arXiv:1611.03287 · Physical review. B./Physical review. B

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

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

Published Apr 11, 2017• 22 citations

Abstract

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 ${\ensuremath{\delta}}_{c}^{<}({\ensuremath{\kappa}}^{>})={\ensuremath{\delta}}_{c}^{>}({\ensuremath{\kappa}}^{>})\ensuremath{\approx}0.25(5)$.

DOI: 10.1103/physrevb.95.134414

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

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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

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Complete phase diagram of the spin- 12 J1 - J2 - J3 model (with J3 = J2 ) on the honeycomb lattice

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Ground-state phases of the frustrated spin- 12 J1 – J2 – J3 Heisenberg ferromagnet ( J1<0 ) on the honeycomb lattice with J3=J2>0

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arXiv:1611.03287 · Physical review. B./Physical review. B

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

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

Published Apr 11, 2017• 22 citations

arXiv HTML PDF

Abstract

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)

DOI: 10.1103/physrevb.95.134414

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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 ${\ensuremath{\alpha}}_{{c}_{2}}(0)>\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2}^{<}\ensuremath{\approx}0.56(1)$.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:1708.06162Physical review. B./Physical review. BDec 12, 201712 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 latticeQuantum 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, $n \to \infty$ .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, 20180 citations

Complete phase diagram of the spin- 12 J1 - J2 - J3 model (with J3 = J2 ) on the honeycomb lattice

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

We use the coupled cluster method (CCM) to investigate the ground-state (GS) properties of the frustrated spin- $\frac{1}{2}$ ${J}_{1}$ - ${J}_{2}$ - ${J}_{3}$ model on the honeycomb lattice, with nearest-neighbor exchange coupling ${J}_{1}$ plus next-nearest-neighbor ( ${J}_{2}$ ) and next-next-nearest-neighbor ( ${J}_{3}$ ) exchanges of equal strength. In particular we find a direct first-order phase transition between the N\'eel-ordered antiferromagnetic phase and the ferromagnetic phase at a value ${J}_{2}/{J}_{1}=\ensuremath{-}1.17\ifmmode\pm\else\textpm\fi{}0.01$ when ${J}_{1}>0$ , compared to the corresponding classical value of $\ensuremath{-}1$. We find no evidence for any intermediate phase. From this and our previous CCM studies of the model we present its full zero-temperature GS phase diagram.We use the coupled cluster method (CCM) to investigate the ground-state (GS) properties of the frustrated spin-$\frac{1}{2}$ ${J}_{1}$-${J}_{2}$-${J}_{3}$ model on the honeycomb lattice, with nearest-neighbor exchange coupling ${J}_{1}$ plus next-nearest-neighbor (${J}_{2}$) and next-next-nearest-neighbor (${J}_{3}$) exchanges of equal strength. In particular we find a direct first-order phase transition between the N\'eel-ordered antiferromagnetic phase and the ferromagnetic phase at a value ${J}_{2}/{J}_{1}=\ensuremath{-}1.17\ifmmode\pm\else\textpm\fi{}0.01$ when ${J}_{1}>0$, compared to the corresponding classical value of $\ensuremath{-}1$. We find no evidence for any intermediate phase. From this and our previous CCM studies of the model we present its full zero-temperature GS phase diagram.

arXiv:1202.6249Physical Review BApr 20, 201227 citations

Ground-state phases of the frustrated spin- 12 J1 – J2 – J3 Heisenberg ferromagnet ( J1<0 ) on the honeycomb lattice with J3=J2>0

P. H. Y. Li, R. F. Bishop, DAMIAN J. J. FARNELL +2 more

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 (${J}_{1}=\ensuremath{-}1$) 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 ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at ${J}_{2}^{\mathrm{cl}}=\frac{1}{7}$ ($\ensuremath{\approx}0.143$) 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 ${J}_{1}=\ensuremath{-}1$) 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 ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with ${J}_{1}=+1$ ) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. 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 N\'eel and striped order) for ${J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}$ , with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, 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 ${J}_{1}=\ensuremath{-}1$) 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 $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range $\frac{1}{10}<{J}_{2}<\frac{1}{5}$ .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 (${J}_{1}=\ensuremath{-}1$) 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 ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at ${J}_{2}^{\mathrm{cl}}=\frac{1}{7}$ ($\ensuremath{\approx}0.143$) 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 ${J}_{1}=\ensuremath{-}1$) 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 ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with ${J}_{1}=+1$) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. 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 N\'eel and striped order) for ${J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}$, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, 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 ${J}_{1}=\ensuremath{-}1$) 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 $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range $\frac{1}{10}<{J}_{2}<\frac{1}{5}$.

arXiv:1109.6229Physical Review BFeb 21, 201231 citations

Ground-state phases of the spin- 1 J 1 − J 2 Heisenberg antiferromagnet on the honeycomb lattice

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

We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling ${J}_{1}>0$ and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling ${J}_{1}>0$ and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.

arXiv:1602.08915Physical review. B./Physical review. BJun 28, 201621 citations

Frustrated spin- 12 J1 – J2 isotropic XY model on the honeycomb lattice

R. F. Bishop, P. H. Y. Li, C. E. Campbell

We study the zero-temperature ground-state (GS) phase diagram of a spin-half ${J}_{1}$ -- ${J}_{2}$ $XY$ model on the honeycomb lattice with nearest-neighbor exchange coupling ${J}_{1}>0$ and frustrating next-nearest-neighbor exchange coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, where both bonds are of the isotropic $XY$ type, using the coupled cluster method. Results are presented for the GS energy per spin, magnetic order parameter, and staggered dimer valence-bond crystalline (SDVBC) susceptibility, for values of the frustration parameter in the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$. In this range, we find phases exhibiting, respectively, N\'eel $xy$ planar [N(p)], N\'eel $z$ -aligned [N( $z$ )], SDVBC, and N\'eel-II $xy$ planar [N-II(p)] orderings. The N\'eel-II states, which break the lattice rotational symmetry, are ones in which the spins of nearest-neighbor pairs along one of the three equivalent honeycomb directions are parallel, while those in the other two directions are antiparallel. The N(p) state, which is stable for the classical version of the model in the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}\frac{1}{6}$, is found to form the GS phase out to a first quantum critical point at ${\ensuremath{\kappa}}_{{c}_{1}}=0.216(5)$, beyond which the stable GS phase has N( $z$ ) order over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}=0.355(5)$. For values $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$, we find a strong competition to form the GS phase between states with N-II(p) and SDVBC forms of order. Our best estimate, however, is that the stable GS phase over the range ${\ensuremath{\kappa}}_{{c}_{2}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{3}}\ensuremath{\approx}0.52(3)$ is a mixed state with both SDVBC and N-II(p) forms of order; and for values $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{3}}$ is the N-II(p) state, which is stable at the classical level only at the highly degenerate point $\ensuremath{\kappa}=\frac{1}{2}$. Over the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$, we find no evidence for any of the spiral phases that are present classically for all values $\ensuremath{\kappa}>\frac{1}{6}$, nor for any quantum spin-liquid state.We study the zero-temperature ground-state (GS) phase diagram of a spin-half ${J}_{1}$--${J}_{2}$ $XY$ model on the honeycomb lattice with nearest-neighbor exchange coupling ${J}_{1}>0$ and frustrating next-nearest-neighbor exchange coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, where both bonds are of the isotropic $XY$ type, using the coupled cluster method. Results are presented for the GS energy per spin, magnetic order parameter, and staggered dimer valence-bond crystalline (SDVBC) susceptibility, for values of the frustration parameter in the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$. In this range, we find phases exhibiting, respectively, N\'eel $xy$ planar [N(p)], N\'eel $z$-aligned [N($z$)], SDVBC, and N\'eel-II $xy$ planar [N-II(p)] orderings. The N\'eel-II states, which break the lattice rotational symmetry, are ones in which the spins of nearest-neighbor pairs along one of the three equivalent honeycomb directions are parallel, while those in the other two directions are antiparallel. The N(p) state, which is stable for the classical version of the model in the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}\frac{1}{6}$, is found to form the GS phase out to a first quantum critical point at ${\ensuremath{\kappa}}_{{c}_{1}}=0.216(5)$, beyond which the stable GS phase has N($z$) order over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}=0.355(5)$. For values $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$, we find a strong competition to form the GS phase between states with N-II(p) and SDVBC forms of order. Our best estimate, however, is that the stable GS phase over the range ${\ensuremath{\kappa}}_{{c}_{2}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{3}}\ensuremath{\approx}0.52(3)$ is a mixed state with both SDVBC and N-II(p) forms of order; and for values $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{3}}$ is the N-II(p) state, which is stable at the classical level only at the highly degenerate point $\ensuremath{\kappa}=\frac{1}{2}$. Over the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$, we find no evidence for any of the spiral phases that are present classically for all values $\ensuremath{\kappa}>\frac{1}{6}$, nor for any quantum spin-liquid state.

arXiv:1403.2252Physical Review BJun 17, 201428 citations

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

\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)

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 ${\ensuremath{\alpha}}_{{c}_{2}}(0)>\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2}^{<}\ensuremath{\approx}0.56(1)$.

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

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

We use the coupled cluster method (CCM) to investigate the ground-state (GS) properties of the frustrated spin-

\frac{1}{2}

{J}_{1}

{J}_{2}

{J}_{3}

model on the honeycomb lattice, with nearest-neighbor exchange coupling

{J}_{1}

plus next-nearest-neighbor (

{J}_{2}

) and next-next-nearest-neighbor (

{J}_{3}

) exchanges of equal strength. In particular we find a direct first-order phase transition between the N\'eel-ordered antiferromagnetic phase and the ferromagnetic phase at a value ${J}_{2}/{J}_{1}=\ensuremath{-}1.17\ifmmode\pm\else\textpm\fi{}0.01$ when

{J}_{1}>0

, compared to the corresponding classical value of $\ensuremath{-}1$. We find no evidence for any intermediate phase. From this and our previous CCM studies of the model we present its full zero-temperature GS phase diagram.

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 (${J}_{1}=\ensuremath{-}1$) 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 ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at

{J}_{2}^{\mathrm{cl}}=\frac{1}{7}

($\ensuremath{\approx}0.143$) 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 ${J}_{1}=\ensuremath{-}1$) 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 ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with

{J}_{1}=+1

) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. 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 N\'eel and striped order) for

{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}

, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, 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 ${J}_{1}=\ensuremath{-}1$) 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 $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, 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}

We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling

{J}_{1}>0

and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.

We study the zero-temperature ground-state (GS) phase diagram of a spin-half

{J}_{1}

{J}_{2}

XY

model on the honeycomb lattice with nearest-neighbor exchange coupling

{J}_{1}>0

and frustrating next-nearest-neighbor exchange coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, where both bonds are of the isotropic

XY

type, using the coupled cluster method. Results are presented for the GS energy per spin, magnetic order parameter, and staggered dimer valence-bond crystalline (SDVBC) susceptibility, for values of the frustration parameter in the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$. In this range, we find phases exhibiting, respectively, N\'eel

xy

planar [N(p)], N\'eel

z

-aligned [N(

z

)], SDVBC, and N\'eel-II

xy

planar [N-II(p)] orderings. The N\'eel-II states, which break the lattice rotational symmetry, are ones in which the spins of nearest-neighbor pairs along one of the three equivalent honeycomb directions are parallel, while those in the other two directions are antiparallel. The N(p) state, which is stable for the classical version of the model in the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}\frac{1}{6}$, is found to form the GS phase out to a first quantum critical point at ${\ensuremath{\kappa}}_{{c}_{1}}=0.216(5)$, beyond which the stable GS phase has N(

z

) order over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}=0.355(5)$. For values $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$, we find a strong competition to form the GS phase between states with N-II(p) and SDVBC forms of order. Our best estimate, however, is that the stable GS phase over the range ${\ensuremath{\kappa}}_{{c}_{2}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{3}}\ensuremath{\approx}0.52(3)$ is a mixed state with both SDVBC and N-II(p) forms of order; and for values $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{3}}$ is the N-II(p) state, which is stable at the classical level only at the highly degenerate point $\ensuremath{\kappa}=\frac{1}{2}$. Over the range $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$, we find no evidence for any of the spiral phases that are present classically for all values $\ensuremath{\kappa}>\frac{1}{6}$, nor for any quantum spin-liquid state.