We study the
S=1 square lattice Heisenberg antiferromagnet with spatially anisotropic nearest-neighbor couplings
J1x and
J1y frustrated by a next-nearest-neighbor coupling
J2 numerically using the density-matrix renormalization-group (DMRG) method and analytically employing the Schwinger-Boson mean-field theory (SBMFT). Up to relatively strong values of the anisotropy, within both methods we find quantum fluctuations to stabilize the N\'eel-ordered state above the classically stable region. Whereas SBMFT suggests a fluctuation-induced first-order transition between the N\'eel state and a stripe antiferromagnet for $1/3\ensuremath{\le}{J}_{1x}/{J}_{1y}\ensuremath{\le}1$ and an intermediate paramagnetic region opening only for very strong anisotropy, the DMRG results clearly demonstrate that the two magnetically ordered phases are separated by a quantum-disordered region for all values of the anisotropy with the remarkable implication that the quantum paramagnetic phase of the spatially isotropic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model is continuously connected to the limit of decoupled Haldane spin chains. Our findings indicate that for
S=1 quantum fluctuations in strongly frustrated antiferromagnets are crucial and not correctly treated on the semiclassical level.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
We study the zero-temperature phase diagram of the two-dimensional quantum
J1XXZ−−J2XXZ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\ensuremath{\Delta}$ on the
z-aligned N\'eel and (collinear) stripe states, as well as on the
xy-planar-aligned N\'eel and collinear stripe states, are examined. All four of these quasiclassical states are chosen in turn as model states, on top of which we systematically include the quantum correlations using a coupled cluster method analysis carried out to very high orders. We find strong evidence for two quantum triple points (QTPs) at $({\ensuremath{\Delta}}^{c}=\ensuremath{-}0.10\ifmmode\pm\else\textpm\fi{}0.15,\text{ }{J}_{2}^{c}/{J}_{1}=0.505\ifmmode\pm\else\textpm\fi{}0.015)$ and $({\ensuremath{\Delta}}^{c}=2.05\ifmmode\pm\else\textpm\fi{}0.15,\text{ }{J}_{2}^{c}/{J}_{1}=0.530\ifmmode\pm\else\textpm\fi{}0.015)$, between which an intermediate magnetically disordered phase emerges to separate the quasiclassical N\'eel and stripe collinear phases. Above the upper QTP $(\ensuremath{\Delta}\ensuremath{\gtrsim}2.0)$ we find a direct first-order phase transition between the N\'eel and stripe phases, exactly as for the classical case. The
z-aligned and
xy-planar-aligned phases meet precisely at $\ensuremath{\Delta}=1$, also as for the classical case. For all values of the anisotropy parameter between those of the two QTPs there exists a narrow range of values of
J2/J1, ${\ensuremath{\alpha}}^{{c}_{1}}(\ensuremath{\Delta})<{J}_{2}/{J}_{1}<{\ensuremath{\alpha}}^{{c}_{2}}(\ensuremath{\Delta})$, centered near the point of maximum classical frustration,
J2/J1=21, for which the intermediate phase exists. This range is widest precisely at the isotropic point, $\ensuremath{\Delta}=1$, where ${\ensuremath{\alpha}}^{{c}_{1}}(1)=0.44\ifmmode\pm\else\textpm\fi{}0.01$ and ${\ensuremath{\alpha}}^{{c}_{2}}(1)=0.59\ifmmode\pm\else\textpm\fi{}0.01$. The two QTPs are characterized by values $\ensuremath{\Delta}={\ensuremath{\Delta}}^{c}$ at which ${\ensuremath{\alpha}}^{{c}_{1}}({\ensuremath{\Delta}}^{c})={\ensuremath{\alpha}}^{{c}_{2}}({\ensuremath{\Delta}}^{c})$.
The phase diagram of a spin-1/2 $J\ensuremath{-}{J}^{\ensuremath{'}}\ensuremath{-}{J}_{2}$ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the
J1−−J2 model on the square lattice with two different nearest-neighbor couplings ${J,J}^{\ensuremath{'}}$ and may also be viewed as an array of coupled Heisenberg chains. The results suggest that the resonating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B 67, 024422 (2003)] for ${J}_{2}{=0.5J}^{\ensuremath{'}}\ensuremath{\ll}J$ is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both sides by a columnar dimer long range order. This columnar order could extend for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}J$ which correspond to the
J1−−J2 model.
Using the density matrix renormalization group on wide cylinders, we study the phase diagram of the spin-1/2 XY model on the honeycomb lattice, with first-neighbor (J1=1) and frustrating second-neighbor (J2>0) interactions. For the intermediate frustration regime 0.22≲J2≲0.36, we find a surprising antiferromagnetic Ising phase, with ordered moments pointing along the z axis, despite the absence of any S(z)S(z) interactions in the Hamiltonian. Surrounding this phase as a function of J2 are antiferromagnetic phases with the moments pointing in the x-y plane for small J2 and a close competition between an x-y plane magnetic collinear phase and a dimer phase for large values of J2. We do not find any spin-liquid phases in this model.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest
(J1)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state
(T=0) and the finite-
T phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.