The one-dimensional ANNNI model in a transverse field: Analytic and numerical study of effective Hamiltonians

We consider a quantum spin-1/2 Ising chain with competing nearest and next-nearest neighbor interactions in a transverse magnetic field, which is known to be equivalent to the classical two-dimensional ANNNI model. Within a perturbation theory for small transverse field (corresponding to low temperatures in the classical ANNNI model) we derive two effective Hamiltonians: the free model describing free fermions on a fictitious lattice that excludes particular heavy excitations of the original system; and the complete model, which incorporates creation and annihilation of these fermions. Whereas the former possesses only three phases (ferromagnetic, floating and anti phase), the latter contains the full physics of the 2d ANNNI model, including a paramagnetic phase between the ferromagnetic and floating phase and a Kosterlitz-Thouless transition. New analytic results are derived for the free model, e.g. the excitation spectrum that turns out to be non-trivial. Our effective Hamiltonians are defined on a restricted Hilbert space so that exact diagonalization calculations can be done for much larger system sizes. Results from extensive Lanczos calculations for system sizes up to L=32 are presented confirming the original predictions from Villain and Bak.