Microscopic theory of phase transitions in a critical region

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.

23 pages

References in corpus (8)

• Finite Temperature Models of Bose-Einstein Condensation
• Superfluid behaviour of a two-dimensional Bose gas
• Critical Behavior of a Trapped Interacting Bose Gas
• Competing Ferromagnetic and Charge-Ordered States in Models for Manganites: the Origin of the CMR Effect
• Analytical theory of mesoscopic Bose-Einstein condensation in ideal gas
• Preparation of atomic Fock states by trap reduction
• Microscopic theory of a phase transition in a critical region: Bose-Einstein condensation in an interacting gas
• Self-similar analytical solution of the critical fluctuations problem for the Bose-Einstein condensation in an ideal gas

Cited by in corpus (4)

• On the energy cascade of 3-wave kinetic equations: Beyond Kolmogorov-Zakharov solutions
• Bose-Einstein condensate fluctuations versus an interparticle interaction
• Quantum supremacy of the many-body fluctuations in the occupations of the excited particle states in a Bose-Einstein-condensed gas
• Exact recursive calculation of circulant permanents: A band of different diagonals inside a uniform matrix