Effects of ring exchange interaction on the Néel phase of two-dimensional, spatially anisotropic, frustrated Heisenberg quantum antiferromagnet

Higher-order quantum effects on the magnetic phase diagram induced by four-spin ring exchange on plaquettes are investigated for a two-dimensional quantum antiferromagnet with $S=1/2$. Spatial anisotropy and frustration are allowed for. Using a perturbative spin-wave expansion up to second order in $1/S$ we obtain the spin-wave energy dispersion, sublattice magnetization, and magnetic phase diagram. We find that for substantial four-spin ring exchange the quantum fluctuations are stronger than in the standard Heisenberg model. A moderate amount of four-spin ring exchange couplings stabilizes the ordered antiferromagnetic Néel state while a large amount renders it unstable. Comparison with inelastic neutron-scattering data for cuprates points toward a moderate ring exchange coupling of 27 to 29$\%$ of the nearest-neighbor exchange coupling.

Figure 1

Classical antiferromagnetic ground state (Néel state) and the various couplings: $J_1$, $J_1^{\prime }$ are nearest-neighbor interactions along the row and column directions, respectively; $J_2$ is the next-nearest-neighbor interaction along the diagonals; and $K$ is the cyclic four-spin ring exchange. All couplings are assumed to be antiferromagnetic, i.e., $J_1,J_1^{\prime },J_2,K>0$.

Figure 2

Spin-wave energy $E_{\mathbf{k}}^{\mathrm{AF}}/J_1$ obtained from LSWT (long-dashed lines), with $1/S$ (dot-dashed lines) and with $1/S^2$ corrections (solid lines) for the Néel ordered phase. We have chosen spatially isotropic coupling $\zeta =1$. In the left panel we show the corrections for relative frustration $\eta =0$ and ring exchange $\mu =0.025$; in the right panel we show them for frustration $\eta =0.2$ and ring exchange $\mu =0.12$. In the latter case, the $1/S^2$ terms in the Hamiltonian provide significant corrections to both the LSWT and $1/S$ results.

Figure 3

The effect of $\mu$ on the spin-wave energy $E_{\mathbf{k}}^{\mathrm{AF}}/J_1$ for the Néel ordered phase with $1/S^2$ corrections shown for two values of $\eta =0,0.2$ and $\zeta =1$.

Figure 4

Spin-wave energy $E_{\mathbf{k}}^{\mathrm{AF}}/J_1$ including $1/S^2$ corrections for $\zeta =1,\mu =0.12$ for various values of $\eta$.

Figure 5

Comparison of the measured spin-wave energy $E_{\mathbf{k}}^{\mathrm{AF}}$ as obtained by inelastic neutron scattering in La${}_2$CuO${}_4$ with the theoretical results including $1/S^2$ corrections for the spatially isotropic model ($\zeta =1$) for $N_L=24$. For given moderate values $\eta$ of relative frustration a value $\mu$ of the four-spin ring exchange can be found such that the dispersions match the experimental data.

Figure 6

The sublattice magnetization $M_{\mathrm{AF}}$ plotted for $\zeta =1$ and for three different values of $\mu =0$ (black), 0.12 (blue/dark gray), and 0.22 (orange/light gray) as a function of the relative magnetic frustration $\eta$. For all three cases, results from linear spin-wave theory (dashed lines) with $1/S$ (dot-dashed lines) and with $1/S^2$ corrections (solid lines) are shown. Magnetization curves with $1/S$ corrections alone diverge in all cases. However, $1/S^2$ corrections compensate the divergence, and the magnetization curves steadily decrease to zero at critical values ${\eta }_c$. We find ${\eta }_c=0.411$ ($\mu =0$), 0.423 ($\mu =0.12$), and 0.399 ($\mu =0.22$).

Figure 7

${\eta }_c$-$\mu$ phase diagram for $\zeta =1$. With increase in $\mu$, ${\eta }_c$ increases up to a maximum value 0.423 at $\mu ={\mu }_t\approx 0.12$ and then sharply decreases. This shows that the ring exchange coupling $\mu$ initially favors the Néel ordering of the NN spins until the turning value ${\mu }_t$ is reached. For $\mu >{\mu }_t$, the four-spin coupling enhancement destabilizes the Néel order.

Figure 8

Sublattice magnetization $M_{\mathrm{AF}}$ with spatial anisotropy $\zeta =0.4$ between the vertical and the horizontal NN couplings for three values of $\mu =0$ (black), 0.08 (blue/dark gray), and 0.13 (orange/light gray) as a function of frustration $\eta$. For all three cases, results from LSWT (dashed lines) with $1/S$ (dot-dashed lines) and with $1/S^2$ corrections (solid lines) are shown. $M_{\mathrm{AF}}$ with $1/S$ corrections alone diverge for $\mu =0$ and 0.08, but not for $\mu =0.13$ where it converges.

Figure 9

${\eta }_c$-$\mu$ phase diagram for $\zeta =0.4$ (left panel) and 0.2 (right panel), to be compared with the phase diagram for the spatially isotropic case $\zeta =1$ in Fig. 7.

Figure 10

(a) The solid and the dashed lines corresponding to the $\alpha$ and $\beta$ propagators. Second-order diagrams for the self-energies ${\Sigma }_{\alpha \alpha }^{\left(2\right)}(\mathbf{k},\omega )$ and ${\Sigma }_{\alpha \beta }^{\left(2\right)}(\mathbf{k},\omega )$ are shown in (b) and (c). The diagrams in (d) contribute only to ${\Sigma }_{\alpha \alpha }^{\left(2\right)}(\mathbf{k},\omega )$. $V^{\left(2\right)},V^{\left(3\right)},V^{\left(5\right)},V^{\left(7\right)},V^{\left(8\right)}$ are the vertex factors. Note that at each vertex two arrows enter the vertex and two leave it, which reflects the conservation of the total $S_z$ component.