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arXiv:cond-mat/0003126 · Physical review. B, Condensed matter

Quantum phase transitions of a square-lattice Heisenberg antiferromagnet with two kinds of nearest-neighbor bonds: A high-order coupled-cluster treatment

Sven E. Krüger, Johannes Richter, Jörg Schulenburg, DAMIAN J. J. FARNELL, R. F. Bishop

Published Jun 1, 2000• 83 citations
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Abstract

We study the zero-temperature phase diagram and the low-lying excitations of a square-lattice spin-half Heisenberg antiferromagnet with two types of regularly distributed nearest-neighbor exchange bonds $[J>0$ (antiferromagnetic) and $\ensuremath{-}\ensuremath{\infty}<{J}^{\ensuremath{'}}<\ensuremath{\infty}]$ using the coupled cluster method (CCM) for high orders of approximation (up to LSUB8). We use a N\'eel model state as well as a helical model state as a starting point for the CCM calculations. We find a second-order transition from a phase with N\'eel order to a finite-gap quantum disordered phase for sufficiently large antiferromagnetic exchange constants ${J}^{\ensuremath{'}}>0.$ For frustrating ferromagnetic couplings ${J}^{\ensuremath{'}}<0$ we find indications that quantum fluctuations favor a first-order phase transition from the N\'eel order to a quantum helical state, by contrast with the corresponding second-order transition in the corresponding classical model. The results are compared to those of exact diagonalizations of finite systems (up to 32 sites) and those of spin-wave and variational calculations. The CCM results agree well with the exact diagonalization data over the whole range of the parameters. The special case of ${J}^{\ensuremath{'}}=0,$ which is equivalent to the honeycomb lattice, is treated more closely.

DOI
10.1103/physrevb.61.14607

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Spin- 12 J1 - J2 Heisenberg model on a cross-striped square lattice

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Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half ($s=\frac{1}{2}$) ${J}_{1}$-${J}_{2}$ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength ${J}_{1}$ and two next-nearest-neighbor (diagonal) bonds of strength ${J}_{2}$. The ${J}_{2}$ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no ${J}_{2}$ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when ${J}_{1}$ and ${J}_{2}$ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum $s=\frac{1}{2}$ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the $s=\frac{1}{2}$ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the $s=\frac{1}{2}$ model are of continuous deconfined type.

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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

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 $\frac{1}{10}<{J}_{2}<\frac{1}{5}$.

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Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 ${J}_{1}--{J}_{2}$ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.

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Magnetic order on a frustrated spin- 12 Heisenberg antiferromagnet on the Union Jack lattice

R. F. Bishop, P. H. Y. Li, DAMIAN J. J. FARNELL +1 more

We use the coupled cluster method (CCM) to study the zero-temperature phase diagram of a two-dimensional frustrated spin-half antiferromagnet, the so-called Union Jack model. It is defined on a square lattice such that all nearest-neighbor pairs are connected by bonds with a strength ${J}_{1}>0$, but only half the next-nearest-neighbor pairs are connected by bonds with a strength ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$. The bonds are arranged such that on the $2\ifmmode\times\else\texttimes\fi{}2$ unit cell they form the pattern of the Union Jack flag. Alternating sites on the square lattice are thus four-connected and eight-connected. We find strong evidence for a first phase transition between a N\'eel antiferromagnetic phase and a canted ferrimagnetic phase at a critical coupling ${\ensuremath{\kappa}}_{{c}_{1}}=0.66\ifmmode\pm\else\textpm\fi{}0.02$. The transition is an interesting one, at which the energy and its first derivative seem continuous, thus providing a typical scenario of a second-order transition (just as in the classical case for the model), although a weakly first-order transition cannot be excluded. By contrast, the average on-site magnetization approaches a nonzero value ${M}_{{c}_{1}}=0.195\ifmmode\pm\else\textpm\fi{}0.005$ on both sides of the transition, which is more typical of a first-order transition. The slope, $dM/d\ensuremath{\kappa}$, of the order parameter curve as a function of the coupling strength $\ensuremath{\kappa}$, also appears to be continuous, or very nearly so, at the critical point ${\ensuremath{\kappa}}_{{c}_{1}}$, thereby providing further evidence of the subtle nature of the transition between the N\'eel and canted phases. Our CCM calculations provide strong evidence that the canted ferrimagnetic phase becomes unstable at large values of $\ensuremath{\kappa}$, and hence we have also used the CCM with a model collinear semistripe-ordered ferrimagnetic state in which alternating rows (and columns) are ferromagnetically and antiferromagnetically ordered, and in which the spins connected by ${J}_{2}$ bonds are antiparallel to one another. We find tentative evidence, based on the relative energies of the two states, for a second zero-temperature phase transition between the canted and semistripe-ordered ferrimagnetic states at a large value of the coupling parameter around ${\ensuremath{\kappa}}_{{c}_{2}}\ensuremath{\approx}125\ifmmode\pm\else\textpm\fi{}5$. This prediction, however, is based on an extrapolation of the CCM results for the canted state into regimes where the solutions have already become unstable and the CCM equations based on the canted state at any level of approximation beyond the lowest have no solutions. Our prediction for ${\ensuremath{\kappa}}_{{c}_{2}}$ is hence less reliable than that for ${\ensuremath{\kappa}}_{{c}_{1}}$. Nevertheless, if this second transition at ${\ensuremath{\kappa}}_{{c}_{2}}$ does exist, our results clearly indicate it to be of first-order type.

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arXiv:1602.08915•Physical review. B./Physical review. B•Jun 28, 2016•21 citations

Quasiclassical magnetic order and its loss in a spin- 1 2 Heisenberg antiferromagnet on a triangular lattice with competing bonds

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

We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-$\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings ${J}_{1}>0$ and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, in the window $0\ensuremath{\le}\ensuremath{\kappa}<1$. The classical version of the model has a single GS phase transition at ${\ensuremath{\kappa}}^{\mathrm{cl}}=\frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) ${120}^{\ensuremath{\circ}}$ N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}^{\mathrm{cl}}$ to an infinitely degenerate family of 4-sublattice AFM N\'eel phases for $\ensuremath{\kappa}>{\ensuremath{\kappa}}^{\mathrm{cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\ensuremath{\kappa}=0$ of the spin-$\frac{1}{2}$ model for the GS energy per spin, $E/N=\ensuremath{-}0.5521(2){J}_{1}$, and the GS magnetic order parameter, $M=0.198(5)$ (in units where the classical value is ${M}^{\mathrm{cl}}=\frac{1}{2}$), which are among the best available. For the spin-$\frac{1}{2}$ ${J}_{1}\ensuremath{-}{J}_{2}$ model we find that the classical transition at $\ensuremath{\kappa}={\ensuremath{\kappa}}^{\mathrm{cl}}$ is split into two quantum phase transitions at ${\ensuremath{\kappa}}_{1}^{c}=0.060(10)$ and ${\ensuremath{\kappa}}_{2}^{c}=0.165(5)$. The two quasiclassical AFM states (viz., the ${120}^{\ensuremath{\circ}}$ N\'eel state and the striped state) are found to be the stable GS phases in the regime $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{1}^{c}$ and $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}^{c}$, respectively, while in the intermediate regimes ${\ensuremath{\kappa}}_{1}^{c}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2}^{c}$ the stable GS phase has no evident long-range magnetic order.

arXiv:1410.6003•Physical Review B•Jan 22, 2015•86 citations
We study the zero-temperature phase diagram and the low-lying excitations of a square-lattice spin-half Heisenberg antiferromagnet with two types of regularly distributed nearest-neighbor exchange bonds [J>0[J>0[J>0 (antiferromagnetic) and \ensuremath−\ensuremath∞<J\ensuremath′<\ensuremath∞]\ensuremath{-}\ensuremath{\infty}<{J}^{\ensuremath{'}}<\ensuremath{\infty}]\ensuremath−\ensuremath∞<J\ensuremath′<\ensuremath∞] using the coupled cluster method (CCM) for high orders of approximation (up to LSUB8). We use a N'eel model state as well as a helical model state as a starting point for the CCM calculations. We find a second-order transition from a phase with N'eel order to a finite-gap quantum disordered phase for sufficiently large antiferromagnetic exchange constants J\ensuremath′>0.{J}^{\ensuremath{'}}>0.J\ensuremath′>0. For frustrating ferromagnetic couplings J\ensuremath′<0{J}^{\ensuremath{'}}<0J\ensuremath′<0 we find indications that quantum fluctuations favor a first-order phase transition from the N'eel order to a quantum helical state, by contrast with the corresponding second-order transition in the corresponding classical model. The results are compared to those of exact diagonalizations of finite systems (up to 32 sites) and those of spin-wave and variational calculations. The CCM results agree well with the exact diagonalization data over the whole range of the parameters. The special case of J\ensuremath′=0,{J}^{\ensuremath{'}}=0,J\ensuremath′=0, which is equivalent to the honeycomb lattice, is treated more closely.
Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half (s=12s=\frac{1}{2}s=21​) J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength J1{J}_{1}J1​ and two next-nearest-neighbor (diagonal) bonds of strength J2{J}_{2}J2​. The J2{J}_{2}J2​ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no J2{J}_{2}J2​ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when J1{J}_{1}J1​ and J2{J}_{2}J2​ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum s=12s=\frac{1}{2}s=21​ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the s=12s=\frac{1}{2}s=21​ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the s=12s=\frac{1}{2}s=21​ model are of continuous deconfined type.
We study the ground-state (GS) properties of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2>0{J}_{2}>0J2​>0) and next-next-nearest-neighbor (J3>0{J}_{3}>0J3​>0) exchanges, for the case J3=J2{J}_{3}={J}_{2}J3​=J2​. 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 J2cl=17{J}_{2}^{\mathrm{cl}}=\frac{1}{7}J2cl​=71​ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​. 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 J1=+1{J}_{1}=+1J1​=+1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2{J}_{2}J2​. 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 J1=+1{J}_{1}=+1J1​=+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 J1=+1{J}_{1}=+1J1​=+1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for J2c1<J2<J2c2{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}J2c1​​<J2​<J2c2​​, 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 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​.
Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2 J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.
We use the coupled cluster method (CCM) to study the zero-temperature phase diagram of a two-dimensional frustrated spin-half antiferromagnet, the so-called Union Jack model. It is defined on a square lattice such that all nearest-neighbor pairs are connected by bonds with a strength J1>0{J}_{1}>0J1​>0, but only half the next-nearest-neighbor pairs are connected by bonds with a strength ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$. The bonds are arranged such that on the $2\ifmmode\times\else\texttimes\fi{}2$ unit cell they form the pattern of the Union Jack flag. Alternating sites on the square lattice are thus four-connected and eight-connected. We find strong evidence for a first phase transition between a N\'eel antiferromagnetic phase and a canted ferrimagnetic phase at a critical coupling ${\ensuremath{\kappa}}_{{c}_{1}}=0.66\ifmmode\pm\else\textpm\fi{}0.02$. The transition is an interesting one, at which the energy and its first derivative seem continuous, thus providing a typical scenario of a second-order transition (just as in the classical case for the model), although a weakly first-order transition cannot be excluded. By contrast, the average on-site magnetization approaches a nonzero value ${M}_{{c}_{1}}=0.195\ifmmode\pm\else\textpm\fi{}0.005$ on both sides of the transition, which is more typical of a first-order transition. The slope, $dM/d\ensuremath{\kappa}$, of the order parameter curve as a function of the coupling strength $\ensuremath{\kappa}$, also appears to be continuous, or very nearly so, at the critical point ${\ensuremath{\kappa}}_{{c}_{1}}$, thereby providing further evidence of the subtle nature of the transition between the N\'eel and canted phases. Our CCM calculations provide strong evidence that the canted ferrimagnetic phase becomes unstable at large values of $\ensuremath{\kappa}$, and hence we have also used the CCM with a model collinear semistripe-ordered ferrimagnetic state in which alternating rows (and columns) are ferromagnetically and antiferromagnetically ordered, and in which the spins connected by J2{J}_{2}J2​ bonds are antiparallel to one another. We find tentative evidence, based on the relative energies of the two states, for a second zero-temperature phase transition between the canted and semistripe-ordered ferrimagnetic states at a large value of the coupling parameter around ${\ensuremath{\kappa}}_{{c}_{2}}\ensuremath{\approx}125\ifmmode\pm\else\textpm\fi{}5$. This prediction, however, is based on an extrapolation of the CCM results for the canted state into regimes where the solutions have already become unstable and the CCM equations based on the canted state at any level of approximation beyond the lowest have no solutions. Our prediction for ${\ensuremath{\kappa}}_{{c}_{2}}$ is hence less reliable than that for ${\ensuremath{\kappa}}_{{c}_{1}}$. Nevertheless, if this second transition at ${\ensuremath{\kappa}}_{{c}_{2}}$ does exist, our results clearly indicate it to be of first-order type.
We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling J1>0{J}_{1}>0J1​>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 use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-12\frac{1}{2}21​ ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings J1>0{J}_{1}>0J1​>0 and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, in the window $0\ensuremath{\le}\ensuremath{\kappa}<1$. The classical version of the model has a single GS phase transition at ${\ensuremath{\kappa}}^{\mathrm{cl}}=\frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) ${120}^{\ensuremath{\circ}}$ N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}^{\mathrm{cl}}$ to an infinitely degenerate family of 4-sublattice AFM N\'eel phases for $\ensuremath{\kappa}>{\ensuremath{\kappa}}^{\mathrm{cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\ensuremath{\kappa}=0$ of the spin-12\frac{1}{2}21​ model for the GS energy per spin, $E/N=\ensuremath{-}0.5521(2){J}_{1}$, and the GS magnetic order parameter, M=0.198(5)M=0.198(5)M=0.198(5) (in units where the classical value is Mcl=12{M}^{\mathrm{cl}}=\frac{1}{2}Mcl=21​), which are among the best available. For the spin-12\frac{1}{2}21​ ${J}_{1}\ensuremath{-}{J}_{2}$ model we find that the classical transition at $\ensuremath{\kappa}={\ensuremath{\kappa}}^{\mathrm{cl}}$ is split into two quantum phase transitions at ${\ensuremath{\kappa}}_{1}^{c}=0.060(10)$ and ${\ensuremath{\kappa}}_{2}^{c}=0.165(5)$. The two quasiclassical AFM states (viz., the ${120}^{\ensuremath{\circ}}$ N\'eel state and the striped state) are found to be the stable GS phases in the regime $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{1}^{c}$ and $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}^{c}$, respectively, while in the intermediate regimes ${\ensuremath{\kappa}}_{1}^{c}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2}^{c}$ the stable GS phase has no evident long-range magnetic order.