Effect of anisotropy on the ground-state magnetic ordering of the spin-one quantum J1XXZJ_{1}^{XXZ}--J2XXZJ_{2}^{XXZ} model on the square lattice

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 ($J_1 \equiv 1$) and next-nearest-neighbour ($J_2 > 0$) bonds. Both bonds have the same $XXZ$-type anisotropy in spin space. The effects on the quasiclassical N\'{e}el-ordered and collinear stripe-ordered states of varying the anisotropy parameter $\Delta$ is investigated using the coupled cluster method carried out to high orders. By contrast with the spin-1/2 case studied previously, we predict no intermediate disordered phase between the N\'{e}el and collinear stripe phases, for any value of the frustration $J_2/J_1$, for either the $z$-aligned ($\Delta > 1$) or $xy$-planar-aligned ($0 \leq \Delta < 1$) states. The quantum phase transition is determined to be first-order for all values of $J_2/J_1$ and $\Delta$. The position of the phase boundary $J_{2}^{c}(\Delta)$ is determined accurately. It is observed to deviate most from its classical position $J_2^c = {1/2}$ (for all values of $\Delta > 0$) at the Heisenberg isotropic point ($\Delta = 1$), where $J_{2}^{c}(1) = 0.55 \pm 0.01$. By contrast, at the XY isotropic point ($\Delta = 0$), we find $J_{2}^{c}(0) = 0.50 \pm 0.01$. In the Ising limit ($\Delta \to \infty$) $J_2^c \to 0.5$ as expected.Comment: 20 pages, 5 figure

The ground-state magnetic ordering of the spin-1/2 frustrated J1-J2 XXZ model on the square lattice

Using the coupled-cluster method for infinite lattices and the exact diagonalization method for finite lattices, we study the influence of an exchange anisotropy Delta on the ground-state phase diagram of the spin-1/2 frustrated J1-J2 XXZ antiferromagnet on the square lattice. We find that increasing Delta>1 (i.e. an Ising type easy-axis anisotropy) as well as decreasing Delta<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 Delta~1.9 this quantum phase disappears, whereas in pure XY limit (Delta=0) there is still a narrow region around J2 =0.5J1 where the quantum paramagnetic ground-state phase exists.Comment: 4 pages, 6 figures, paper accepted for the proceedings of the conference HFM 200

A frustrated quantum spin-{\boldmath s} model on the Union Jack lattice with spins {\boldmath s>1/2}

The zero-temperature phase diagrams of a two-dimensional frustrated quantum antiferromagnetic system, namely the Union Jack model, are studied using the coupled cluster method (CCM) for the two cases when the lattice spins have spin quantum number $s=1$ and $s=3/2$. The system is defined on a square lattice and the spins interact via isotropic Heisenberg interactions such that all nearest-neighbour (NN) exchange bonds are present with identical strength $J_{1}>0$, and only half of the next-nearest-neighbour (NNN) exchange bonds are present with identical strength $J_{2} \equiv \kappa J_{1} > 0$. The bonds are arranged such that on the $2 \times 2$ unit cell they form the pattern of the Union Jack flag. Clearly, the NN bonds by themselves (viz., with $J_{2}=0$) produce an antiferromagnetic N\'{e}el-ordered phase, but as the relative strength $\kappa$ of the frustrating NNN bonds is increased a phase transition occurs in the classical case ($s \rightarrow \infty$) at $\kappa^{\rm cl}_{c}=0.5$ to a canted ferrimagnetic phase. In the quantum cases considered here we also find strong evidence for a corresponding phase transition between a N\'{e}el-ordered phase and a quantum canted ferrimagnetic phase at a critical coupling $\kappa_{c_{1}}=0.580 \pm 0.015$ for $s=1$ and $\kappa_{c_{1}}=0.545 \pm 0.015$ for $s=3/2$. In both cases the ground-state energy $E$ and its first derivative $dE/d\kappa$ seem continuous, thus providing a typical scenario of a second-order phase transition at $\kappa=\kappa_{c_{1}}$, although the order parameter for the transition (viz., the average ground-state on-site magnetization) does not go to zero there on either side of the transition.Comment: 1

Effect of anisotropy on the ground-state magnetic ordering of the spin-half quantum J1XXZJ_1^{XXZ}--J2XXZJ_2^{XXZ} model on the square lattice

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe states, as well as on the $xy$-planar-aligned N\'{e}el 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 {\it quantum triple points} (QTP's) at ($\Delta ^{c} = -0.10 \pm 0.15, J_{2}^{c}/J_{1} = 0.505 \pm 0.015$) and ($\Delta ^{c} = 2.05 \pm 0.15, J_{2}^{c}/J_{1} = 0.530 \pm 0.015$), between which an intermediate magnetically-disordered phase emerges to separate the quasiclassical N\'{e}el and stripe collinear phases. Above the upper QTP ($\Delta \gtrsim 2.0$) we find a direct first-order phase transition between the N\'{e}el and stripe phases, exactly as for the classical case. The $z$-aligned and $xy$-planar-aligned phases meet precisely at $\Delta = 1$, also as for the classical case. For all values of the anisotropy parameter between those of the two QTP's there exists a narrow range of values of $J_{2}/J_{1}$, $\alpha^{c_1}(\Delta)<J_{2}/J_{1} <\alpha^{c_2}(\Delta)$, centered near the point of maximum classical frustration, $J_{2}/J_{1} = {1/2}$, for which the intermediate phase exists. This range is widest precisely at the isotropic point, $\Delta = 1$, where $\alpha^{c_1}(1) = 0.44 \pm 0.01$ and $\alpha^{c_2}(1) = 0.59 \pm 0.01$. The two QTP's are characterized by values $\Delta = \Delta^{c}$ at which $\alpha^{c_1}(\Delta^{c})=\alpha^{c_2}(\Delta^{c})$.Comment: 28 pages, 5 figure

High-Order Coupled Cluster Method (CCM) Calculations for Quantum Magnets with Valence-Bond Ground States

In this article, we prove that exact representations of dimer and plaquette valence-bond ket ground states for quantum Heisenberg antiferromagnets may be formed via the usual coupled cluster method (CCM) from independent-spin product (e.g. N\'eel) model states. We show that we are able to provide good results for both the ground-state energy and the sublattice magnetization for dimer and plaquette valence-bond phases within the CCM. As a first example, we investigate the spin-half $J_1$--$J_2$ model for the linear chain, and we show that we are able to reproduce exactly the dimerized ground (ket) state at $J_2/J_1=0.5$. The dimerized phase is stable over a range of values for $J_2/J_1$ around 0.5. We present evidence of symmetry breaking by considering the ket- and bra-state correlation coefficients as a function of $J_2/J_1$. We then consider the Shastry-Sutherland model and demonstrate that the CCM can span the correct ground states in both the N\'eel and the dimerized phases. Finally, we consider a spin-half system with nearest-neighbor bonds for an underlying lattice corresponding to the magnetic material CaV$_4$O$_9$ (CAVO). We show that we are able to provide excellent results for the ground-state energy in each of the plaquette-ordered, N\'eel-ordered, and dimerized regimes of this model. The exact plaquette and dimer ground states are reproduced by the CCM ket state in their relevant limits.Comment: 34 pages, 13 figures, 2 table

Magnetic order in spin-1 and spin-3/2 interpolating square-triangle Heisenberg antiferromagnets

Using the coupled cluster method we investigate spin-$s$ $J_{1}$-$J_{2}'$ Heisenberg antiferromagnets (HAFs) on an infinite, anisotropic, triangular lattice when the spin quantum number $s=1$ or $s=3/2$. With respect to a square-lattice geometry the model has antiferromagnetic ($J_{1} > 0$) bonds between nearest neighbours and competing ($J_{2}' > 0$) bonds between next-nearest neighbours across only one of the diagonals of each square plaquette, the same one in each square. In a topologically equivalent triangular-lattice geometry, we have two types of nearest-neighbour bonds: namely the $J_{2}' \equiv \kappa J_{1}$ bonds along parallel chains and the $J_{1}$ bonds producing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at $\kappa = 0$ and a set of decoupled chains at $\kappa \rightarrow \infty$, with the isotropic HAF on the triangular lattice in between at $\kappa = 1$. For both the $s=1$ and the $s=3/2$ models we find a second-order quantum phase transition at $\kappa_{c}=0.615 \pm 0.010$ and $\kappa_{c}=0.575 \pm 0.005$ respectively, between a N\'{e}el antiferromagnetic state and a helical state. In both cases the ground-state energy $E$ and its first derivative $dE/d\kappa$ are continuous at $\kappa=\kappa_{c}$, while the order parameter for the transition (viz., the average on-site magnetization) does not go to zero on either side of the transition. The transition at $\kappa = \kappa_{c}$ for both the $s=1$ and $s=3/2$ cases is analogous to that observed in our previous work for the $s=1/2$ case at a value $\kappa_{c}=0.80 \pm 0.01$. However, for the higher spin values the transition is of continuous (second-order) type, as in the classical case, whereas for the $s=1/2$ case it appears to be weakly first-order in nature (although a second-order transition could not be excluded).Comment: 17 pages, 8 figues (Figs. 2-7 have subfigs. (a)-(d)

The spin-1/2 J1-J2 Heisenberg antiferromagnet on the square lattice: Exact diagonalization for N=40 spins

We present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N=40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J2^{c1} and J2 > J2^{c2} separated by a gapful quantum paramagnetic phase. We estimate J2^{c1} \approx 0.35J1 and J2^{c2} \approx 0.66J1.Comment: 16 pages, 2 tables, 11 figure

Optimisation de la performance environnementale des activités agricoles à l'échelle d'un espace à enjeux : le cas du bassin versant de la Boulouze

We introduce in this thesis the notion of environmental performance as an analytical framework aimed at studying the impacts of agricultural activities on the environment. We outline first a need to formalize this notion and then define it as the distance between the environmental state of a system at a specific time and a state of high environmental performance to achieve. Focus is on water management at the watershed level with three criteria: nitrogen, sediments (water quality) and water flows (water quantity). Our aim is to optimize the environmental performance of agricultural activities at the watershed scale. First the initial status of the watershed is described, then a method is proposed. This method is based on a coupling between an agro-hydrological model, the Soil and Water Assessment Tool (SWAT), and the Weighted Goal Programming optimization method. Our purpose is the reallocation of farming systems within the watershed when considering the optimization criteria simultaneously. We implement this method on the Boulouze watershed (coteaux de Gascogne, in the southwestern part of France). Results outline an improvement of the environmental performance. The analysis of the new land-use plan emphasizes the fact that the changes of farming systems are not only due to evolutions of the surfaces where they are implemented, but that the environmental performance at watershed scale is also affected by their location. Finally we explore the applicability of the method used highlighting its attributes as a relevant tool for modeling scenarios and for communicating.Les travaux présentés dans cette thèse portent sur la notion de performance environnementale comme cadre analytique pour l'étude des impacts des activités agricoles sur l'environnement. Après avoir mis en évidence un manque de formalisation de cette notion dans la littérature, nous la définissons comme la distance entre l'état environnemental d'un écosystème à un moment donné et un état environnemental à atteindre pour cet écosystème, dit « de haute performance environnementale ». Dans cette étude, l'état environnemental est approché par trois critères liés à la ressource « eau » : les concentrations en nitrates et en matières en suspension, et les débits. Suite à la description de l'état initial, nous proposons une méthode d'optimisation de la performance environnementale des activités agricoles basée sur un couplage entre un modèle agro-hydrologique (SWAT) et un modèle d'optimisation multicritère (Weighted Goal Programming). La démarche d'optimisation considère la réaffectation spatialisée de systèmes de culture sur l'espace considéré comme facteur d'amélioration, les différents critères de l'optimisation étant examinés simultanément. La méthode est implémentée sur le bassin versant de la Boulouze (coteaux de Gascogne). Les résultats montrent une amélioration de la performance environnementale du système étudié. L'analyse de la nouvelle occupation des sols souligne qu'au-delà des évolutions quantitatives des surfaces allouées aux systèmes de culture, la spatialisation des changements d'un système vers un autre influe également sur la performance environnementale à l'échelle du bassin versant. Enfin, nous interrogeons l'applicabilité de cette méthode et mettons en évidence l'intérêt qu'elle présente comme outil à la fois de modélisation de scénarios et de communication