Spacetime Approach to Phase Transitions

TL;DR

This paper explores a spacetime-based geometric approach to understanding phase transitions in various models, linking critical phenomena to fractal structures and percolation theory.

Contribution

It introduces a novel geometric framework using Feynman's sum-over-paths and high-temperature expansion to analyze phase transitions across multiple physical systems.

Findings

Critical exponents relate to fractal structures of graphs.

Graphs percolate at the critical point, enabling geometric analysis.

Applicable to spin models, $\,\, ext{phi}^4$ theory, Bose-Einstein condensation, Higgs, and U(1) gauge transitions.

Abstract

In these notes, the application of Feynman's sum-over-paths approach to thermal phase transitions is discussed. The paradigm of such a spacetime approach to critical phenomena is provided by the high-temperature expansion of spin models. This expansion, known as the hopping expansion in the context of lattice field theory, yields a geometric description of the phase transition in these models, with the thermal critical exponents being determined by the fractal structure of the high-temperature graphs. The graphs percolate at the thermal critical point and can be studied using purely geometrical observables known from percolation theory. Besides the phase transition in spin models and in the closely related $\phi^4$ theory, other transitions discussed from this perspective include Bose-Einstein condensation, and the transitions in the Higgs model and the pure U(1) gauge theory.