Search

We use the coupled cluster method implemented at high orders of approximation to study the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ model on the triangular lattice with Heisenberg interactions between nearest-neighbour and next-nearest-neighbour pairs of spins, with coupling strengths $J_{1}>0$ and $J_{2} \equiv \kappa J_{1} >0$, respectively. In the window $0 \leq \kappa \leq 1$ we find that the 3-sublattice 120$^{\circ}$ N\'{e}el-ordered and 2-sublattice 180$^{\circ}$ stripe-ordered antiferromagnetic states form the stable ground-state phases in the regions $\kappa < \kappa^{c}_{1} = 0.060(10)$ and $\kappa > \kappa^{c}_{2} = 0.165(5)$, respectively. The spin-triplet gap is found to vanish over essentially the entire region $\kappa^{c}_{1} < \kappa < \kappa^{c}_{2}$ of the intermediate phase.

We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings $J_{1}>0$ and $J_{2} \equiv \kappa J_{1}>0$, respectively, in the window $0 \leq \kappa \kappa^{{\rm cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\kappa = 0$ of the spin-$\frac{1}{2}$ model for the GS energy per spin, $E/N=-0.5521(2)J_{1}$, and the GS magnetic order parameter, $M=0.198(5)$ (in units where the classical value is $M^{{\rm cl}}=\frac{1}{2}$), which are among the best available. For the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ model we find that the classical transition at $\kappa=\kappa^{{\rm cl}}$ is split into two quantum phase transition at $\kappa^{c}_{1}=0.060(10)$ and $\kappa^{c}_{2}=0.165(5)$. The two quasiclassical AFM states (viz., the 120$^{\circ}$ N\'{e}el state and the striped state) are found to be the stable GS phases in the regime $\kappa \kappa^{c}_{2}$, respectively, while in the intermediate regimes $\kappa^{c}_{1}

We study the zero-temperature ground-state (GS) properties of the spin-$\frac{1}{2}$ anisotropic planar pyrochlore, using the coupled cluster method (CCM) implemented to high orders of approximation. The system comprises a $J_{1}$--$J_{2}$ model on the checkerboard lattice, with isotropic Heisenberg interactions of strength $J_{1}$ between all nearest-neighbour pairs of spins on the square lattice, and of strength $J_{2}$ between half of the next-nearest-neighbour pairs (in the checkerboard pattern). We calculate results for the GS energy and average local GS on-site magnetization, using various antiferromagnetic classical ground states as CCM model states. We also give results for the susceptibility of one of these states against the formation of crossed-dimer valence-bond crystalline (CDVBC) ordering. The complete GS phase diagram is presented for arbitrary values of the frustration parameter $\kappa \equiv J_{2}/J_{1}$, and when each of the exchange couplings can take either sign.

We study the zero-temperature ($T = 0$) ground-state (GS) properties of a frustrated spin-half $J_{1}^{XXZ}$--$J_{2}^{XXZ}$model on the honeycomb lattice with nearest-neighbor and next-nearest-neighbor interactions with exchange couplings $J_{1} > 0$ and $J_{2} \equiv \kappa J_{1} > 0$, respectively, using the coupled cluster method. Both interactions are of the anisotropic XXZ type. We present the $T = 0$ GS phase diagram of the model in the ranges $0 \leq \Delta \leq 1$ of the spin-space anisotropy parameter and $0 \leq \kappa \leq 1$ of the frustration parameter. A possible quantum spin-liquid region is identified.

Using the coupled cluste method (CCM )we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half (s = 1/2) J1-J2 Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength J1 and two next-nearest-neighbor (diagonal) bonds of strength J2. The J2 bonds are arranged so that the basic square plaquettes in alternating columns have either both or no J2 bonds included. The classical (s →∞) version of the model has four collinear phases when J1 and J2 can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N´eel, double N´eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum s = 1/2 model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the s = 1/2 model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the s = 1/2 model are of continuous deconfined type.

We study the ground-state (GS) properties of the frustrated spin-1/2 J1–J2–J3 Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (J1 = −1) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2 > 0) and next-next-nearest-neighbor (J3 > 0) exchanges, for the case J3 = J2. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (“striped”) collinear order at $J_{2}^{c}$ ≈ 0.1095 ± 0.0005, which is below the corresponding hypothetical transition point at $J_{2}^{cl}$ = 1/7(≈ 0.143) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 1/10 < J2 < 1/5 . Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with J1 = −1) to previous results for the corresponding antiferromagnetic case (with J1 = +1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2. However, there are considerable differences in the behavior of the order parameters for the two models for $J_{2}/|J_{1}| \lesssim 0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with J1 = +1) breaks down at $J_{2}^{c_2}$ ≈ 0.60, whereas the “equivalent” point for the ferromagnetic model (with J1 = −1) occurs at $J_{2}^{c}$ ≈ 0.11. Unlike in the antiferromagnetic model (with J1 = +1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N´eel and striped order) for $J_{2}^{c_1} < J_{2} < J_{2}^{c_2}$, with $J_{2}^{c_1}$ ≈ 0.47, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with J1 = −1) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10 \lesssim J_{2} \lesssim 0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range 1/10 < J2 < 1/5.

We study the zero-temperature ground-state (gs) phase diagram of the spin-half anisotropic planar pyrochlore (or crossed chain) model using the coupled cluster method (CCM). The model is equivalently described as a frustrated $J_{1}$--$J_{2}$ antiferromagnet on the two-dimensional checkerboard lattice, with nearest-neighbor exchange bonds of strength $J_{1}>0$ and next-nearest-neighbor bonds of strength $J_{2} \equiv \kappa J_{1} > 0$. Using various antiferromagnetic (AFM) classical ground states as CCM reference states, we present results for the gs energy, average local on-site magnetization, and the susceptibilities of these states against the formation of plaquette valence-bond crystal (PVBC) and crossed-dimer valence-bond crystal (CDVBC) ordering. We show that the AFM quasiclassical state with Neel ordering is the gs phase for $\kappa < \kappa_{c_1} \approx 0.80 \pm 0.01$, that none of the infinitely degenerate set of AFM states that form the gs phase for the classical version ($s \to \infty$) of the model (for $\kappa > 1$) survive the quantum fluctuations to form a stable magnetically ordered gs phase for the s = 1/2 case. Instead, our calculations indicate a PVBC-ordered phase for $\kappa_{c_1} < \kappa < \kappa_{c_2} \approx 1.22 \pm 0.02$, and a CDVBC-ordered phase for all $\kappa > \kappa_{c_2}$. Both transitions are likely to be direct ones, although we cannot exclude very narrow coexistence regions confined to $0.79 \lesssim \kappa \lesssim 0.81$ and $1.20 \lesssim \kappa \lesssim 1.22$ respectively.

We use the coupled cluster method (CCM) to investigate the ground-state (GS) properties of the frustrated spin-1/2 J1-J2-J3 model on the honeycomb lattice, with nearest-neighbor exchange coupling J1 plus next-nearest-neighbor (J2) and next-next-nearest-neighbor (J3) exchanges of equal strength. In particular we find a direct first-order phase transition between the N´eel-ordered antiferromagnetic phase and the ferromagnetic phase at a value J2/J1 = −1.17 ± 0.01 when J1 > 0, compared to the corresponding classical value of −1. We find no evidence for any intermediate phase. From this and our previous CCM studies of the model we present its full zero-temperature GS phase diagram.

We use the coupled cluster method (CCM) to study the zero-temperature phase diagram of a two-dimensional frustrated spin-half antiferromagnet, the so-called Union Jack model. It is defined on a square lattice such that all nearest-neighbor pairs are connected by bonds with a strength J1>0, but only half the next-nearest-neighbor pairs are connected by bonds with a strength J2≡κJ1>0. The bonds are arranged such that on the 2×2 unit cell they form the pattern of the Union Jack flag. Alternating sites on the square lattice are thus four-connected and eight-connected. We find strong evidence for a first phase transition between a Néel antiferromagnetic phase and a canted ferrimagnetic phase at a critical coupling κc1=0.66±0.02. The transition is an interesting one, at which the energy and its first derivative seem continuous, thus providing a typical scenario of a second-order transition (just as in the classical case for the model), although a weakly first-order transition cannot be excluded. By contrast, the average on-site magnetization approaches a nonzero value Mc1=0.195±0.005 on both sides of the transition, which is more typical of a first-order transition. The slope, dM/dκ, of the order parameter curve as a function of the coupling strength κ, also appears to be continuous, or very nearly so, at the critical point κc1, thereby providing further evidence of the subtle nature of the transition between the Néel and canted phases. Our CCM calculations provide strong evidence that the canted ferrimagnetic phase becomes unstable at large values of κ, and hence we have also used the CCM with a model collinear semistripe-ordered ferrimagnetic state in which alternating rows (and columns) are ferromagnetically and antiferromagnetically ordered, and in which the spins connected by J2 bonds are antiparallel to one another. We find tentative evidence, based on the relative energies of the two states, for a second zero-temperature phase transition between the canted and semistripe-ordered ferrimagnetic states at a large value of the coupling parameter around κc2≈125±5. This prediction, however, is based on an extrapolation of the CCM results for the canted state into regimes where the solutions have already become unstable and the CCM equations based on the canted state at any level of approximation beyond the lowest have no solutions. Our prediction for κc2 is hence less reliable than that for κc1. Nevertheless, if this second transition at κc2 does exist, our results clearly indicate it to be of first-order type.

The coupled cluster method is applied to a spin-half model at zero temperature (T=0), which interpolates between Heisenberg antiferromagnets (HAFs) on a kagome and a square lattice. With respect to an underlying triangular lattice the strengths of the Heisenberg bonds joining the nearest-neighbor (NN) kagome sites are J1≥0 along two of the equivalent directions and J2≥0 along the third. Sites connected by J2 bonds are themselves connected to the missing NN non-kagome sites of the triangular lattice by bonds of strength J1′≥0. When J1′=J1 and J2=0 the model reduces to the square-lattice HAF. The magnetic ordering of the system is investigated and its T=0 phase diagram discussed. Results for the kagome HAF limit are among the best available.

The zero-temp. phase transitions of a spin-half Heisenberg system on the square lattice were evaluated using the coupled cluster method (CCM). Quantum fluctuations along with competition without frustration destroyed Neel long-range order by local singlet format, indicating pure quantum effect and have no classical counterpart. The control parameter is the strength of the competition and the breakdown of Neel ordering is accompanied by the opening of a spin gap. in which CCM described both the order parameter and the gap satisfactorily. Competition due to frustration was found to give more complex magnetic properties. The CCM provides a consistent description of collinear, noncollinear and disordered phases of the quantum spin-half model. A strong impact of quantum fluctuations on the nature of the collinear-noncollinear transition was obsd., and quantum fluctuations may change the second-order classical transition to a first-order quantum transition. [on SciFinder (R)]

Using the coupled cluster method (CCM), we present a simple accurate calculation for the energies of the ground- and first excited states of the linear E x e Jahn-Teller and pseudo-Jahn-Teller Hamiltonians. From the solution of a single transcendental equation, we obtain results with a maximal error of 1.2%. These results are notably better than previous results obtained both via the CCM and other many-body approximations.

We illustrate how the systematic inclusion of multi-spin correlations of the quantum spin-lattice systems can be efficiently implemented within the frame-work of the coupled-cluster method by examining the ground-state properties of both the square-lattice and the frustrated triangular-lattice quantum antiferromagnets. The ground-state energy and the sublattice magnetization are calculated for the square-lattice and triangular-lattice Heisenberg antiferromagnets, and our best estimates give values for the sublattice magnetization which are 62% and 51% of the classical results for the square and triangular lattices, respectively. We furthermore make a conjecture as to why previous series expansion calculations have not indicated Neel-like long-range order for the triangular-lattice Heisenberg antiferromagnet. We investigate the critical behavior of the anisotropic systems by obtaining approximate values for the positions of phase transition points.

We extend our own previous applications of the microscopic coupled-cluster method (CCM) to quantum antiferromagnets. In particular, we carry out a systematic calculation involving high-order multispin correlations for the spin-1/2 anisotropic Heisenberg models on the one-dimensional chain and the two-dimensional square lattice. Their ground-state properties are obtained as functions of the anisotropy parameter. Our CCM analysis not only produces accurate results for such physical quantities as the ground-state energy which are comparable to the best results from other techniques, but it also enables us to study the quantum phase transitions of the spin models in a systematic and unbiased manner.

We apply the microscopic coupled-cluster method (CCM) to the spin-½ XXZ models on both the one-dimensional chain and the two-dimensional square lattice. Based on a systematic approximation scheme of the CCM developed by us previously, we carry out high-order ab initio calculations using computer-algebraic techniques. The ground-state properties of the models are obtained with high accuracy as functions of the anisotropy parameter. Furthermore, our CCM analysis enables us to study their quantum critical behavior in a systematic and unbiased manner.

We have applied the microscopic coupled-cluster method (CCM) of many-body quantum theory to the U(1) lattice gauge models in (1+1) and (2+1) dimensions. The mode couplings and plaquette correlations are studied in detail by means of a hierarchical truncation scheme. Good numerical results for the ground-state energy have been obtained for a range of coupling constants varying from strong to weak. The systematic and nonperturbative natures of the CCM are emphasized.

We have applied the microscopic coupled-cluster method (CCM) of many-body quantum theory to the U(1) lattice gauge models in (1+1) and (2+1) dimensions. The mode couplings and plaquette correlations are studied in detail by means of a hierarchical truncation scheme. Good numerical results for the ground-state energy have been obtained for a range of coupling constants varying from strong to weak. The systematic and nonperturbative natures of the CCM are emphasized.

The Coupled Cluster Method (CCM) is one of the most powerful and universally applied techniques of quantum many-body theory. In particular, it has been used extensively in order to investigate many types of lattice quantum spin system at zero temperature. The ground- and excited-state properties of these systems may now be determined routinely to great accuracy. In this Chapter we present an overview of the CCM formalism and we describe how the CCM is applied in detail. We illustrate the power and versatility of the method by presenting results for four different spin models. These are, namely, the XXZ model, a Heisenberg model with bonds of differing strengths on the square lattice, a model which interpolates between the Kagomé- and triangular-lattice antiferromagnets, and a frustrated ferrimagnetic spin system on the square lattice. We consider the ground-state properties of all of these systems and we present accurate results for the excitation energies of the spin-half square-lattice XXZ model. We utilise an ldquoextendedrdquo SUB2 approximation scheme, and we demonstrate how this approximation may be solved exactly by using Fourier transform methods or, alternatively, by determining and solving the SUB2-m problem. We also present the results of ldquolocalisedrdquo approximation schemes called the LSUBm or SUBm-m schemes. We note that we must utilise computational techniques in order to solve these localised approximation schemes to ldquohigh order.rdquo We show that we are able to determine the positions of quantum phase transitions with much accuracy, and we demonstrate that we are able to determine their quantum criticality by using the CCM in conjunction with the coherent anomaly method (CAM). Also, we illustrate that the CCM may be used in order to determine the ldquonodal surfacesrdquo of lattice quantum spin systems. Finally, we show how connections to cumulant series expansions may be made by determining the perturbation series of a spin-half triangular-lattice antiferromagnet using the CCM at various levels of LSUBm approximation. References

We discuss the influence of strong quantum fluctuations on zero-temperature phase transitions in a two-dimensional spin-half Heisenberg system. Using a high-order coupled cluster treatment, we study competition of magnetic bonds with and without frustration. We find that the coupled cluster treatment is able to describe the zero-temperature transitions in a qualitatively correct way, even if frustration is present and other methods such as quantum Monte Carlo fail., Comment: 8 pages, 12 Postscipt figures; Accepted for publication in World Scientific