Ground-state phases of the frustrated spin-$\frac{1}{2}$ $J_1$–$J_2$–$J_3$ Heisenberg ferromagnet ($J_1<0$) on the honeycomb lattice with $J_3=J_2>0$

We study the ground-state (GS) properties of the frustrated spin-$\frac{1}{2}$ $J_1$–$J_2$–$J_3$ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor ($J_1=-1$) exchange and frustrating antiferromagnetic next-nearest-neighbor ($J_2>0$) and next-next-nearest-neighbor ($J_3>0$) exchanges, for the case $J_3=J_2$. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (“striped”) collinear order at $J_2^c\approx 0.1095\pm 0.0005$, which is below the corresponding hypothetical transition point at $J_2^{\mathrm{cl}}=\frac{1}{7}$ ($\approx 0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region $\frac{1}{10}<J_2<\frac{1}{5}$. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with $J_1=-1$) to previous results for the corresponding antiferromagnetic case (with $J_1=+1$). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter $J_2$. However, there are considerable differences in the behavior of the order parameters for the two models for $J_2/\left|J_1\right|\lesssim 0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with $J_1=+1$) breaks down at $J_2^{c_2}\approx 0.60$, whereas the “equivalent” point for the ferromagnetic model (with $J_1=-1$) occurs at $J_2^c\approx 0.11$. Unlike in the antiferromagnetic model (with $J_1=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with Néel and striped order) for $J_2^{c_1}<J_2<J_2^{c_2}$, with $J_2^{c_1}\approx 0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with $J_1=-1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\lesssim J_2\lesssim 0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range $\frac{1}{10}<J_2<\frac{1}{5}$.

Figure 1

The $J_1$–$J_2$–$J_3$ honeycomb model, showing (a) the ferromagnetic (FM) state and (b) the classical striped antiferromagnetic (AFM) state. The arrows represent spins located on lattice sites •.

Figure 2

Ground-state energy per spin, $E/N$, versus $J_2$ for the striped phase of the spin-$\frac{1}{2}$$J_1$–$J_2$–$J_3$ honeycomb model with $J_1=-1$ compared with those for $J_1=+1$ (with $J_3=J_2>0$ for both cases). The CCM results using the striped model state for various LSUB$m$ approximations with $m=\{6,8,10,12\}$, together with the ED ($N=32$) results are shown in (a). We show in (b) the extrapolated CCM LSUB$\infty$ results using the $m=\{6,8,10,12\}$ data points fitted to Eq. (6). In all cases curves without symbols attached refer to the case $J_1=-1$, whereas the corresponding curves with symbols refer to the case $J_1=+1$. The FM result from Eq. (2) with $s=\frac{1}{2}$ is also displayed.

Figure 3

Ground-state magnetic order parameter $M$ for the striped AFM state of the spin-$\frac{1}{2}$ $J_1$–$J_2$–$J_3$ honeycomb model with $J_1=-1$, compared with those for $J_1=+1$ (with $J_3=J_2>0$ for both cases). The CCM results for various LSUB$m$ approximations with $m=\{6,8,10,12\}$ are shown in (a). We also show in (b) the extrapolated LSUB$\infty$ results using the $m=\{6,8,10,12\}$ data points. The curves labeled LSUB$\infty \left(1\right)$ and LSUB$\infty \left(2\right)$ are obtained using Eqs. (7) and (8), respectively. In all cases curves without symbols attached refer to the case $J_1=-1$, whereas the corresponding curves with symbols refer to the case $J_1=+1$.

Figure 4

Selected spin-spin correlation functions $\langle \mathbf{s}\left(0\right)·\mathbf{s}\left(\mathbf{r}\right)\rangle$ for the spin-$\frac{1}{2}$$J_1$–$J_2$–$J_3$ model on the honeycomb lattice with $J_3=J_2$, for both the FM case (with $J_1=-1$) and the AFM case (with $J_1=+1$), using (a) the CCM with the striped collinear state as model state, at the LSUB10 level of approximation; and (b) the ED method on a lattice of size $N=32$. Values $r=1,1.732$ and 2 correspond respectively to NN, NNN, and NNNN pairs of spins. In all cases curves without symbols attached refer to the case $J_1=-1$, whereas the corresponding curves with symbols refer to the case $J_1=+1$.

Figure 5

An expanded view near the FM boundary of the same spin-spin correlation functions $\langle \mathbf{s}\left(0\right)·\mathbf{s}\left(\mathbf{r}\right)\rangle$ shown in Fig. 4b for the spin-$\frac{1}{2}$ $J_1$–$J_2$–$J_3$ model on the honeycomb lattice for the FM case with $J_1=-1$ and $J_3=J_2$ using the ED method on a lattice of size $N=32$.

Figure 6

The phase diagram of the spin-$\frac{1}{2}$ $J_1$–$J_2$–$J_3$ honeycomb model in the $J_1$–$J_2$ plane, for the case $J_3=J_2$. The continuous transition between the AFM Néel and PVBC phases at $J_2/J_1\approx 0.47$ is shown by a broken line, while the first-order transition between the PVBC and AFM striped phases at $J_2/J_1\approx 0.60$ is shown by a solid line. Our results indicate that the transition between the striped AFM and FM phases is either a first-order one at $J_2/J_1\approx -0.11$ or occurs via an intermediate phase, probably with noncollinear spiral order, which exists in the region $-0.12\lesssim J_2/J_1\lesssim 0.10$. The region between the FM and AFM Néel phases with $J_3=J_2<0$ has not been investigated by us.