Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2006-12-27
Physics
Condensed Matter
Statistical Mechanics
59 pages, 18 figures. Write-up of Ising Lectures presented at the National Academy of Sciences, Lviv, Ukraine, 2004. 2nd versi
Scientific paper
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.
Janke Wolfhard
Schakel Adriaan M. J.
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