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The Scaling Closure Theorem: Fisher Curvature Exponents from CFT Structure at Statistical Critical Points

Maxim Zhuravlev

Published Mar 10, 2026• 0 citations
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Abstract

Proves the Scaling Closure Theorem: d_R = (d*nu + 2*eta)/(d*nu + eta) from five structural assumptions (A1-A5) about the microscopic Fisher manifold at criticality. A1 (Ricci decomposition) and A2 (Z_2 fusion) and A4 (eigenvalue anisotropy) are proved as theorems. A3 (diagonal block cancellation) at 99.95%+ and A5 (triangle non-cancellation) at 99.9%+. Universality verified across 8 CFT classes including 2D Ising, Potts q=3,4, 3D Ising/XY/Heisenberg, BKT, and Gaussian. Companion proof paper for the Prediction Letter (arXiv:2603.07651).

DOI
10.5281/zenodo.18942520

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Finite-size scaling in the canonical ensemble

Youjin Deng, Henk W. J. Blöte

We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent $y_1$ that depends on the temperature exponent $y_t$ and the dimensionality $d$ as $y_1=-|2y_t-d|$. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.

arXiv:cond-mat/0508348•Aug 15, 2005•1 citation
Proves the Scaling Closure Theorem: d_R = (dnu + 2eta)/(d*nu + eta) from five structural assumptions (A1-A5) about the microscopic Fisher manifold at criticality. A1 (Ricci decomposition) and A2 (Z_2 fusion) and A4 (eigenvalue anisotropy) are proved as theorems. A3 (diagonal block cancellation) at 99.95%+ and A5 (triangle non-cancellation) at 99.9%+. Universality verified across 8 CFT classes including 2D Ising, Potts q=3,4, 3D Ising/XY/Heisenberg, BKT, and Gaussian. Companion proof paper for the Prediction Letter (arXiv:2603.07651).
We study the scalar curvature of the Fisher information metric on the microscopic coupling-parameter manifold of lattice spin models at criticality. For a ddd-dimensional lattice with periodic boundary conditions and n=Ldn = L^dn=Ld sites, the Fisher manifold has m=d⋅nm = d \cdot nm=d⋅n dimensions (one per bond), and we find ∣R(Jc)∣∼ndR|\mathcal{R}(J_c)| \sim n^{d_R}∣R(Jc​)∣∼ndR​ with dR=(dν+2η)/(dν+η)d_R = (dν+ 2η)/(dν+ η)dR​=(dν+2η)/(dν+η), where ννν and ηηη are the correlation-length and anomalous-dimension critical exponents. For 2D Ising (ν=1ν= 1ν=1, η=1/4η= 1/4η=1/4), this predicts dR=10/9d_R = 10/9dR​=10/9, confirmed by exact transfer-matrix computations (L=6L = 6L=6--999: dR=1.1115±0.0002d_R = 1.1115 \pm 0.0002dR​=1.1115±0.0002) and multi-seed MCMC through L=24L = 24L=24. For 3D Ising (ν=0.630ν= 0.630ν=0.630, η=0.0363η= 0.0363η=0.0363), the prediction dR=1.019d_R = 1.019dR​=1.019 is consistent with MCMC on L3L^3L3 tori up to L=10L = 10L=10 (power-law fit: dR=1.040d_R = 1.040dR​=1.040). For 2D Potts q=3q = 3q=3 (predicted 33/29≈1.13833/29 \approx 1.13833/29≈1.138), FFT-MCMC through L=40L = 40L=40 shows deffd_\mathrm{eff}deff​ oscillating non-monotonically around ∼1.20\sim 1.20∼1.20, consistent with O(1/(ln⁡L)2)O(1/(\ln L)^2)O(1/(lnL)2) logarithmic corrections. For q=4q = 4q=4 (predicted 22/1922/1922/19), effective exponents oscillate with strong logarithmic corrections. The Ricci decomposition identity R3=−R1/2R_3 = -R_1/2R3​=−R1​/2, R4=−R2/2R_4 = -R_2/2R4​=−R2​/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.
We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a finite-size effect governed by an exponent y1y_1y1​ that depends on the temperature exponent yty_tyt​ and the dimensionality ddd as y1=−∣2yt−d∣y_1=-|2y_t-d|y1​=−∣2yt​−d∣. We verify this prediction by a Monte Carlo analysis of several two-dimensional lattice models in the percolation, the Ising and the 3-state Potts universality classes.