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The topological gap at criticality: scaling exponent d + η, universality, and scope

Matthew Loftus

Abstract

The topological gap $Δ= TP_{H_1}^{real} - TP_{H_1}^{shuf}$ -- 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: $Δ(L,T) = A L^{d+η} G_-(L|t/T_c|)$, 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 $0.03σ$; the $G_-$ exponent $γ= 1.089 \pm 0.077$ is consistent with $1+β/ν= 9/8$ ($ΔR^2 < 10^{-5}$). For 2D Potts $q=3$ with $L$ up to 1024, $α= 2.272 \pm 0.024$ ($0.2σ$ 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 $q=4$ ($α= 2.347 \pm 0.017$, $9.3σ$ from $d+η= 5/2$) where logarithmic corrections prevent convergence, and for raw 3D Ising ($4σ$ from $d+η$), but density normalization $Δ/|M|^{1/2}$ recovers $α= 3.06 \pm 0.04$ ($0.6σ$). The framework fails for first-order, BKT, and percolation. The criterion: $α= d+η$ holds when corrections to scaling are algebraic ($ω> 0$) but fails when logarithmic ($ω\to 0$).

The topological gap at criticality: scaling exponent d + η, universality, and scope

Abstract

The topological gap -- the excess total persistence of the majority-spin alpha complex over a density-matched null -- encodes critical correlations in spin models. We establish finite-size scaling: , with . For 2D Ising, , matching to ; the exponent is consistent with (). For 2D Potts with up to 1024, ( from ), with two-term corrections to scaling (). The exponent (68% CI ) matches . Scope boundaries: the law fails for 2D Potts (, from ) where logarithmic corrections prevent convergence, and for raw 3D Ising ( from ), but density normalization recovers (). The framework fails for first-order, BKT, and percolation. The criterion: holds when corrections to scaling are algebraic () but fails when logarithmic ().

Paper Structure

This paper contains 17 sections, 8 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Models
  4. The scaling exponent is $d + \eta$
  5. 2D Ising: precise measurement
  6. Corrections to scaling
  7. Temperature dependence
  8. Per-statistic decomposition
  9. The $G_-$ scaling function
  10. Universality: 2D Potts $q = 3$
  11. Scope: where the scaling fails
  12. The marginal case: Potts $q = 4$
  13. 3D Ising: density dilution and its resolution
  14. Berezinskii-Kosterlitz-Thouless transitions
  15. 2D percolation
  16. ...and 2 more sections

Figures (4)

  • Figure 1: (a) Log-log plot of $\Delta(\mathrm{TP}_{H_1})$ vs $L$ for the 2D Ising model at $T_c$. The red line shows the best fit $L^{2.249}$; the blue dashed line is the extensive reference $L^2$; the green dotted line is the original value $L^{2.42}$. (b) Running exponent (local log-log slope between consecutive $L$ pairs). The effective exponent converges to $d + \eta = 9/4$ (red dashed) at large $L$.
  • Figure 2: Potts $q = 3$ scaling collapse: $\Delta/L^{2.506}$ versus $L|t|$ for four system sizes. The black curve is the free $G_-$ fit with $\gamma = 1.114$.
  • Figure 3: 3D Ising: density dilution and its resolution. (a) Log-log $\Delta$ vs $L$ at $T_c$: the raw gap (blue) scales as $L^{2.78}$ ($4\sigma$ from $d+\eta$); the normalized gap $\Delta/|M|^{1/2}$ (red) scales as $L^{3.06}$ ($0.6\sigma$ from $d+\eta = 3.036$). (b) Running exponent: the raw (blue squares) drops from 3.17 to 1.79; the normalized (red circles) stays flat near $d+\eta$. (c) $\alpha(p)$ as a function of the normalization power $p$: the exponent crosses $d+\eta$ near $p = 1/2$, matching the prediction $\alpha_{\rm raw} + p\,\beta/\nu$.
  • Figure 4: Measured $\alpha$ (circles/squares with error bars) vs. predicted $d + \eta$ (triangles) for all models. Green: confirmed ($<1\sigma$). Grey: rejected. The 2D Ising and Potts $q = 3$ results match $d + \eta$ to $0.03\sigma$ and $0.2\sigma$ respectively; the $q = 4$ model is rejected at $9.3\sigma$.