Spectral Signatures of Third-Order Pseudo-Transitions in Finite Systems: An Eigen-Microstate Approach

TL;DR

This paper introduces a spectral method to identify third-order pseudo-transitions in finite systems by analyzing eigen-microstates, providing a geometric and order-parameter-free approach to structural criticality.

Contribution

The authors develop a spectral generalized response framework that detects higher-order anomalies and distinguishes fluctuation modes without relying on microcanonical entropy.

Findings

Extrema of the ratio R_3 track higher-order anomalies in models.

Spectral projection differentiates dependent and independent fluctuation branches.

Effective spectral dimension R_eff characterizes participation background in anomalies.

Abstract

Third-order pseudo-transitions in finite systems reflect reorganization beyond conventional criticality, yet their identification usually relies on microcanonical entropy, which is often inaccessible in practice. Here we introduce a spectral generalized response within the eigen-microstate framework. From the distribution of normalized spectral weights, we construct the third-order ratio $R_3=K_3/(K_2)^3$, which probes asymmetric redistribution among fluctuation modes beyond leading-mode condensation. Across Ising and Potts models on regular lattices and random regular networks, extrema of $R_3$ consistently track higher-order anomalies. Combined with spectral projection, the method further distinguishes dependent and independent branches: the former remain tied to the dominant ordering channel, whereas the latter arise from redistribution within the subleading fluctuation subspace. The effective spectral dimension $R_{\mathrm{eff}}$ provides the participation background in which these anomalies develop. These results establish a geometric characterization of third-order pseudo-transitions as reorganizations of statistical weight in configuration space and provide an order-parameter-free route to finite-size structural criticality.