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:1204.2390 · Physical Review B

Phase diagram of a frustrated Heisenberg antiferromagnet on the honeycomb lattice: The J1 - J2 - J3 model

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

Published Oct 9, 2012• 99 citations
arXivHTMLPDF

Abstract

We use the coupled-cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2 ${J}_{1}$-${J}_{2}$-${J}_{3}$ model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local onsite magnetization. With the nearest-neighbor coupling strength ${J}_{1}\ensuremath{\equiv}1$, we find four magnetically ordered phases in the parameter window ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, namely, the N\'eel, striped, and N\'eel-II collinear AFM phases, plus a spiral phase. The N\'eel-II phase appears as a stable GS phase in the classical version of the model only for values ${J}_{3}<0$. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at (a) $({J}_{2}^{{c}_{1}},{J}_{3}^{{c}_{1}})=(0.51\ifmmode\pm\else\textpm\fi{}0.01,0.69\ifmmode\pm\else\textpm\fi{}0.01)$ between the N\'eel, striped, and QP phases; (b) $({J}_{2}^{{c}_{2}},{J}_{3}^{{c}_{2}})=(0.65\ifmmode\pm\else\textpm\fi{}0.02,0.55\ifmmode\pm\else\textpm\fi{}0.01)$ between the striped, spiral, and QP phases; and (c) $({J}_{2}^{{c}_{3}},{J}_{3}^{{c}_{3}})=(0.69\ifmmode\pm\else\textpm\fi{}0.01,0.12\ifmmode\pm\else\textpm\fi{}0.01)$ between the spiral, N\'eel-II, and QP phases.

DOI
10.1103/physrevb.86.144404

Related papers

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

arXiv:1109.6229•Physical Review B•Feb 21, 2012•31 citations

Frustrated Heisenberg antiferromagnet on the honeycomb lattice: A candidate for deconfined quantum criticality

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

We study the ground-state (gs) phase diagram of the frustrated spin-$\frac{1}{2}$ ${J}_{1}$--${J}_{2}$--${J}_{3}$ antiferromagnet with ${J}_{2}={J}_{3}=\ensuremath{\kappa}{J}_{1}$ on the honeycomb lattice using coupled-cluster theory and exact diagonalization methods. We present results for the gs energy, magnetic order parameter, spin-spin correlation function, and plaquette valence-bond crystal (PVBC) susceptibility. We find a N\'eel antiferromagnetic (AFM) phase for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}\ensuremath{\approx}0.47$, a collinear striped AFM phase for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}\ensuremath{\approx}0.60$, and a paramagnetic PVBC phase for ${\ensuremath{\kappa}}_{{c}_{1}}\ensuremath{\lesssim}\ensuremath{\kappa}\ensuremath{\lesssim}{\ensuremath{\kappa}}_{{c}_{2}}$. The transition at ${\ensuremath{\kappa}}_{{c}_{2}}$ appears to be of first-order type, while that at ${\ensuremath{\kappa}}_{{c}_{1}}$ is continuous. Since the N\'eel and PVBC phases break different symmetries our results favor the deconfinement scenario for the transition at ${\ensuremath{\kappa}}_{{c}_{1}}$.

arXiv:1103.3856•Physical Review B•Jul 8, 2011•65 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

Frustrated Heisenberg antiferromagnet on the honeycomb lattice with spin quantum numbers≥ 1

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

The ground-state (GS) phase diagram of the frustrated spin- s J 1 -J 2 -J 3 Heisenberg antiferromagnet on the honeycomb lattice is studied using the coupled cluster method implemented to high orders of approximation, for spin quantum numbers s = 1, 3/2, 2 , 5/2. The model has antiferromagnetic (AFM) nearest-neighbour, next-nearest-neighbour and next-next-nearest-neighbour exchange couplings (with strength J 1 > 0, J 2 > 0 and J 3 > 0, respectively). We specifically study the case J 3 = J 2 = κJ 1 , in the range 0 < κ < 1 of the frustration parameter, which includes the point of maximum classical (s → ∞ ) frustration, viz., the classical critical point at κ cl = 1/2 , which separates the Neel phase for κ < κ cl and the collinear striped AFM phase for κ > κ cl . Results are presented for the GS energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. For all spins s > 3/2 we find a quantum phase diagram very similar to the classical one, with a direct first-order transition between the two collinear AFM states at a value κ c (s) which is slightly greater than κ c l [e.g., κ c (3/2) ≈ 0.53(1)] and which approaches it monotonically as s → ∞ . By contrast, for the case s = 1 the transition is split into two such that the stable GS phases are one with Néel AFM order for κ < κ c1 = 0.485(5) and one with striped AFM order for κ > κ c2 = 0.528(5), just as in the case s = 1/2 (for which κ c1 ≈ 0.47 and κ c2 ≈ 0.60). For both the s = 1/2 and s = 1 models the transition at κ c2 appears to be of first-order type, while that at κ c1 appears to be continuous. However, whereas in the s = 1/2 case the intermediate phase appears to have PVBC order over the entire range κ c1 < κ < κ c2 , in the s = 1 case PVBC ordering either exists only over a very small part of the region or, more likely, is absent everywhere.

arXiv:1511.00499•Journal of Physics Conference Series•Mar 1, 2016•7 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.

arXiv:1202.6249•Physical Review B•Apr 20, 2012•27 citations

Quantum phases of the planar antiferromagnetic J1−J2−J3 Heisenberg model

Johannes Reuther, P. Wölfle, R. Darradi +3 more

We present results for a complementary analysis of the frustrated planar ${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ spin-1/2 quantum antiferromagnet (AFM). Using dynamical functional renormalization group, high-order-coupled cluster calculations, and series expansion based on the flow equation method, we have calculated generalized momentum-resolved susceptibilities, the ground-state energy, the magnetic-order parameter, and the elementary excitation gap. From these, we determine a quantum phase diagram that shows a large window of a quantum paramagnetic (QP) phase situated among the N\'eel, spiral, and collinear states, which are present already in the classical ${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ AFM. Our findings are consistent with substantial plaquette correlations in the QP phase. The extent of the QP region is found to be in satisfying agreement between the three different approaches we have employed.

arXiv:1010.3098•Physical Review B•Feb 17, 2011•76 citations
We use the coupled-cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2 J1{J}_{1}J1​-J2{J}_{2}J2​-J3{J}_{3}J3​ model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local onsite magnetization. With the nearest-neighbor coupling strength J1\ensuremath≡1{J}_{1}\ensuremath{\equiv}1J1​\ensuremath≡1, we find four magnetically ordered phases in the parameter window J2,J3\ensuremath∈[0,1]{J}_{2},{J}_{3}\ensuremath{\in}[0,1]J2​,J3​\ensuremath∈[0,1], namely, the N'eel, striped, and N'eel-II collinear AFM phases, plus a spiral phase. The N'eel-II phase appears as a stable GS phase in the classical version of the model only for values J3<0{J}_{3}<0J3​<0. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at (a) (J2c1,J3c1)=(0.51\ifmmode±\else\textpm\fi0.01,0.69\ifmmode±\else\textpm\fi0.01)({J}_{2}^{{c}_{1}},{J}_{3}^{{c}_{1}})=(0.51\ifmmode\pm\else\textpm\fi{}0.01,0.69\ifmmode\pm\else\textpm\fi{}0.01)(J2c1​​,J3c1​​)=(0.51\ifmmode±\else\textpm\fi0.01,0.69\ifmmode±\else\textpm\fi0.01) between the N'eel, striped, and QP phases; (b) (J2c2,J3c2)=(0.65\ifmmode±\else\textpm\fi0.02,0.55\ifmmode±\else\textpm\fi0.01)({J}_{2}^{{c}_{2}},{J}_{3}^{{c}_{2}})=(0.65\ifmmode\pm\else\textpm\fi{}0.02,0.55\ifmmode\pm\else\textpm\fi{}0.01)(J2c2​​,J3c2​​)=(0.65\ifmmode±\else\textpm\fi0.02,0.55\ifmmode±\else\textpm\fi0.01) between the striped, spiral, and QP phases; and (c) (J2c3,J3c3)=(0.69\ifmmode±\else\textpm\fi0.01,0.12\ifmmode±\else\textpm\fi0.01)({J}_{2}^{{c}_{3}},{J}_{3}^{{c}_{3}})=(0.69\ifmmode\pm\else\textpm\fi{}0.01,0.12\ifmmode\pm\else\textpm\fi{}0.01)(J2c3​​,J3c3​​)=(0.69\ifmmode±\else\textpm\fi0.01,0.12\ifmmode±\else\textpm\fi0.01) between the spiral, N'eel-II, and QP phases.
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​.
We study the ground-state (gs) phase diagram of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ antiferromagnet with ${J}_{2}={J}_{3}=\ensuremath{\kappa}{J}_{1}$ on the honeycomb lattice using coupled-cluster theory and exact diagonalization methods. We present results for the gs energy, magnetic order parameter, spin-spin correlation function, and plaquette valence-bond crystal (PVBC) susceptibility. We find a N\'eel antiferromagnetic (AFM) phase for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}\ensuremath{\approx}0.47$, a collinear striped AFM phase for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}\ensuremath{\approx}0.60$, and a paramagnetic PVBC phase for ${\ensuremath{\kappa}}_{{c}_{1}}\ensuremath{\lesssim}\ensuremath{\kappa}\ensuremath{\lesssim}{\ensuremath{\kappa}}_{{c}_{2}}$. The transition at ${\ensuremath{\kappa}}_{{c}_{2}}$ appears to be of first-order type, while that at ${\ensuremath{\kappa}}_{{c}_{1}}$ is continuous. Since the N\'eel and PVBC phases break different symmetries our results favor the deconfinement scenario for the transition at ${\ensuremath{\kappa}}_{{c}_{1}}$.
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 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 ${\ensuremath{\alpha}}_{{c}_{2}}(0)>\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2}^{<}\ensuremath{\approx}0.56(1)$.
The ground-state (GS) phase diagram of the frustrated spin- s J 1 -J 2 -J 3 Heisenberg antiferromagnet on the honeycomb lattice is studied using the coupled cluster method implemented to high orders of approximation, for spin quantum numbers s = 1, 3/2, 2 , 5/2. The model has antiferromagnetic (AFM) nearest-neighbour, next-nearest-neighbour and next-next-nearest-neighbour exchange couplings (with strength J 1 > 0, J 2 > 0 and J 3 > 0, respectively). We specifically study the case J 3 = J 2 = κJ 1 , in the range 0 < κ < 1 of the frustration parameter, which includes the point of maximum classical (s → ∞ ) frustration, viz., the classical critical point at κ cl = 1/2 , which separates the Neel phase for κ < κ cl and the collinear striped AFM phase for κ > κ cl . Results are presented for the GS energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. For all spins s > 3/2 we find a quantum phase diagram very similar to the classical one, with a direct first-order transition between the two collinear AFM states at a value κ c (s) which is slightly greater than κ c l [e.g., κ c (3/2) ≈ 0.53(1)] and which approaches it monotonically as s → ∞ . By contrast, for the case s = 1 the transition is split into two such that the stable GS phases are one with Néel AFM order for κ < κ c1 = 0.485(5) and one with striped AFM order for κ > κ c2 = 0.528(5), just as in the case s = 1/2 (for which κ c1 ≈ 0.47 and κ c2 ≈ 0.60). For both the s = 1/2 and s = 1 models the transition at κ c2 appears to be of first-order type, while that at κ c1 appears to be continuous. However, whereas in the s = 1/2 case the intermediate phase appears to have PVBC order over the entire range κ c1 < κ < κ c2 , in the s = 1 case PVBC ordering either exists only over a very small part of the region or, more likely, is absent everywhere.
We use the coupled cluster method (CCM) to investigate 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​ model on the honeycomb lattice, with nearest-neighbor exchange coupling J1{J}_{1}J1​ plus next-nearest-neighbor (J2{J}_{2}J2​) and next-next-nearest-neighbor (J3{J}_{3}J3​) 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 J1>0{J}_{1}>0J1​>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 present results for a complementary analysis of the frustrated planar ${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ spin-1/2 quantum antiferromagnet (AFM). Using dynamical functional renormalization group, high-order-coupled cluster calculations, and series expansion based on the flow equation method, we have calculated generalized momentum-resolved susceptibilities, the ground-state energy, the magnetic-order parameter, and the elementary excitation gap. From these, we determine a quantum phase diagram that shows a large window of a quantum paramagnetic (QP) phase situated among the N\'eel, spiral, and collinear states, which are present already in the classical ${J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3}$ AFM. Our findings are consistent with substantial plaquette correlations in the QP phase. The extent of the QP region is found to be in satisfying agreement between the three different approaches we have employed.