The topological gap at criticality: scaling exponent d + η, universality, and scope
This work provides a novel topological framework for analyzing critical phenomena in statistical physics, though it is incremental in refining scaling laws for specific spin models.
The paper tackled the problem of characterizing critical correlations in spin models using a topological gap, establishing finite-size scaling laws and validating them with high precision for 2D Ising and Potts models, achieving deviations as low as 0.03σ and 0.2σ from theoretical predictions. It identified scope boundaries where the law fails, such as for models with logarithmic corrections.
The topological gap $Î= TP_{H_1}^{real} - TP_{H_1}^{shuf}$ -- the excess $H_1$ total persistence of the majority-spin alpha complex over a density-matched null -- encodes critical correlations in spin models. We establish finite-size scaling: $Î(L,T) = A L^{d+η} G_-(L|t/T_c|)$, with $G_-(x) \sim (1+x/x_0)^{-(1+β/ν)}$. For 2D Ising, $α= 2.249 \pm 0.038$, matching $d+η= 9/4$ to $0.03Ï$; the $G_-$ exponent $γ= 1.089 \pm 0.077$ is consistent with $1+β/ν= 9/8$ ($ÎR^2 < 10^{-5}$). For 2D Potts $q=3$ with $L$ up to 1024, $α= 2.272 \pm 0.024$ ($0.2Ï$ from $d+η= 2.267$), with two-term corrections to scaling ($R^2 = 0.9999$). The $G_-$ exponent $γ= 1.114$ (68% CI $[1.053, 1.173]$) matches $1+β/ν= 17/15$. Scope boundaries: the law fails for 2D Potts $q=4$ ($α= 2.347 \pm 0.017$, $9.3Ï$ from $d+η= 5/2$) where logarithmic corrections prevent convergence, and for raw 3D Ising ($4Ï$ from $d+η$), but density normalization $Î/|M|^{1/2}$ recovers $α= 3.06 \pm 0.04$ ($0.6Ï$). The framework fails for first-order, BKT, and percolation. The criterion: $α= d+η$ holds when corrections to scaling are algebraic ($Ï> 0$) but fails when logarithmic ($Ï\to 0$).