Motivated by recent numerical work, we use the boson-vortex duality to study the possible phases of the frustrated spin-$\frac{1}{2}\phantom{\rule{4pt}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}$ XXZ models on the honeycomb lattice. By condensing the vortices, we obtain various gapped phases that either break certain lattice symmetry or preserve all the symmetries. The gapped phases breaking lattice symmetries occur when the vortex band structure has two minima. Condensing one of the two vortex flavors leads to an Ising ordered phase, while condensing both vortex flavors gives rise to a valence-bond-solid state. Both of those phases have been observed in the numerical studies of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ XXZ honeycomb model. Furthermore, by tuning the band structure of vortex and condensing it at the $\mathrm{\ensuremath{\Gamma}}$ point, we obtain a featureless paramagnet. But the precise nature of this featureless state is still unclear and needs future study.
We study the zero-temperature ground-state (GS) phase diagram of a spin-half
J1--
J2 XY model on the honeycomb lattice with nearest-neighbor exchange coupling
J1>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.
Strongly correlated systems with geometric frustrations can host the emergent phases of matter with unconventional properties. Here, we study the spin
S=1 Heisenberg model on the honeycomb lattice with the antiferromagnetic first- (
J1) and second-neighbor (
J2) interactions ($0.0\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$) by means of density-matrix renormalization group. In the parameter regime ${J}_{2}/{J}_{1}\ensuremath{\lesssim}0.27$, the system sustains a N\'eel antiferromagnetic phase. At the large
J2 side ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.32$, a stripe antiferromagnetic phase is found. Between the two magnetic ordered phases $0.27\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.32$, we find a nonmagnetic intermediate region with a plaquette valence-bond order. Although our calculations are limited within a six unit-cell width on cylinder, we present evidence that this plaquette state could be a strong candidate for this nonmagnetic region in the thermodynamic limit. We also briefly discuss the nature of the quantum phase transitions in the system. We gain further insight into the nonmagnetic phases in the spin-1 system by comparing its phase diagram with the spin-
1/2 system.
We study the ground-state (gs) phase diagram of the frustrated spin-
21 J1--
J2--
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 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.
We present a comprehensive computational study of the phase diagram of the frustrated
S=1/2 Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest
(J2) and third-neighbor
(J3) couplings. Using a combination of exact diagonalizations (EDs) of the original spin model, of the Hamiltonian projected into the nearest-neighbor short-range valence-bond basis, and of an effective quantum dimer model, as well as a self-consistent cluster mean-field theory, we determine the boundaries of several magnetically ordered phases in the region ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a plaquette valence-bond crystal phase. At larger
J2, we locate a sizable region in which staggered valence-bond crystal correlations are found to be important, either due to genuine valence-bond crystal (VBC) ordering or as a consequence of magnetically ordered phases, which break lattice rotational symmetry. Furthermore, we find that a particular parameter-free Gutzwiller projected tight-binding wave function has remarkably accurate energies compared to finite-size extrapolated ED energies along the transition line from conventional N\'eel to plaquette VBC phases, a fact that points to possibly interesting critical behavior---such as a deconfined critical point---across this transition. We also comment on the relevance of this spin model to model the spin liquid region found in the half filled Hubbard model on the honeycomb lattice.