For the Quantum Heisenberg Ferromagnet, Tao to the Proof of a Phase Transition

TL;DR

This paper outlines a proof of a phase transition in the 3-dimensional quantum Heisenberg ferromagnet, using a polymer lattice gas model and physical intuition, with implications for mathematical physics.

Contribution

It introduces a novel approach to proving phase transitions in quantum spin systems using polymer models and elementary arguments, bridging physics intuition and rigorous mathematics.

Findings

Phase transition occurs in 3-d but not in 2-d.

Polymer model effectively represents the trace of the Hamiltonian.

Physical arguments provide insight into the nature of the phase transition.

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 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 lattice 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 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.