# ANNNI Model in Transverse Field

## Abstract

Competing interactions between spins or frustration give rise to intriguing many-body states with spatially modulated spin structures. The simplest model with the regular frustration is the classical axial next nearest neighbour Ising (ANNNI) model. The classical ANNNI model is described by a system of Ising spins with nearest neighbour interactions along all the lattice directions (*x*, *y* and *z*) as well as a competing next nearest neighbour interaction in one axial direction (*z* for instance). Chapter 4 discusses detailed results on quantum ANNNI models in a transverse field with a brief introduction to the classical model. Analytic studies on the basis of an interacting fermion representation of the model, real-space and field theoretic renormalisation group techniques, the numerical exact diagonalisation method, and Monte Carlo simulations have revealed a variety of ground-state phases of the quantum ANNNI chain. These are mentioned in Chap. 4. Studies of higher dimensional quantum ANNNI models are also presented there. Appendices of this chapter include details of approximate methods used in the study of the quantum ANNNI chain.

## Keywords

Paramagnetic Phase Quantum Fluctuation Neighbour Interaction Transverse Field Ising System## References

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