Finding an exact solution for a realistic interacting quantum many-body problem is often challenging. There are only a few problems where an exact solution can be found, usually in a narrow parameter space. Here, we propose a spin-
21 Heisenberg model on a square lattice with spatial anisotropy and bond depletion for the nearest-neighbor antiferromagnetic interactions but not for the next-nearest-neighbor interactions. This model has an exact and unique dimer ground state at
J2/J1=21; a dimer state is a product state of spin singlets on dimers (here, staggered nearest-neighbor bonds). We examine this model by employing the bond-operator mean-field theory and exact diagonalization. These analytical and numerical methods precisely affirm the correctness of the dimer ground state at the exact point
(J2/J1=21). As one moves away from the exact point, the dimer order melts and vanishes when the spin gap becomes zero. The mean-field theory with harmonic approximation indicates that the dimer order persists for $\ensuremath{-}0.35\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}1.35$. However, in nonharmonic approximation, the upper critical point lowers by 0.28 to 1.07, but the lower critical point remains intact. The exact diagonalization results suggest that the latter approximation fares better. The model reveals N\'eel order below the lower critical point and stripe magnetic order above the upper critical point. It has a topologically equivalent model on a honeycomb lattice where the nearest-neighbor interactions are still spatial anisotropic, but the bond depletion shifts into the isotropic next-neighbor interactions. Moreover, these models can also be generalized in the three dimensions.
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 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.
Using the density matrix renormalization group, we determine the phase diagram of the spin-1/2 Heisenberg antiferromagnet on a honeycomb lattice with a nearest-neighbor interaction J(1) and a frustrating, next-nearest-neighbor exchange J(2). As frustration increases, the ground state exhibits Néel, plaquette, and dimer orders, with critical points at J(2)/J(1) = 0.22 and 0.35. We observe that both the spin gap and the corresponding order parameters vanish continuously at both the critical points, indicating the presence of deconfined quantum criticality.
We study the ground-state (GS) properties of the frustrated spin-
21 J1--
J2--
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) and next-next-nearest-neighbor (
J3>0) exchanges, for the case
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=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
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). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter
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) 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), 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, 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
101<J2<51.
In this paper, we investigate the ground-state phase diagram of the
S=1/2 Heisenberg-$\mathrm{\ensuremath{\Gamma}}$ model on a honeycomb lattice by dimer series expansion and exact diagonalization. We focus on the effects of the anisotropy of the interactions; by tuning the coupling constants, the system changes between the isolated dimer and the spin-chain models. We find that, in the spin-chain limit, there are three kinds of states: A Tomonaga-Luttinger liquid and two magnetically long-range-ordered states. All three states become two-dimensional long-range-ordered states by the infinitesimal interchain interaction except for the case where the Heisenberg interaction is much weaker than the off-diagonal symmetric $(\mathrm{\ensuremath{\Gamma}})$ interaction. Starting from the isolated dimer limit, a triplet dimer phase survives up to the isotropically interacting system in a large part of the phase diagram where the Heisenberg and $\mathrm{\ensuremath{\Gamma}}$ interactions are ferromagnetic and antiferromagnetic, respectively. Otherwise, a phase transition to a magnetically ordered phase occurs before the interaction becomes isotropic. This indicates that the quantum spin liquid proposed in the $\mathrm{\ensuremath{\Gamma}}$ model [A. Catuneanu et al., npj Quantum Mater. 3, 23 (2018)] is unstable against the anisotropy of the interactions.
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.