The Zero Temperature Heisenberg Ferromagnet as a Field Theory

Abstract

We envision an infinite lattice in N dimensions whose points are isomorphic to ℒ^⊗ N. At each lattice site there is a degree of freedom known as spin. The spin can take two values which we call ±½ħ. Hereafter ħ = 1, the natural units. A state of the system is given by specifying the spin (±½) at each lattice point together with an amplitude there, i.e., a real number at each site. The idealized interaction is given through a generally unbounded operator, the Hamiltonian, H, which is, in the algebraic framework, the generator of the time evolution automorphism. This simplified picture, together with a particular form for H, due originally to Heisenberg, is an idealization of the zero temperature ferromagnet where the crystal ions are frozen in place.

work partially supported by a USAFOSR grant