Microscopic theory of phase transitions in a critical region

TL;DR

This paper develops a comprehensive microscopic theory for phase transitions at critical points, providing exact equations for Bose-Einstein condensation and magnetic transitions, including an outline for the 3D Ising model.

Contribution

It introduces a general, exact microscopic framework for phase transitions, extending existing equations to the critical region for Bose-Einstein condensation and ferromagnetism.

Findings

Exact equations for Green functions and order parameters across the critical region.

Critical-region extension of Beliaev-Popov and Gross-Pitaevskii equations.

Outline of an exact solution for the 3D Ising model.

Abstract

The problem of finding a microscopic theory of phase transitions across a critical point is a central unsolved problem in theoretical physics. We find a general solution to that problem and present it here for the cases of Bose-Einstein condensation in an interacting gas and ferromagnetism in a lattice of spins, interacting via a Heisenberg or Ising Hamiltonian. For Bose-Einstein condensation, we present the exact, valid for the entire critical region, equations for the Green functions and order parameter, that is a critical-region extension of the Beliaev-Popov and Gross-Pitaevskii equations. For the magnetic phase transition, we find an exact theory in terms of constrained bosons in a lattice and obtain similar equations for the Green functions and order parameter. In particular, we outline an exact solution for the three-dimensional Ising model.