Quantum $J_1\mathrm{\text{−}}{J}_2$ Antiferromagnet on a Stacked Square Lattice: Influence of the Interlayer Coupling on the Ground-State Magnetic Ordering

Using the coupled-cluster method and the rotation-invariant Green’s function method, we study the influence of the interlayer coupling $J_{\perp }$ on the magnetic ordering in the ground state of the spin-$1/2$ $J_1\mathrm{\text{−}}{J}_2$ frustrated Heisenberg antiferromagnet ($J_1\mathrm{\text{−}}{J}_2$ model) on the stacked square lattice. In agreement with known results for the $J_1\mathrm{\text{−}}{J}_2$ model on the strictly two-dimensional square lattice ($J_{\perp }=0$), we find that the phases with magnetic long-range order at small $J_2<J_{c_1}$ and large $J_2>J_{c_2}$ are separated by a magnetically disordered (quantum paramagnetic) ground-state phase. Increasing the interlayer coupling $J_{\perp }>0$, the parameter region of this phase decreases, and, finally, the quantum paramagnetic phase disappears for quite small $J_{\perp }\sim (0.2–0.3)J_1$.

Figure 1

Magnetic order parameter $M$ versus $J_2$ for various strengths of the interlayer coupling $J_{\perp }$. (a) RGM, (b) CCM.

Figure 2

Ground-state phase diagram. (a) RGM, (b) CCM. The solid lines show those values of $J_2$ where the order parameters vanish. The dashed line in (b) represents those values of $J_2$ where the two energies calculated for the Néel and collinear reference states become equal.

Figure 3

CCM results for the energy per spin $e$ for both reference states (inset) and the order parameter $M$ for $J_{\perp }=0.2$. Both quantities are obtained by extrapolation of the raw $\mathrm{LSUB}n$ results to the limit $n\to \infty$ as explained in the text. The energies calculated with the Néel and collinear reference states become equal at $J_2\approx 0.58$, indicating a first-order transition. For the order parameter $M$, we take that value calculated with the reference state of lower CCM energy.