We study the zero-temperature phase diagram of the J 1 XXZ – J 2 XXZ Heisenberg model for spin-1 particles on an infinite square lattice interacting via nearest-neighbour ( ) and next-nearest-neighbour ( J 2 >0) bonds. The two bonds have the same XXZ -type anisotropy in spin space. The effects on the quasiclassical Néel-ordered and collinear stripe-ordered states of varying the anisotropy parameter Δ are investigated using the coupled cluster method carried out up to high orders. By contrast with the case for spin- particles studied previously, no intermediate disordered phase between the Néel and collinear stripe phases, for any value of the frustration J 2 / J 1 , for either the z -aligned (Δ>1) or xy -planar-aligned (0≤Δ<1) states, is predicted here. The quantum phase transition is determined as first order for all values of J 2 / J 1 and Δ. The position of the phase boundary J 2 c (Δ) is determined accurately. It is observed to deviate most from its classical position (for all values of Δ>0) at the Heisenberg isotropic point (Δ = 1), where J 2 c (1) = 0.55 ± 0.01. By contrast, at the XY isotropic point (Δ = 0), we find J 2 c (0) = 0.50 ± 0.01. In the Ising limit ( ), as expected.
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 ${\ensuremath{\alpha}}^{{c}_{1}}({\ensuremath{\Delta}}^{c})={\ensuremath{\alpha}}^{{c}_{2}}({\ensuremath{\Delta}}^{c})$.
We use a variational mean-field-like approach, the coupled cluster method (CCM), and exact diagonalization to investigate the ground-state order–disorder transition for the square lattice spin-half XXZ model with two different nearest-neighbour couplings J and . On increasing the model shows in the isotropic Heisenberg limit a second-order transition from semi-classical Néel order to a quantum paramagnetic phase with enhanced local dimer correlations on the bonds at about . This transition is driven by the quantum competition between and J . On increasing the anisotropy parameter Δ>1 we diminish the quantum fluctuations and thus the degree of competition. As a result the transition point is shifted to larger values. We find indications for a linear increase of with Δ, i.e. the transition disappears in the Ising limit .
We study the spin-1/2 and spin-1 [Formula: see text] Heisenberg antiferromagnets (HAFs) on an infinite, anisotropic, two-dimensional triangular lattice, using the coupled cluster method. With respect to an underlying square-lattice geometry the model contains antiferromagnetic (J1 > 0) bonds between nearest neighbours and competing [Formula: see text] bonds between next-nearest-neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry the model has two sorts of nearest-neighbour bonds, with [Formula: see text] bonds along parallel chains and with J1 bonds providing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one extreme (κ = 0) and a set of decoupled chains at the other (κ → ∞), with the isotropic HAF on the triangular lattice in between at κ = 1. For the spin-1/2 [Formula: see text] model, we find a weakly first-order (or possibly second-order) quantum phase transition from a Néel-ordered state to a helical state at a first critical point at κc1 = 0.80 ± 0.01, and a second critical point at κc2 = 1.8 ± 0.4 where a first-order transition occurs between the helical state and a collinear stripe-ordered state. For the corresponding spin-1 model we find an analogous transition of the second-order type at κc1 = 0.62 ± 0.01 between states with Néel and helical ordering, but we find no evidence of a further transition in this case to a stripe-ordered phase.
We study the phase diagram of the isotropic J 1 – J 1 '– J 2 Heisenberg model for spin-1 particles on an anisotropic square lattice, using the coupled-cluster method. We find no evidence for an intermediate phase between the Néel and stripe states, as compared with all previous results for the corresponding spin-1/2 case. However, we find a quantum tricritical point at J 1 '/ J 1 ≈0.66±0.03, J 2 / J 1 ≈0.35±0.02, where a line of second-order phase transitions between the quasi-classical Néel and stripe-ordered phases (for J 1 '/ J 1 ≲0.66) meets a line of first-order phase transitions between the same two states (for J 1 '/ J 1 ≳0.66).
The zero-temperature ground-state (GS) properties and phase diagram of a frustrated spin-1 J 1 - J 2 Heisenberg model on the checkerboard square lattice are studied, using the coupled cluster method. We consider the case where the nearest-neighbour exchange bonds have strength J 1 > 0 and the next-nearest-neighbour exchange bonds present (viz., in the checkerboard pattern of the planar pyrochlore) have strength J 2 = κ J 1 > 0. We find significant differences from both the spin-1/2 and classical versions of the model. We find that the spin-1 model has a first phase transition at κ C1 ≈ 1.00 ± 0.01 (as does the classical model at κ cl = 1) between two antiferromagnetic phases, viz., a quasiclassical Néel phase (for κ < κ C1 ) and one of the infinitely degenerate family of quasiclassical phases (for k > κ C1 ) that exists in the classical model for κ > κ c1 , which is now chosen by the order by disorder mechanism as (probably) the "doubled Néel" (or Néel*) state. By contrast, none of this family survives quantum fluctuations to form a stable GS phase in the spin-1/2 case. We also find evidence for a second quantum critical point at κ C2 ≈ 2.0 ± 0.5 in the spin-1 model, such that for κ > κ C2 the quasiclassical (Néel*) ordering melts and a nonclassical phase appears, which, on the basis of preliminary evidence, appears unlikely to have crossed-dimer valence-bond crystalline (CDVBC) ordering, as in the spin-1/2 case. Unlike in the spin-1/2 case, where the Néel and CDVBC phases are separated by a phase with plaquette valence-bond crystalline (PVBC) ordering, we find very preliminary evidence for such a PVBC state in the spin-1 model for all κ > κ C2 .
Using the coupled-cluster method for infinite lattices and the exact diagonalization method for finite lattices, we study the influence of an exchange anisotropy Δ on the ground-state phase diagram of the spin-1/2 frustrated J 1 - J 2 XXZ antiferromagnet on the square lattice. We find that increasing Δ > 1 (i.e. an Ising type easy-axis anisotropy) as well as decreasing Δ < 1 (i.e. an XY type easy-plane anisotropy) both lead to a monotonic shrinking of the parameter region of the magnetically disordered quantum phase. Finally, at Δ c ≈ 1.9 this quantum phase disappears, whereas in the pure XY limit (Δ = 0) there is still a narrow region around J 2 = 0.5 J 1 where the quantum paramagnetic ground-state phase exists.