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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.

A review of "The Quantum World of Ultra-Cold Atoms and Light. Book I: Foundations of Quantum Optics," by C. Gardiner and P. Zoller, Singapore, World Scientific Publ. Co., 2014, xviii + 310 pp., £42.00 (paperback), ISBN 978-1-78326-461-2.

We study the zero-temperature ground-state (GS) phase diagram of a spin-half J1–J2 XY model on the honeycomb lattice with nearest-neighbor exchange coupling J1 > 0 and frustrating next-nearest-neighbor exchange coupling $J_{2} \equiv \kappa J_{1} > 0$, where both bonds are of the isotropic XY type, using the coupled cluster method. Results are presented for the GS energy per spin, magnetic order parameter, and staggered dimer valence-bond crystalline (SDVBC) susceptibility, for values of the frustration parameter in the range $o \leq \kappa \leq 1$. In this range, we find phases exhibiting, respectively, Neel xy planar [N(p)], Neel z-aligned [N(z)], SDVBC,and Neel-II xy planar [N-II(p)] orderings. The Neel-II states, which break the lattice rotational symmetry, are ones in which the spins of nearest-neighbor pairs along one of the three equivalent honeycomb directions are parallel, while those in the other two directions are antiparallel. The N(p) state, which is stable for the classical version of the model in the range $0 \leq \kappa \leq \frac{1}{6}$, is found to form the GS phase out to a first quantum critical point at $\kappa_{c_1} = 0.216(5)$, beyond which the stable GS phase has N(z) order over the range $\kappa_{c_1} < \kappa < \kappa_{c_2}= 0.355(5)$. For values $\kappa > \kappa_{c_2}$ , we find a strong competition to form the GS phase between states with N-II(p) and SDVBC forms of order. Our best estimate, however, is that the stable GS phase over the rannge $\kappa_{c_2} < \kappa < \kappa_{c_3} ≈ 0.52(3$) is a mixed state with both SDVBC and N-II(p) forms of order; and for values $\kappa > \kappa_{c_3}$ is the N-II(p) state, which is stable at the classical level only at the highly degenerate point $kappa = \frac{1}{2}$ . Over the range $0 \leq \kappa \leq 1$, we find no evidence for any of the spiral phases that are present classically for all values $\kappa > \frac{1}{6}$, nor for any quantum spin-liquid state.

We investigate the ground states of s = 1/2 Heisenberg antiferromagnets on the 11 two-dimensional (2D) Archimedean lattices by using the coupled-cluster method. Magnetic interactions and quantum fluctuations play against each other subtly in 2D quantum magnets and the magnetic ordering is thus sensitive to the features of lattice topology. Archimedean lattices are those lattices that have 2D arrangements of regular polygons and they often build the underlying magnetic lattices of insulating quasi-two-dimensional quantum magnetic materials. Hence they allow a systematic study of the relationship between lattice topology and magnetic ordering. We find that the Archimedean lattices fall into three groups: those with semiclassical magnetic ground-state long-range order, those with a magnetically disordered (cooperative quantum paramagnetic) ground state, and those with a fragile magnetic order. The most relevant parameters affecting the magnetic ordering are the coordination number and the degree of frustration present.

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 use the coupled cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2 $J_{1}$--$J_{2}$--$J_{3}$ model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local on-site magnetization. With the nearest-neighbor coupling strength $J_{1} \equiv 1$ we find four magnetically ordered phases in the parameter window $J_{2},J_{3} \in [0,1]$, namely the N\'{e}el (N), striped (S), and anti-N\'{e}el (aN) collinear AFM phases, plus a spiral phase. The aN phase appears as a stable GS phase in the classical version of the model only for values $J_{3}

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.

Using the coupled cluster method (CCM) we study the zero-temperature phase diagram of a spin-half Heisenberg antiferromagnet (HAF), the so-called J1–J2′ model, defined on an anisotropic two-dimensional lattice. With respect to an underlying square-lattice geometry the model contains antiferromagnetic (J1>0) bonds between nearest neighbors and competing (J2′>0) bonds between next-nearest neighbors across only one of the diagonals of each square plaquette, the same diagonal in every square. Considered on an equivalent triangular-lattice geometry the model may be regarded as having two sorts of nearest-neighbor bonds, with J2′≡κJ1 bonds along parallel chains and J1 bonds providing an interchain coupling. Each triangular plaquette thus contains two J1 bonds and one J2′ bond. Hence, the model interpolates between a spin-half HAF on the square lattice at one extreme (κ=0) and a set of decoupled spin-half chains at the other (κ→∞), with the spin-half HAF on the triangular lattice in between at κ=1. We use a Néel state, a helical state, and a collinear stripe-ordered state as separate starting model states for the CCM calculations that we carry out to high orders of approximation (up to eighth order, n=8, in the localized subsystem set of approximations, LSUBn). The interplay between quantum fluctuations, magnetic frustration, and varying dimensionality leads to an interesting quantum phase diagram. We find strong evidence that quantum fluctuations favor a weakly first-order or possibly second-order transition from Néel order to a helical state at a first critical point at κc1=0.80±0.01 by contrast with the corresponding second-order transition between the equivalent classical states at κcl=0.5. We also find strong evidence for a second critical point at κc2=1.8±0.4 where a first-order transition occurs, this time from the helical phase to a collinear stripe-ordered phase. This latter result provides quantitative verification of a recent qualitative prediction of and Starykh and Balents [Phys. Rev. Lett. 98, 077205 (2007)] based on a renormalization group analysis of the J1–J2′ model that did not, however, evaluate the corresponding critical point.

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 N´eel 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 N´eel and stripe-ordered phases (for $J'_1/J_1 \gtrsim 0.66$) meets a line of first-order phase transitions between the same two states (for $J'_1/J_1 \lesssim 0.66$).

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)

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)]

The authors used the coupled cluster method (CCM) to investigate the evolution of the Rabi Hamiltonian from an initial state |0,| > composed of an empty field mode and an unexcited atom. Within the rotation-wave approxn. (RWA) the system would remain in this state indefinitely, since it is an eigenstate of the excitation no. N with corresponding eigenvalue zero. However, including of the otherwise neglected \"counter-rotating\" terms means that we allow \"energy non-conserving\" processes to occur, and their presence drives the temporal evolution of the Rabi system. The authors present results for the at. inversion and field occupation as function of time. They also show that no anti-bunching occurs, although the \"virtual\" processes do cause the field to become squeezed. [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 study the ground state and low-lying excited states in the nonlinear sigma model with the coupled cluster method of quantum many-body theory.

This work describes recent progress of the UMIST-VALENCIA collaboration on the ab initio study of ground states of light nuclei using realistic forces. The method presented here constructs trial variational wave functions by superimposing a central Jastrow correlation on a state-dependent translationally invariant linearly correlated state, with very promising results.

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 use the coupled-cluster method to study ground-state properties of anisotropic S=1/2 antiferromagnets on square and triangular lattices, with the inclusion of arbitrarily long-ranged two-spin correlations. We detect the singularities of various quantities associated with the quantum phase transitions and also compute their critical exponents. The two-spin correlation coefficients for the triangular lattice are found to exhibit an interesting oscillatory behavior in their signs, knowledge of which could assist the implementation of quantum Monte Carlo simulations.

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.

Coherent mixed states (or thermal coherent states) associated with the displaced harmonic oscillator at finite temperature, are introduced as a 'random' (or 'thermal' or 'noisy') basis in Hilbert space. A resolution of the identity for these states is proved and used to generalize the usual coherent state formalism for the finite temperature case. The Bargmann representation of an operator is introduced and its relation to the P and Q representations is studied. Generalized P and Q representations for the finite temperature case are also considered and several interesting relations among them are derived.