The topological gap at criticality: scaling exponent d + {\eta}, universality, and scope

TL;DR

This paper introduces a finite-size scaling framework for the topological gap in spin models, revealing critical exponents consistent with known universality classes and identifying scope limitations related to corrections to scaling.

Contribution

It establishes a new scaling law for the topological gap at criticality, connecting it to universality classes and clarifying scope boundaries based on correction types.

Findings

The topological gap scaling exponent matches theoretical predictions for 2D Ising and Potts models.

The framework fails for models with logarithmic corrections like 2D Potts q=4.

Density normalization can recover scaling behavior in some cases.

Abstract

The topological gap $\Delta = TP_{H_1}^{real} - TP_{H_1}^{shuf}$ -- 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: $\Delta(L,T) = A L^{d+\eta} G_-(L|t/T_c|)$, with $G_-(x) \sim (1+x/x_0)^{-(1+\beta/\nu)}$. For 2D Ising, $\alpha = 2.249 \pm 0.038$, matching $d+\eta = 9/4$ to $0.03\sigma$; the $G_-$ exponent $\gamma = 1.089 \pm 0.077$ is consistent with $1+\beta/\nu = 9/8$ ($\Delta R^2 < 10^{-5}$). For 2D Potts $q=3$ with $L$ up to 1024, $\alpha = 2.272 \pm 0.024$ ($0.2\sigma$ from $d+\eta = 2.267$), with two-term corrections to scaling ($R^2 = 0.9999$). The $G_-$ exponent $\gamma = 1.114$ (68% CI $[1.053, 1.173]$) matches $1+\beta/\nu = 17/15$. Scope boundaries: the law fails for 2D Potts $q=4$ ($\alpha = 2.347 \pm 0.017$, $9.3\sigma$ from $d+\eta = 5/2$) where logarithmic corrections prevent convergence, and for raw 3D Ising ($4\sigma$ from $d+\eta$), but density normalization $\Delta/|M|^{1/2}$ recovers $\alpha = 3.06 \pm 0.04$ ($0.6\sigma$). The framework fails for first-order, BKT, and percolation. The criterion: $\alpha = d+\eta$ holds when corrections to scaling are algebraic ($\omega > 0$) but fails when logarithmic ($\omega \to 0$).