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

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 have 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 on 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.

Figure 1

(a) Schematic diagram of the three-dimensional spin ordering in the Fe pnictides. Here we show the $a$-$b$ plane for the considered model with nearest-neighbor coupling $J_1$ and next-to-nearest-neighbor coupling $J_2$. The interplane coupling $J_c$ is directed into the page. In addition we show in (b) the real space positions of the lattice points $A$, $a$, $B$, and $b$ which make up the unit cell.

Figure 2

Quantum correction parameters $\mu$, $\nu$, $\eta$, and $\chi$ as a function of $J_1/J_2$. Calculations were performed by numerical iteration for a range of $y$ values for both $S=1/2$ and $S=1$. Here we show the results for $y=0.01$ (solid line) and $0.10$ (dashed line).

Figure 3

Staggered magnetization in the three-dimensional magnetic Brillouin zone as a function of the ratio of the exchange couplings $J_1$ and $J_2$. The calculated three-dimensional staggered magnetizations were obtained using (a) linear spin-wave theory and (b) self-consistent spin-wave theory. In each plot we show results for $S=1/2$ and $S=1$ for $y=0.01$ (solid line) and $0.10$ (dashed line).

Figure 4

The phase diagram of the $S=1/2$ 3D $J_1$-$J_2$-$J_c$ Heisenberg model; $y=J_c/J_2$ measures the interlayer coupling. The region above the curve corresponds to the columnar spin-stripe phase and the region below to possibly the columnar spin-dimerized phase.

Figure 5

The phase diagram of the $S=1$ 3D $J_1$-$J_2$-$J_c$ Heisenberg model; $y=J_c/J_2$ measures the interlayer coupling. The region above the curve corresponds to the columnar spin-stripe phase and the region below to possibly either a columnar spin-dimerized phase like for $S=1/2$ or a quadrupolar ordered phase. Note the very different scales compared to Fig. 4.

Figure 6

Spin-wave velocities along the crystal axes as functions of the ratio $J_1/J_2$ for $y=0.01$ (solid line) and $0.10$ (dashed line). Results are presented for $S=1/2$ and $S=1$. Note that the ordinate values are divided by $\sqrt{y}$ for $v_c$ only.

Figure 7

Spin excitation spectra along high-symmetry cuts through the Brillouin zone for $S=1$. We show the excitation spectra for both a system deep in the columnar phase ($J_1/J_2=1$) and one near the quantum phase transition ($J_1/J_2=1.98$). Calculations were performed using self-consistent spin-wave theory and here we compare results for $y=0.01$ (solid and dotted lines) and $0.10$ (dashed and dot-dashed lines).

Figure 8

Spin-wave dispersion in the plane of the spin stripes. The spin-wave dispersions $\omega (\mathbf{k})$ at $\omega (k_a,k_b,\pi )$ in units of $J_2$ is shown for $J_1/J_2=0.76$ (top) and $J_1/J_2=1.972$ (bottom) for $y=0.10$. Values were obtained using self-consistent spin-wave theory for $S=1$.

Figure 9

Spin-wave dispersion in the plane along and through the spin stripes. The spin-wave dispersions $\omega (\mathbf{k})$ at $\omega (\pi ,k_b,k_c)$ in units of $J_2$ is shown for $J_1/J_2=0.76$ (top) and $J_1/J_2=1.972$ (bottom) for $y=0.10$. Values were obtained using self-consistent spin-wave theory for $S=1$.

Figure 10

Comparison of fitted dispersions for the critical and noncritical scenarios with inelastic neutron-scattering data. We see that the noncritical scenario agrees nicely with the experimental data, whereas the critcal scenario does not describe the disperison at higher energies due to problems with the reduced staggered moment and the anisotropic spin-wave velocities.