How much of persistent homology is topology? A quantitative decomposition for spin model phase transitions

Matthew Loftus

cond-mat.stat-mechMar 30, 2026

How much of persistent homology is topology? A quantitative decomposition for spin model phase transitions

Authors: Matthew Loftus

Organizations: Cedar Loop LLC, Washington, USA

Abstract

Point-cloud persistent homology (PH) -- computing alpha or Rips complexes on spin-position point clouds -- has been widely applied to detect phase transitions in classical spin models since Donato et al. (2016), with subsequent studies attributing the detection to the topological content of the persistence diagram. We ask a simple question that has not been posed: what fraction of the PH signal is genuinely topological? We introduce f_topo, a quantitative decomposition that separates the density-driven and topological contributions to any PH statistic by comparing real spin configurations against density-matched shuffled null models. Across the 2D Ising model (system sizes L = 16-128, ten temperatures) and Potts models (q = 3, 5), we find that H_0 statistics -- total persistence, persistence entropy, feature count -- are 94-100% density-driven (f_topo < 0.07). The density-matched shuffled null detects T_c at the identical location and with comparable peak height as real configurations, showing that density alone is sufficient for phase transition detection. However, H_1 statistics are partially topological: the topological fraction grows with system size as delta(TP_{H_1}) ~ L^{0.53} and follows a finite-size scaling collapse delta(T, L) = L^{0.53} g(tL^{1/nu}) with collapse quality CV = 0.27. The longest persistence bar is strongly topological (f_topo > 1) and scales with the correlation length. A scale-resolved analysis reveals that the topological excess shifts from large-scale to small-scale features as L increases. We propose that the TDA-for-phase-transitions community adopt shuffled null models as standard practice, and that H_1 rather than H_0 statistics be used when genuine topological information is sought.

Figures & tables

Figure 1: Topological fraction $f_{\mathrm{topo}}$ for each PH statistic at $T_{\mathrm{c}}$ , $L=64$ . Blue bars: standard PH statistics. Red/green/yellow: per-bar split. $H_{0}$ statistics (TP, PE, $n$ ) have $f_{\mathrm{topo}}<0.06$ . The maximum bar has $f_{\mathrm{topo}}>1$ (spatial correlations enhance it beyond the random baseline). $H_{1}$ count has $f_{\mathrm{topo}}=0.28$ .

Statistic	$L\!=\!16$	$L\!=\!24$	$L\!=\!32$	$L\!=\!48$	$L\!=\!64$	$L\!=\!128$
$\mathrm{TP}_{H_{0}}$	0.008	0.015	0.021	0.040	0.037	0.065
$\mathrm{TP}_{H_{1}}$	0.044	0.071	0.093	0.123	0.133	0.186
PE	0.055	0.050	0.052	0.056	0.053	0.051
$n_{H_{0}}$	0.000	0.000	0.000	0.000	0.000	0.000
$n_{H_{1}}$	0.148	0.187	0.224	0.270	0.282	0.345
$\delta(\max)$	0.77	1.77	2.58	3.50	4.67	8.07

Table 1: Topological fraction

f_{\mathrm{topo}}

T_{\mathrm{c}}

for the 2D Ising model. Errors (not shown for readability) are

\pm

0.003–0.010 from bootstrap. The max bar row uses the relative excess

\delta

[Eq. ( 2 )] instead of

f_{\mathrm{topo}}

, since

f_{\mathrm{topo}}>1

when

S_{\mathrm{shuf}}<S_{\mathrm{rand}}

(see text).

$L$	Peak $T$		Peak height		Ratio
	Real	Shuf	Real	Shuf	S/R
32	2.325	2.325	0.328	0.329	1.005
64	2.284	2.284	0.544	0.614	1.128
128	2.284	2.284	1.009	1.107	1.097

Table 2: Peak location and height of

|d(\mathrm{TP}_{H_{0}}/L^{2})/dT|

for real and shuffled configurations.

Figure 2: The shuffled null detects $T_{\mathrm{c}}$ . Top: $\mathrm{TP}_{H_{0}}/L^{2}$ vs. temperature for real (blue), shuffled (orange), and random (green) point clouds. Bottom: $|d(\mathrm{TP}_{H_{0}}/L^{2})/dT|$ peaks at the same temperature bin for real and shuffled at all three system sizes. Peak heights agree to within 13%. The random null (fixed $\rho=0.5$ ) also peaks near $T_{\mathrm{c}}$ at $L=128$ .

Figure 3: Scaling of the topological fraction with system size at $T_{\mathrm{c}}$ . Left: $f_{\mathrm{topo}}(\mathrm{TP}_{H_{1}})$ grows with $L$ ; a power law $L^{0.67}$ (red dashed) and saturation model (blue dotted) both fit the data and predict topology dominates $H_{1}$ at $L\to\infty$ . Center: $f_{\mathrm{topo}}(n_{H_{1}})$ shows the same growth. Right: $f_{\mathrm{topo}}(\mathrm{TP}_{H_{0}})$ remains small at all $L$ . Triangles mark extrapolated values at $L=256$ and $512$ .

Figure 4: Left: $\mathrm{TP}_{H_{0}}^{\mathrm{shuf}}/L^{2}$ vs. majority fraction $\rho$ for all 60 $(L,T)$ combinations (six system sizes, ten temperatures). All points collapse onto a single curve $0.245\,\rho^{0.83}$ (dashed, $R^{2}=0.996$ ). Center: real, shuffled, and theoretical prediction for $L=64$ . Right: residuals from the universal curve, all below 2.5%.

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

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

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

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