The continuum limit of the infinite isotropic Heisenberg chain in its ground state representation

Abstract

The Hamiltonian of the infinite spin- 1 2 isotropic Heisenberg chain with lattice spacing ϵ and nearest-neighbor interaction, corresponding to a ground state representation of the quasilocal algebra, acts on a Hilbert space that can be naturally embedded in a continuum Fock space. It is proved that as ϵ → 0 this Hamiltonian converges on the latter space in the strong resolvent sense to a Hamiltonian describing a gas of free non-relativistic equal mass bosons. It is also shown that the projection on the spin wave sectors containing bound magnons strongly converges to zero as ϵ → 0, even though the bound magnon energies converge as ϵ → 0.