).
• Figure 2: Effective exponent $d_\mathrm{eff}$ versus consecutive pair index for four universality classes. Horizontal lines mark theoretical predictions $d_{R}$. 2D Ising (circles) has converged to its prediction to $<0.01\%$. 3D Ising (squares) shows monotonic convergence toward its prediction. 2D Potts $q = 3$ (triangles) and $q = 4$ (diamonds) are still converging toward their respective predictions.
• Figure 3: Ashkin-Teller continuous-family test. Predicted $d_{R}(\lambda)$ from the exact Coulomb gas $\nu(\lambda)$ (solid curve) compared with exact TM $d_\mathrm{eff}$ at $L = 3$--$5$ (symbols) for five $\lambda$ values along the self-dual line. All $d_\mathrm{eff}$ are above the prediction and monotonically decreasing with $L$, consistent with convergence.