The One-Dimensional ANNNI model in a Transverse Field: Analytic and numerical study of Effective Hamiltonians

TL;DR

This paper develops an effective fermion Hamiltonian approach to analyze the 1D ANNNI model in a transverse field, enabling exact diagonalization and detailed study of phase transitions and critical properties.

Contribution

It introduces a proper fermion scheme for the ANNNI model, improving the analysis of phase transitions and critical behavior beyond previous free fermion approximations.

Findings

Confirmed predictions of the effective fermion Hamiltonian with systems up to size 32.

Analyzed phase diagram and critical properties of the model.

Provided insights into the floating-phase/anti-phase transition.

Abstract

We consider a spin-$\frac{1}{2}$ chain with competing nearest and next-nearest neighbor interactions within a transverse magnetic field, which is known to be an equiavelent to the ANNNI model. When studing thermodynamics of the 2D ANNNI model Villain and Bak arrived to a free fermion approximation that neglects heavy excitations from the ferromagnetic ground state, which is an appropriate description close to the paramagnetic-ferromagnetic transition. In the vicinity of the floating-phase/anti-phase transition another sort of quasiparticles, but free fermions too, appears to be convenient. Although free fermions are a suitable tool for investigation of the phase diagram and the critical properties, they are defined on the fictitious lattice which makes the analysis non-rigorous. Here we deal with a proper fermion scheme which is especially effective %devised to describe the floating-phase/anti-phase transition. for performing exact diagonalization calculations for cyclic chains. Systems up to size $L=32$ has been analysed and the predictions of the effective fermion Hamiltonian has been confirmed. Various predictions for the infinite system and the critical properties are derived.