Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1312.6324 · Physical Review B

Quantum phases in the frustrated Heisenberg model on the bilayer honeycomb lattice

Hao Zhang, Marcelo Arlego, Carlos Alberto Lamas

Published Jan 6, 2014• 31 citations
arXivHTMLPDF

Abstract

We use a combination of analytical and numerical techniques to study the phase diagram of the frustrated Heisenberg model on the bilayer honeycomb lattice. Using the Schwinger-boson description of the spin operators followed by a mean-field decoupling, the magnetic phase diagram is studied as a function of the frustration coupling ${J}_{2}$ and the interlayer coupling ${J}_{\ensuremath{\perp}}$. The presence of both magnetically ordered and disordered phases is investigated by means of the evaluation of ground-state energy, spin gap, local magnetization, and spin-spin correlations. We observe a phase with a spin gap and short-range N\'eel correlations that survives for nonzero next-nearest-neighbor interaction and interlayer coupling. Furthermore, we detect signatures of a reentrant behavior in the melting of the N\'eel phase and symmetry restoring when the system undergoes a transition from an on-layer nematic valence-bond crystal phase to an interlayer valence-bond crystal phase. We complement our work with exact diagonalization on small clusters and dimer-series expansion calculations, together with a linear spin-wave approach to study the phase diagram as a function of the spin $S$, the frustration, and the interlayer couplings.

DOI
10.1103/physrevb.89.024403

Related papers

Self consistent study of the quantum phases in a frustrated antiferromagnet on the bilayer honeycomb lattice

Arlego, Marcelo José Fabián, Carlos Alberto Lamas, Zhang, H.

We study the frustrated Heisenberg model on the bilayer honeycomb lattice. The ground-state energy and spin gap are calculated, using different bosonic representations at mean field level and numerical calculations, to explore different sectors of the phase diagram. In particular we make use of a bond operator formalism and series expansion calculations to study the extent of dimer inter-layer phase. On the other hand we use the Schwinger boson method and exact diagonalization on small systems to analyze the evolution of on-layer phases. In this case we specifically observe a phase that presents a spin gap and short range Neel correlations that survives even in the presence of non-zero next-nearest-neighbor interaction and inter-layer coupling.

•Apr 1, 2014•9 citations

Quantum phase diagram of a frustrated antiferromagnet on the bilayer honeycomb lattice

Hao Zhang, Carlos Alberto Lamas, Marcelo Arlego +1 more

We study the spin-1/2 Heisenberg antiferromagnet on a bilayer honeycomb lattice including interlayer frustration. Using a set of complementary approaches, namely, Schwinger bosons, dimer series expansion, bond operators, and exact diagonalization, we map out the quantum phase diagram. Analyzing ground-state energies and elementary excitation spectra, we find four distinct phases, corresponding to three collinear magnetic long-range ordered states, and one quantum disordered interlayer dimer phase. We detail that the latter phase is adiabatically connected to an exact singlet product ground state of the bilayer, which exists along a line of maximum interlayer frustration. The order within the remaining three phases will be clarified.

arXiv:1512.03030•Physical review. B./Physical review. B•Jun 27, 2016•19 citations

Phase diagram of a frustrated quantum antiferromagnet on the honeycomb lattice: Magnetic order versus valence-bond crystal formation

A. Fabricio Albuquerque, David R. Schwandt, Balázs Hetényi +3 more

We present a comprehensive computational study of the phase diagram of the frustrated $S=1/2$ Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest $({J}_{2})$ and third-neighbor $({J}_{3})$ couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger ${J}_{2}$, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.

arXiv:1102.5325•Physical Review B•Jul 6, 2011•204 citations

Quantum phase diagram of a frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg model on an $AA$-stacked bilayer honeycomb lattice

P. H. Y. Li, R. F. Bishop

The zero-temperature quantum phase diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on an $AA$-stacked bilayer honeycomb lattice is investigated using the coupled cluster method (CCM). The model comprises two monolayers in each of which the spins, residing on honeycomb-lattice sites, interact via both nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg exchange iteractions, with respective strengths $J_{1} > 0$ and $J_{2} \equiv \kappa J_{1}>0$. The two layers are coupled via a comparable Heisenberg exchange interaction between NN interlayer pairs, with a strength $J_{1}^{\perp} \equiv \delta J_{1}$. The complete phase boundaries of two quasiclassical collinear AFM phases, namely the Neel and Neel-II phases, are calculated in the $\kappa \delta$ half-plane with $\kappa > 0$. Whereas on each monolayer in the Neel state all NN pairs of spins are antiparallel, in the Neel-II state NN pairs of spins on zigzag chains along one of the three equivalent honeycomb-lattice directions are antiparallel, while NN interchain spins are parallel. We calculate directly in the thermodynamic (infinite-lattice) limit both the magnetic order parameter $M$ and the excitation energy $\Delta$ from the $s^{z}_{T}=0$ ground state to the lowest-lying $|s^{z}_{T}|=1$ excited state (where $s^{z}_{T}$ is the total $z$ component of spin for the system as a whole, and where the collinear ordering lies along the $z$ direction) for both quasiclassical states used (separately) as the CCM model state, on top of which the multispin quantum correlations are then calculated to high orders ($n \leq 10$) in a systematic series of approximations involving $n$-spin clusters. The sole approximation made is then to extrapolate the sequences of $n$th-order results for $M$ and $\Delta$ to the exact limit, $n \to \infty$.

•May 3, 2018•0 citations

Frustrated honeycomb-bilayer Heisenberg antiferromagnet: The spin- 12 J1−J2−J1⊥ model

R. F. Bishop, P. H. Y. Li

We use the coupled cluster method to study the zero-temperature quantum phase diagram of the spin-$\frac{1}{2}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ model on the honeycomb bilayer lattice. In each layer, we include both nearest-neighbor and frustrating next-nearest-neighbor antiferromagnetic exchange couplings, of strength ${J}_{1}>0$ and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively. The two layers are coupled by an interlayer nearest-neighbor exchange, with coupling constant ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}>0$. We calculate directly in the infinite-lattice limit both the ground-state energy per spin and the N\'eel magnetic order parameter, as well as the triplet spin gap. By implementing the method to very high orders of approximation we obtain an accurate estimate for the full boundary of the N\'eel phase in the $\ensuremath{\kappa}\ensuremath{\delta}$ plane. For each value $\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(0)\ensuremath{\approx}1.70(5)$, we find an upper critical value ${\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$, such that N\'eel order is present for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$. Conversely, for each value $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(0)\ensuremath{\approx}0.19(1)$, we find an upper critical value ${\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, such that N\'eel order persists for $0<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$. Most interestingly, for values of $\ensuremath{\kappa}$ in the range ${\ensuremath{\kappa}}_{c}(0)<\ensuremath{\kappa}<{\ensuremath{\kappa}}^{>}\ensuremath{\approx}0.215(2)$, we find a reentrant behavior such that N\'eel order exists only in the range ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, with ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})>0$. These latter upper and lower critical values coalesce when $\ensuremath{\kappa}={\ensuremath{\kappa}}^{>}$, such that ${\ensuremath{\delta}}_{c}^{<}({\ensuremath{\kappa}}^{>})={\ensuremath{\delta}}_{c}^{>}({\ensuremath{\kappa}}^{>})\ensuremath{\approx}0.25(5)$.

arXiv:1611.03287•Physical review. B./Physical review. B•Apr 11, 2017•22 citations

High-order study of the quantum critical behavior of a frustrated spin- 1 2 antiferromagnet on a stacked honeycomb bilayer

R. F. Bishop, P. H. Y. Li

We study a frustrated spin-$\frac{1}{2}\phantom{\rule{0.16em}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ Heisenberg antiferromagnet on an $AA$-stacked bilayer honeycomb lattice. In each layer we consider nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor antiferromagnetic (AFM) exchange couplings ${J}_{1},\phantom{\rule{0.16em}{0ex}}{J}_{2}$, and ${J}_{3}$, respectively. The two layers are coupled with an AFM NN exchange coupling ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}$. The model is studied for arbitrary values of $\ensuremath{\delta}$ along the line ${J}_{3}={J}_{2}\ensuremath{\equiv}\ensuremath{\alpha}{J}_{1}$ that includes the most highly frustrated point at $\ensuremath{\alpha}=\frac{1}{2}$, where the classical ground state is macroscopically degenerate. The coupled cluster method is used at high orders of approximation to calculate the magnetic order parameter and the triplet spin gap. We are thereby able to give an accurate description of the quantum phase diagram of the model in the $\ensuremath{\alpha}\ensuremath{\delta}$ plane in the window $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1,\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1$. This includes two AFM phases with N\'eel and striped order, and an intermediate gapped paramagnetic phase that exhibits various forms of valence-bond crystalline order. We obtain accurate estimations of the two phase boundaries, $\ensuremath{\delta}={\ensuremath{\delta}}_{{c}_{i}}(\ensuremath{\alpha})$, or equivalently, $\ensuremath{\alpha}={\ensuremath{\alpha}}_{{c}_{i}}(\ensuremath{\delta})$, with $i=1$ (N\'eel) and 2 (striped). The two boundaries exhibit an ``avoided crossing'' behavior with both curves being re-entrant. Thus, in this $\ensuremath{\alpha}\ensuremath{\delta}$ window, N\'eel order exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{1}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{{c}_{1}}(0)\ensuremath{\approx}0.46(2)$ and ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{1}}(0)<\ensuremath{\alpha}<{\ensuremath{\alpha}}_{1}^{>}\ensuremath{\approx}0.49(1)$, and striped order similarly exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{2}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{{c}_{2}}(0)\ensuremath{\approx}0.600(5)$ and ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{2}}(0)>\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2}^{<}\ensuremath{\approx}0.56(1)$.

arXiv:1708.06162•Physical review. B./Physical review. B•Dec 12, 2017•12 citations
We use a combination of analytical and numerical techniques to study the phase diagram of the frustrated Heisenberg model on the bilayer honeycomb lattice. Using the Schwinger-boson description of the spin operators followed by a mean-field decoupling, the magnetic phase diagram is studied as a function of the frustration coupling J2{J}_{2}J2​ and the interlayer coupling J\ensuremath⊥{J}_{\ensuremath{\perp}}J\ensuremath⊥​. The presence of both magnetically ordered and disordered phases is investigated by means of the evaluation of ground-state energy, spin gap, local magnetization, and spin-spin correlations. We observe a phase with a spin gap and short-range N'eel correlations that survives for nonzero next-nearest-neighbor interaction and interlayer coupling. Furthermore, we detect signatures of a reentrant behavior in the melting of the N'eel phase and symmetry restoring when the system undergoes a transition from an on-layer nematic valence-bond crystal phase to an interlayer valence-bond crystal phase. We complement our work with exact diagonalization on small clusters and dimer-series expansion calculations, together with a linear spin-wave approach to study the phase diagram as a function of the spin SSS, the frustration, and the interlayer couplings.
We study the frustrated Heisenberg model on the bilayer honeycomb lattice. The ground-state energy and spin gap are calculated, using different bosonic representations at mean field level and numerical calculations, to explore different sectors of the phase diagram. In particular we make use of a bond operator formalism and series expansion calculations to study the extent of dimer inter-layer phase. On the other hand we use the Schwinger boson method and exact diagonalization on small systems to analyze the evolution of on-layer phases. In this case we specifically observe a phase that presents a spin gap and short range Neel correlations that survives even in the presence of non-zero next-nearest-neighbor interaction and inter-layer coupling.
We study the spin-1/2 Heisenberg antiferromagnet on a bilayer honeycomb lattice including interlayer frustration. Using a set of complementary approaches, namely, Schwinger bosons, dimer series expansion, bond operators, and exact diagonalization, we map out the quantum phase diagram. Analyzing ground-state energies and elementary excitation spectra, we find four distinct phases, corresponding to three collinear magnetic long-range ordered states, and one quantum disordered interlayer dimer phase. We detail that the latter phase is adiabatically connected to an exact singlet product ground state of the bilayer, which exists along a line of maximum interlayer frustration. The order within the remaining three phases will be clarified.
We present a comprehensive computational study of the phase diagram of the frustrated S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest (J2)({J}_{2})(J2​) and third-neighbor (J3)({J}_{3})(J3​) couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger J2{J}_{2}J2​, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.
Quantum phase diagram of a frustrated spin-12\frac{1}{2}21​ J1J_{1}J1​--J2J_{2}J2​--J1⊥J_{1}^{\perp}J1⊥​ Heisenberg model on an AAAAAA-stacked bilayer honeycomb lattice
The zero-temperature quantum phase diagram of the spin-12\frac{1}{2}21​ J1J_{1}J1​--J2J_{2}J2​--J1⊥J_{1}^{\perp}J1⊥​ model on an AAAAAA-stacked bilayer honeycomb lattice is investigated using the coupled cluster method (CCM). The model comprises two monolayers in each of which the spins, residing on honeycomb-lattice sites, interact via both nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg exchange iteractions, with respective strengths J1>0J_{1} > 0J1​>0 and J2≡κJ1>0J_{2} \equiv \kappa J_{1}>0J2​≡κJ1​>0. The two layers are coupled via a comparable Heisenberg exchange interaction between NN interlayer pairs, with a strength J1⊥≡δJ1J_{1}^{\perp} \equiv \delta J_{1}J1⊥​≡δJ1​. The complete phase boundaries of two quasiclassical collinear AFM phases, namely the Neel and Neel-II phases, are calculated in the κδ\kappa \deltaκδ half-plane with κ>0\kappa > 0κ>0. Whereas on each monolayer in the Neel state all NN pairs of spins are antiparallel, in the Neel-II state NN pairs of spins on zigzag chains along one of the three equivalent honeycomb-lattice directions are antiparallel, while NN interchain spins are parallel. We calculate directly in the thermodynamic (infinite-lattice) limit both the magnetic order parameter MMM and the excitation energy Δ\DeltaΔ from the sTz=0s^{z}_{T}=0sTz​=0 ground state to the lowest-lying ∣sTz∣=1|s^{z}_{T}|=1∣sTz​∣=1 excited state (where sTzs^{z}_{T}sTz​ is the total zzz component of spin for the system as a whole, and where the collinear ordering lies along the zzz direction) for both quasiclassical states used (separately) as the CCM model state, on top of which the multispin quantum correlations are then calculated to high orders (n≤10n \leq 10n≤10) in a systematic series of approximations involving nnn-spin clusters. The sole approximation made is then to extrapolate the sequences of nnnth-order results for MMM and Δ\DeltaΔ to the exact limit, n→∞n \to \inftyn→∞.
We use the coupled cluster method to study the zero-temperature quantum phase diagram of the spin-$\frac{1}{2}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ model on the honeycomb bilayer lattice. In each layer, we include both nearest-neighbor and frustrating next-nearest-neighbor antiferromagnetic exchange couplings, of strength J1>0{J}_{1}>0J1​>0 and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively. The two layers are coupled by an interlayer nearest-neighbor exchange, with coupling constant ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}>0$. We calculate directly in the infinite-lattice limit both the ground-state energy per spin and the N\'eel magnetic order parameter, as well as the triplet spin gap. By implementing the method to very high orders of approximation we obtain an accurate estimate for the full boundary of the N\'eel phase in the $\ensuremath{\kappa}\ensuremath{\delta}$ plane. For each value $\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(0)\ensuremath{\approx}1.70(5)$, we find an upper critical value ${\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$, such that N\'eel order is present for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$. Conversely, for each value $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(0)\ensuremath{\approx}0.19(1)$, we find an upper critical value ${\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, such that N\'eel order persists for $0<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$. Most interestingly, for values of $\ensuremath{\kappa}$ in the range ${\ensuremath{\kappa}}_{c}(0)<\ensuremath{\kappa}<{\ensuremath{\kappa}}^{>}\ensuremath{\approx}0.215(2)$, we find a reentrant behavior such that N\'eel order exists only in the range ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, with ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})>0$. These latter upper and lower critical values coalesce when $\ensuremath{\kappa}={\ensuremath{\kappa}}^{>}$, such that ${\ensuremath{\delta}}_{c}^{<}({\ensuremath{\kappa}}^{>})={\ensuremath{\delta}}_{c}^{>}({\ensuremath{\kappa}}^{>})\ensuremath{\approx}0.25(5)$.
We study a frustrated spin-$\frac{1}{2}\phantom{\rule{0.16em}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ Heisenberg antiferromagnet on an AAAAAA-stacked bilayer honeycomb lattice. In each layer we consider nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor antiferromagnetic (AFM) exchange couplings J1,J2{J}_{1},\phantom{\rule{0.16em}{0ex}}{J}_{2}J1​,J2​, and J3{J}_{3}J3​, respectively. The two layers are coupled with an AFM NN exchange coupling ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}$. The model is studied for arbitrary values of $\ensuremath{\delta}$ along the line ${J}_{3}={J}_{2}\ensuremath{\equiv}\ensuremath{\alpha}{J}_{1}$ that includes the most highly frustrated point at $\ensuremath{\alpha}=\frac{1}{2}$, where the classical ground state is macroscopically degenerate. The coupled cluster method is used at high orders of approximation to calculate the magnetic order parameter and the triplet spin gap. We are thereby able to give an accurate description of the quantum phase diagram of the model in the $\ensuremath{\alpha}\ensuremath{\delta}$ plane in the window $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1,\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1$. This includes two AFM phases with N\'eel and striped order, and an intermediate gapped paramagnetic phase that exhibits various forms of valence-bond crystalline order. We obtain accurate estimations of the two phase boundaries, $\ensuremath{\delta}={\ensuremath{\delta}}_{{c}_{i}}(\ensuremath{\alpha})$, or equivalently, $\ensuremath{\alpha}={\ensuremath{\alpha}}_{{c}_{i}}(\ensuremath{\delta})$, with i=1i=1i=1 (N\'eel) and 2 (striped). The two boundaries exhibit an ``avoided crossing'' behavior with both curves being re-entrant. Thus, in this $\ensuremath{\alpha}\ensuremath{\delta}$ window, N\'eel order exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{1}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{{c}_{1}}(0)\ensuremath{\approx}0.46(2)$ and ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{1}}(0)<\ensuremath{\alpha}<{\ensuremath{\alpha}}_{1}^{>}\ensuremath{\approx}0.49(1)$, and striped order similarly exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{2}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{{c}_{2}}(0)\ensuremath{\approx}0.600(5)$ and ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{2}}(0)>\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2}^{<}\ensuremath{\approx}0.56(1)$.