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arXiv:0805.3107 · Physical Review B

Effect of anisotropy on the ground-state magnetic ordering of the spin-half quantum J 1 X X Z – J 2 X X Z model on the square lattice

R. F. Bishop, P. H. Y. Li, R. Darradi, J. Schulenburg, Johannes Richter

Published Aug 11, 2008• 70 citations
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Abstract

We study the zero-temperature phase diagram of the two-dimensional quantum ${J}_{1}^{XXZ}--{J}_{2}^{XXZ}$ 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 ${J}_{2}/{J}_{1}$, ${\ensuremath{\alpha}}^{{c}_{1}}(\ensuremath{\Delta})<{J}_{2}/{J}_{1}<{\ensuremath{\alpha}}^{{c}_{2}}(\ensuremath{\Delta})$, centered near the point of maximum classical frustration, ${J}_{2}/{J}_{1}=\frac{1}{2}$, 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 ${\ensuremath{\alpha}}^{{c}_{1}}({\ensuremath{\Delta}}^{c})={\ensuremath{\alpha}}^{{c}_{2}}({\ensuremath{\Delta}}^{c})$.

DOI
10.1103/physrevb.78.054412

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We study the J1-J1'-J2 quantum spin model on the two-dimensional square lattice using the coupled cluster method. We compare and contrast the influence of the interchain coupling J1' on the zero-temperature phase diagrams for the two spin values s = 1/2 and s = 1. Our most important result for the s = 1/2 case is the predicted existence of a quantum triple point (QTP) at (J1' ≈ 0.60 ± 0.03, J2 ≈ 0.33 ± 0.02) when J1 = 1. Below the QTP (J1'/J1 ≤ 0.60) we predict a second-order phase transition between the quasi-classical Neel and stripe-ordered phases, whereas the corresponding classical model, which contains only these two phases for all spin values s, yields a first-order transition. Above the QTP (J1'/J1 ≥ 0.60) an intermediate disordered phase emerges, which has no classical counterpart. By contrast, the situation for s = 1 is qualitatively different. Instead of a QTP where three phases co-exist, we now predict a quantum tricritical point at (J1' ≈ 0.66 ± 0.03, J2 ≈ 0.35 ± 0.02) when J1 = 1, where a line of second-order phase transitions between the quasi classical Neel and stripe-ordered phases (for J1'/J1 ≤ 0.66) meets a line of first-order phase transitions between the same two states (for J1'/J1 ≥ 0.66). Surprisingly, we find no evidence at all for any intermediate disordered phase in the s = 1 case.
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We study the S=1S=1S=1 square lattice Heisenberg antiferromagnet with spatially anisotropic nearest-neighbor couplings J1x{J}_{1x}J1x​ and J1y{J}_{1y}J1y​ frustrated by a next-nearest-neighbor coupling J2{J}_{2}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=1S=1S=1 quantum fluctuations in strongly frustrated antiferromagnets are crucial and not correctly treated on the semiclassical level.
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