Criticality Beyond Nonanalyticity: Intrinsic Microcanonical Signatures of Phase Transitions

TL;DR

This paper demonstrates that criticality in phase transitions can be identified through intrinsic features in microcanonical entropy derivatives at finite sizes, providing an order-parameter-free perspective that precedes traditional singularities.

Contribution

It introduces a microcanonical inflection-point analysis method to detect phase transition signatures at finite sizes, challenging the conventional reliance on nonanalyticities in thermodynamic potentials.

Findings

Microcanonical entropy derivatives reveal intrinsic structures indicating criticality.

Inflection points and peaks in entropy derivatives define pseudocritical trajectories.

These structures sharpen and evolve with system size, culminating in the thermodynamic limit.

Abstract

Phase transitions are conventionally defined by nonanalyticities of thermodynamic potentials in the thermodynamic limit. In this Letter, we show that the singularity is not the definition of criticality but its asymptotic outcome: criticality is already written in the microcanonical entropy derivatives at any finite size as intrinsic morphological structures -- inflection points and extrema. The singularity is then the endpoint of a sharpening process that evolves with increasing system size. Combining microcanonical inflection-point analysis (MIPA) with the Berlin-Kac spherical model -- for which the microcanonical density of states is known in closed form at every finite $N$ -- we systematically identify these structures in the energy profiles of entropy derivatives that encode the transition. An inflection point in the inverse temperature $\beta_N(\epsilon)=\partial_\epsilon S_N$ and a pronounced peak in its derivative $\gamma_N(\epsilon)=\partial^2_\epsilon S_N$ define a well-controlled pseudocritical trajectory whose controlled sharpening and drift culminate in the macroscopic cusp at the critical energy $\epsilon_c$ in the thermodynamic limit. This establishes an intrinsic, order-parameter-free notion of criticality that precedes its singular asymptotic representation.