Three dimensional generalization of the $J_1$-$J_2$ Heisenberg model on a square lattice and role of the interlayer coupling $J_c$

TL;DR

This study extends the two-dimensional $J_1$-$J_2$ Heisenberg model to three dimensions by including interlayer coupling $J_c$, analyzing its effects on magnetic order and fluctuations, with implications for iron pnictide compounds.

Contribution

The paper introduces a three-dimensional $J_1$-$J_2$-$J_c$ model and investigates its magnetic properties using self-consistent spin wave theory, highlighting the role of interlayer coupling.

Findings

Interlayer coupling $J_c$ suppresses quantum fluctuations.

Two scenarios for phase instability: vanishing magnetization or spin wave velocity.

Quantum fluctuations are stronger for $S=1/2$ than for $S=1$.

Abstract

A possibility to describe magnetism in the iron pnictide parent compounds in terms of the two-dimensional frustrated Heisenberg $J_1$-$J_2$ model has been actively discussed recently. However, recent neutron scattering data has shown that the pnictides have a relatively large spin wave dispersion in the direction perpendicular to the planes. This indicates that the third dimension is very important. Motivated by this observation we study the $J_1$-$J_2$-$J_c$ model that is the three dimensional generalization of the $J_1$-$J_2$ Heisenberg model for $S = 1/2$ and S = 1. Using self-consistent spin wave theory we present a detailed description of the staggered magnetization and magnetic excitations in the collinear state. We find that the introduction of the interlayer coupling $J_c$ suppresses the quantum fluctuations and strengthens the long range ordering. In the $J_1$-$J_2$-$J_c$ model, we find two qualitatively distinct scenarios for how the collinear phase becomes unstable upon increasing $J_1$. Either the magnetization or one of the spin wave velocities vanishes. For $S = 1/2$ renormalization due to quantum fluctuations is significantly stronger than for S=1, in particular close to the quantum phase transition. Our findings for the $J_1$-$J_2$-$J_c$ model are of general theoretical interest, however, the results show that it is unlikely that the model is relevant to undoped pnictides.