Paul Federbush
For the Quantum Heisenberg Ferromagnet, Tao to the
Proof of a Phase Transition
(15K, LaTeX)
ABSTRACT. We present the outline of a proof for the 3-d
phase transition which we hope to carry forth.
At the same time this paper provides some physical
understanding of the phase transition, in the flavor
of relatively simple arguments from an undergraduate
statistical mechanics course. A number of directions
for mathematical research, interesting in their own
right, will be suggested by aspects of the development.
We hope and believe that readers will be enticed by
the naturalness and beauty of the path; some perhaps
even, big game veterans, sniffing the quarry, will be
ready to join the hunt.
The central construct views the trace, Tr(exp(-beta*H)),
as a gas of polymers, each representing a cycle in the
permutation group, with hard core interactions. The
activities of the polymers have expressions as arising
from the main conjecture of the paper. The estimates lead
to a phase transition in 3-d, but not in 2-d. This occurs
via the same argument that a random walk in 2-d has certain
return to the origin, but not so for a random walk in 3-d.