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Arrays of nonlinear resonators offer a fertile ground for a wide range of complex phenomena and opportunities for advanced photonic sensing and computing. Recently, significant attention has focused on studying coupled resonators in special-purpose configurations either on chips or in table-top experiments. However, a path to realizing a large-scale programmable network of nonlinear photonic resonators remains elusive because of the challenges associated with simultaneously achieving strong nonlinearity, independent operation of the resonators, and programmability of the couplings. In this work, we break these barriers by realizing large-scale, time-multiplexed optical parametric oscillators (OPOs) on a single lithium niobate nanophotonic chip. We show independent operation of 70 identical OPOs in an ultrafast nanophotonic circuit. The OPOs exhibit an ultra-low threshold of a few picojoules, substantially surpassing the strength of nonlinearity of other platforms. Using our ultrafast nanophotonic circuit, a network of N OPOs with programmable all-to-all couplings requires only a few additional components. The time-multiplexed nanophotonic OPOs can enable myriad applications, including ultrafast classical and quantum information processing., Comment: 19 pages, 5 figures

Rapid advancements in deep learning over the past decade have fueled an insatiable demand for efficient and scalable hardware. Photonics offers a promising solution by leveraging the unique properties of light. However, conventional neural network architectures, which typically require dense programmable connections, pose several practical challenges for photonic realizations. To overcome these limitations, we propose and experimentally demonstrate Photonic Neural Cellular Automata (PNCA) for photonic deep learning with sparse connectivity. PNCA harnesses the speed and interconnectivity of photonics, as well as the self-organizing nature of cellular automata through local interactions to achieve robust, reliable, and efficient processing. We utilize linear light interference and parametric nonlinear optics for all-optical computations in a time-multiplexed photonic network to experimentally perform self-organized image classification. We demonstrate binary classification of images in the fashion-MNIST dataset using as few as 3 programmable photonic parameters, achieving an experimental accuracy of 98.0% with the ability to also recognize out-of-distribution data. The proposed PNCA approach can be adapted to a wide range of existing photonic hardware and provides a compelling alternative to conventional photonic neural networks by maximizing the advantages of light-based computing whilst mitigating their practical challenges. Our results showcase the potential of PNCA in advancing photonic deep learning and highlights a path for next-generation photonic computers.

Cellular automata are a class of computational models based on simple rules and algorithms that can simulate a wide range of complex phenomena. However, when using conventional computers, these 'simple' rules are only encapsulated at the level of software. This can be taken one step further by simplifying the underlying physical hardware. Here, we propose and implement a simple photonic hardware platform for simulating complex phenomena based on cellular automata. Using this special-purpose computer, we experimentally demonstrate complex phenomena including fractals, chaos, and solitons, which are typically associated with much more complex physical systems. The flexibility and programmability of our photonic computer presents new opportunities to simulate and harness complexity for efficient, robust, and decentralized information processing schemes using light.

Mode-locked lasers play a key role in modern science and technology. Not only do they lay the foundation for ultrafast optics and play a central role in nonlinear optics, but also they have important applications in imaging, telecommunications, sensing, and computing. While substantial efforts have focused on mode-locking the spectral modes of lasers, relatively little attention has been paid to mode-locking their temporal modes. However, temporal mode-locking presents ample opportunities to develop new technologies and to study the intersection of nonlinear, non-Hermitian, and topological phenomena, which, in recent years, has been a priority for the field of topological physics. Here, we theoretically predict and experimentally realize topological temporal mode-locking in a laser cavity with time-delayed intracavity couplings. These couplings introduce non-Hermitian point-gap topology into the temporal modes of our laser and conspire with an effective nonlocal nonlinearity to generate mode-locked temporal structures in our laser cavity, whose form can be tailored by suitably engineering the underlying couplings. We harness this approach to realize a nonlinearity-driven non-Hermitian skin effect, and we show that the topological temporal modes of our laser are robust against disorder-induced localization. The flexibility and programmability of our topological temporal mode-locking scheme reveals new opportunities to study nonlinear and non-Hermitian topological phenomena in mode-locked photonic resonators and could enable new applications of mode-locked lasers to sensing and optical computing., Comment: 11 pages, 6 figures

In recent years, the computational demands of deep learning applications have necessitated the introduction of energy-efficient hardware accelerators. Optical neural networks are a promising option; however, thus far they have been largely limited by the lack of energy-efficient nonlinear optical functions. Here, we experimentally demonstrate an all-optical Rectified Linear Unit (ReLU), which is the most widely used nonlinear activation function for deep learning, using a periodically-poled thin-film lithium niobate nanophotonic waveguide and achieve ultra-low energies in the regime of femtojoules per activation with near-instantaneous operation. Our results provide a clear and practical path towards truly all-optical, energy-efficient nanophotonic deep learning.

The zero-temperature phase diagram of the frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg magnet on an $AA$-stacked honeycomb bilayer lattice is studied using the coupled cluster method implemented to very high orders. On each monolayer the spins interact via nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic Heisenberg interactions with respective strength parameters $J_{1}>0$ and $J_{2}\equiv\kappa J_{1}>0$. The two layers are coupled such that NN interlayer pairs of spins also interact via a similar isotropic Heisenberg interaction of strength $J_{1}^{\perp}\equiv \delta J_{1}$, which may be of either sign. In particular, we locate with high accuracy the complete phase boundaries in the $\kappa$-$\delta$ half-plane with $\kappa>0$ of the two quasiclassical collinear antiferromagnetic phases with N\'{e}el or N\'{e}el-II magnetic order in each monolayer, and the interlayer NN pairs of spins either aligned (for $\delta<0$) or anti-aligned (for $\delta > 0$) to one another. Compared to the two-sublattice N\'{e}el order, in which all NN intralayer pairs of spins are antiparallel to one another, the four-sublattice N\'{e}el-II order is characterized by NN intralayer pairs of spins on the honeycomb lattice being antiparallel to one another along zigzag (or sawtooth) chains in a specified direction from among the three equivalent honeycomb-lattice directions, and parallel to one another for the corresponding interchain pairs.

Bulk materials possessing a relative electric permittivity $\varepsilon$ close to zero exhibit giant Kerr nonlinearities. However, harnessing this response in guided-wave geometries is not straightforward, due to the extreme and counter-intuitive properties of epsilon-near-zero materials. Here we investigate, through rigorous calculations of the Kerr nonlinear coefficient, how the remarkable nonlinear properties of such materials can be exploited in several different types of structures, including bulk films, plasmonic nanowires, and metal nanoapertures. We find the largest Kerr nonlinear response when both the modal area and the group velocity are simultaneously minimized, corresponding to omnidirectional field enhancement. The physical insights developed will be key for understanding and engineering nonlinear nanophotonic systems with extreme nonlinearities and point to new design paradigms.

The zero-temperature phase diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on an $AA$-stacked square-lattice bilayer is studied using the coupled cluster method implemented to very high orders. Both nearest-neighbor (NN) and frustrating next-nearest-neighbor Heisenberg exchange interactions, of strengths $J_{1}>0$ and $J_{2} \equiv \kappa J_{1}>0$, respectively, are included in each layer. The two layers are coupled via a NN interlayer Heisenberg exchange interaction with a strength $J_{1}^{\perp} \equiv \delta J_{1}$. The magnetic order parameter $M$ (viz., the sublattice magnetization) is calculated directly in the thermodynamic (infinite-lattice) limit for the two cases when both layers have antiferromagnetic ordering of either the N\'{e}el or the striped kind, and with the layers coupled so that NN spins between them are either parallel (when $\delta < 0$) or antiparallel (when $\delta > 0$) to one another. Calculations are performed at $n$th order in a well-defined sequence of approximations, which exactly preserve both the Goldstone linked cluster theorem and the Hellmann-Feynman theorem, with $n \leq 10$. The sole approximation made is to extrapolate such sequences of $n$th-order results for $M$ to the exact limit, $n \to \infty$. By thus locating the points where $M$ vanishes, we calculate the full phase boundaries of the two collinear AFM phases in the $\kappa$--$\delta$ half-plane with $\kappa > 0$. In particular, we provide the accurate estimate, ($\kappa \approx 0.547,\delta \approx -0.45$), for the position of the quantum triple point (QTP) in the region $\delta < 0$. We also show that there is no counterpart of such a QTP in the region $\delta > 0$, where the two quasiclassical phase boundaries show instead an ``avoided crossing'' behavior, such that the entire region that contains the nonclassical paramagnetic phases is singly connected.

The interplay between lattice topology, frustration, and spin quantum number, $s$, is explored for the Heisenberg antiferromagnet (HAFM) on the eleven two-dimensional Archimedean lattices (square, honeycomb, CaVO, SHD, SrCuBO, triangle, bounce, trellis, maple-leaf, star, and kagome). We show the CCM provides consistently accurate results when compared to the results of other approximate methods. The $\sqrt{3}\times\sqrt{3}$ model state provides lower ground-state energies than those of the $q=0$ model state for the kagome and star lattices for most values of $s$. The $q=0$ model state provides lower ground-state energies only for $s=1/2$ for the kagome lattice and $s=1/2$ and $s=1$ for the star lattice. The kagome and star lattices demonstrate the least amount of magnetic ordering and the unfrustrated lattices (square, honeycomb, SHD, and CaVO) demonstrate the most magnetic ordering for all values of $s$. The SrCuBO and triangular lattices also demonstrate high levels of magnetic ordering, while the remaining lattices (bounce, maple-leaf, and trellis) tend to lie between these extremes, again for all values of $s$. These results also clearly reflect the strong increase in magnetic order with increasing spin quantum number $s$ for all lattices. The ground-state energy, $E_g/(NJs^2)$, scales with $s^{-1}$ to first order, as expected from spin-wave theory, although the order parameter, $M/s$, scales with $s^{-1}$ for most of the lattices only. Self-consistent spin-wave theory calculations indicated previously that $M/s$ scales with $s^{-2/3}$ for the kagome lattice HAFM, whereas previous CCM results (replicated here also) suggested that $M/s$ scales with $s^{-1/2}$. By using similar arguments, we find here also that $M/s$ scales with $s^{-1/3}$ on the star lattice and with $s^{-2/3}$ on the SrCuBO lattice., Comment: 29 pages, 4 figures, 4 tables

The zero-temperature quantum phase diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on an $AA$-stacked bilayer honeycomb lattice is investigated using the coupled cluster method (CCM). The model comprises two monolayers in each of which the spins, residing on honeycomb-lattice sites, interact via both nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg exchange iteractions, with respective strengths $J_{1} > 0$ and $J_{2} \equiv \kappa J_{1}>0$. The two layers are coupled via a comparable Heisenberg exchange interaction between NN interlayer pairs, with a strength $J_{1}^{\perp} \equiv \delta J_{1}$. The complete phase boundaries of two quasiclassical collinear AFM phases, namely the N\'{e}el and N\'{e}el-II phases, are calculated in the $\kappa \delta$ half-plane with $\kappa > 0$. Whereas on each monolayer in the N\'{e}el state all NN pairs of spins are antiparallel, in the N\'{e}el-II state NN pairs of spins on zigzag chains along one of the three equivalent honeycomb-lattice directions are antiparallel, while NN interchain spins are parallel. We calculate directly in the thermodynamic (infinite-lattice) limit both the magnetic order parameter $M$ and the excitation energy $\Delta$ from the $s^{z}_{T}=0$ ground state to the lowest-lying $|s^{z}_{T}|=1$ excited state (where $s^{z}_{T}$ is the total $z$ component of spin for the system as a whole, and where the collinear ordering lies along the $z$ direction) for both quasiclassical states used (separately) as the CCM model state, on top of which the multispin quantum correlations are then calculated to high orders ($n \leq 10$) in a systematic series of approximations involving $n$-spin clusters. The sole approximation made is then to extrapolate the sequences of $n$th-order results for $M$ and $\Delta$ to the exact limit, $n \to \infty$.

The coupled cluster method is implemented at high orders of approximation to investigate the zero-temperature $(T=0)$ phase diagram of the frustrated spin-$s$ $J_{1}$--$J_{2}$--$J_{3}$ antiferromagnet on the honeycomb lattice. The system has isotropic Heisenberg interactions of strength $J_{1}>0$, $J_{2}>0$ and $J_{3}>0$ between nearest-neighbour, next-nearest-neighbour and next-next-nearest-neighbour pairs of spins, respectively. We study it in the case $J_{3}=J_{2}\equiv \kappa J_{1}$, in the window $0 \leq \kappa \leq 1$ that contains the classical tricritical point (at $\kappa_{{\rm cl}}=\frac{1}{2}$) of maximal frustration, appropriate to the limiting value $s \to \infty$ of the spin quantum number. We present results for the magnetic order parameter $M$, the triplet spin gap $\Delta$, the spin stiffness $\rho_{s}$ and the zero-field transverse magnetic susceptibility $\chi$ for the two collinear quasiclassical antiferromagnetic (AFM) phases with N\'{e}el and striped order, respectively. Results for $M$ and $\Delta$ are given for the three cases $s=\frac{1}{2}$, $s=1$ and $s=\frac{3}{2}$, while those for $\rho_{s}$ and $\chi$ are given for the two cases $s=\frac{1}{2}$ and $s=1$. On the basis of all these results we find that the spin-$\frac{1}{2}$ and spin-1 models both have an intermediate paramagnetic phase, with no discernible magnetic long-range order, between the two AFM phases in their $T=0$ phase diagrams, while for $s > 1$ there is a direct transition between them. Accurate values are found for all of the associated quantum critical points. While the results also provide strong evidence for the intermediate phase being gapped for the case $s=\frac{1}{2}$, they are less conclusive for the case $s=1$. On balance however, at least the transition in the latter case at the striped phase boundary seems to be to a gapped intermediate state.

We use the coupled cluster method (CCM) to study a frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg antiferromagnet on a bilayer honeycomb lattice with $AA$ stacking. Both nearest-neighbor (NN) and frustrating next-nearest-neighbor antiferromagnetic (AFM) exchange interactions are present in each layer, with respective exchange coupling constants $J_{1}>0$ and $J_{2} \equiv \kappa J_{1} > 0$. The two layers are coupled with NN AFM exchanges with coupling strength $J_{1}^{\perp}\equiv \delta J_{1}>0$. We calculate to high orders of approximation within the CCM the zero-field transverse magnetic susceptibility $\chi$ in the N\'eel phase. We thus obtain an accurate estimate of the full boundary of the N\'eel phase in the $\kappa\delta$ plane for the zero-temperature quantum phase diagram. We demonstrate explicitly that the phase boundary derived from $\chi$ is fully consistent with that obtained from the vanishing of the N\'eel magnetic order parameter. We thus conclude that at all points along the N\'eel phase boundary quasiclassical magnetic order gives way to a nonclassical paramagnetic phase with a nonzero energy gap. The N\'eel phase boundary exhibits a marked reentrant behavior, which we discuss in detail.

We study a frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{3}$--$J_{1}^{\perp}$ Heisenberg antiferromagnet on an $AA$-stacked bilayer honeycomb lattice. In each layer we consider nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor antiferromagnetic (AFM) exchange couplings $J_{1}$, $J_{2}$, and $J_{3}$, respectively. The two layers are coupled with an AFM NN exchange coupling $J_{1}^{\perp}\equiv\delta J_{1}$. The model is studied for arbitrary values of $\delta$ along the line $J_{3}=J_{2}\equiv\alpha J_{1}$ that includes the most highly frustrated point at $\alpha=\frac{1}{2}$, where the classical ground state is macroscopically degenerate. The coupled cluster method is used at high orders of approximation to calculate the magnetic order parameter and the triplet spin gap. We are thereby able to give an accurate description of the quantum phase diagram of the model in the $\alpha\delta$ plane in the window $0 \leq \alpha \leq 1$, $0 \leq \delta \leq 1$. This includes two AFM phases with N\'eel and striped order, and an intermediate gapped paramagnetic phase that exhibits various forms of valence-bond crystalline order. We obtain accurate estimations of the two phase boundaries, $\delta = \delta_{c_{i}}(\alpha)$, or equivalently, $\alpha = \alpha_{c_{i}}(\delta)$, with $i=1$ (N\'eel) and 2 (striped). The two boundaries exhibit an "avoided crossing" behavior with both curves being reentrant.

We implement the coupled cluster method to very high orders of approximation to study the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$ Heisenberg model on a cross-striped square lattice. Every nearest-neighbour pair of sites on the square lattice has an isotropic antiferromagnetic exchange bond of strength $J_{1}>0$, while the basic square plaquettes in alternate columns have either both or neither next-nearest-neighbour (diagonal) pairs of sites connected by an equivalent frustrating bond of strength $J_{2} \equiv \alpha J_{1} > 0$. By studying the magnetic order parameter (i.e., the average local on-site magnetization) in the range $0 \leq \alpha \leq 1$ of the frustration parameter we find that the quasiclassical antiferromagnetic N\'{e}el and (so-called) double N\'{e}el states form the stable ground-state phases in the respective regions $\alpha < \alpha_{1a}^{c} = 0.46(1)$ and $\alpha > \alpha_{1b}^{c} = 0.615(5)$. The double N\'{e}el state has N\'{e}el ($\cdots\uparrow\downarrow\uparrow\downarrow\cdots$) ordering along the (column) direction parallel to the stripes of squares with both or no $J_{2}$ bonds, and spins alternating in a pairwise ($\cdots\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\downarrow\downarrow\cdots$) fashion along the perpendicular (row) direction, so that the parallel pairs occur on squares with both $J_{2}$ bonds present. Further explicit calculations of both the triplet spin gap and the zero-field uniform transverse magnetic susceptibility provide compelling evidence that the ground-state phase over all or most of the intermediate regime $\alpha_{1a}^{c} < \alpha < \alpha_{1b}^{c}$ is a gapped state with no discernible long-range magnetic order.

We use the coupled cluster method to study the zero-temperature quantum phase diagram of the spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ model on the honeycomb bilayer lattice. In each layer we include both nearest-neighbor and frustrating next-nearest-neighbor antiferromagnetic exchange couplings, of strength $J_{1}>0$ and $J_{2} \equiv \kappa J_{1} > 0$, respectively. The two layers are coupled by an interlayer nearest-neighbor exchange, with coupling constant $J_{1}^{\perp} \equiv \delta J_{1}>0$. We calculate directly in the infinite-lattice limit both the ground-state energy per spin and the N\'{e}el magnetic order parameter, as well as the triplet spin gap. By implementing the method to very high orders of approximation we obtain an accurate estimate for the full boundary of the N\'{e}el phase in the $\kappa\delta$ plane. For each value $\delta < \delta_{c}^{>}(0) \approx 1.70(5)$ we find an upper critical value $\kappa_{c}(\delta)$, such that N\'{e}el order is present for $\kappa < \kappa_{c}(\delta)$. Conversely, for each value $\kappa < \kappa_{c}(0) \approx 0.19(1)$ we find an upper critical value $\delta_{c}^{>}(\kappa)$, such that N\'{e}el order persists for $0 < \delta < \delta_{c}^{>}(\kappa)$. Most interestingly, for values of $\kappa$ in the range $\kappa_{c}(0) < \kappa < \kappa^{>} \approx 0.215(2)$ we find a reentrant behavior such that N\'{e}el order exists only in the range $\delta_{c}^{<}(\kappa) < \delta < \delta_{c}^{>}(\kappa)$, with $\delta_{c}^{<}(\kappa)>0$. These latter upper and lower critical values coalesce when $\kappa = \kappa^{>}$, such that $\delta_{c}^{<}(\kappa^{>}) = \delta_{c}^{>}(\kappa^{>}) \approx 0.25(5)$.

We use the coupled cluster method (CCM) to study the ground-state properties and lowest-lying triplet excited state of the spin-half {\it XXZ} antiferromagnet on the square lattice. The CCM is applied to it to high orders of approximation by using an efficient computer code that has been written by us and which has been implemented to run on massively parallelized computer platforms. We are able therefore to present precise data for the basic quantities of this model over a wide range of values for the anisotropy parameter $\Delta$ in the range $-1 \leq \Delta < \infty$ of interest, including both the easy-plane $(-1 < \Delta < 1)$ and easy-axis $(\Delta > 1)$ regimes, where $\Delta \rightarrow \infty$ represents the Ising limit. We present results for the ground-state energy, the sublattice magnetization, the zero-field transverse magnetic susceptibility, the spin stiffness, and the triplet spin gap. Our results provide a useful yardstick against which other approximate methods and/or experimental studies of relevant antiferromagnetic square-lattice compounds may now compare their own results. We also focus particular attention on the behaviour of these parameters for the easy-axis system in the vicinity of the isotropic Heisenberg point ($\Delta = 1$), where the model undergoes a phase transition from a gapped state (for $\Delta > 1$) to a gapless state (for $\Delta \leq 1$), and compare our results there with those from spin-wave theory (SWT). Interestingly, the nature of the criticality at $\Delta=1$ for the present model with spins of spin quantum number $s=\frac{1}{2}$ that is revealed by our CCM results seems to differ qualitatively from that predicted by SWT, which becomes exact only for its near-classical large-$s$ counterpart.

We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling $J_{1}>0$ and frustrating next-nearest-neighbor coupling $J_{2} \equiv \kappa J_{1} > 0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0 < \kappa < 1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with 2-sublattice N\'{e}el order for $\kappa < \kappa_{c_{1}} = 0.250(5)$, and another with 4-sublattice N\'{e}el-II order for $\kappa > \kappa_{c_{2}} = 0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range $\kappa_{c_{1}} < \kappa < \kappa^{i}_{c} = 0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range $\kappa^{i}_{c} < \kappa < \kappa_{c_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.

The coupled cluster method (CCM) is employed to very high orders of approximation to study the ground-state (GS) properties of the spin-$s$ Heisenberg antiferromagnet (with isotropic interactions, all of equal strength, between nearest-neighbour pairs only) on the honeycomb lattice. We calculate with high accuracy the complete set of GS parameters that fully describes the low-energy behaviour of the system, in terms of an effective magnon field theory, viz., the energy per spin, the magnetic order parameter (i.e., the sublattie magnetization), the spin stiffness and the zero-field (uniform, transverse) magnetic susceptibility, for all values of the spin quantum number $s$ in the range $\frac{1}{2} \leq s \leq \frac{9}{2}$. The CCM data points are used to calculate the leading quantum corrections to the classical ($s \rightarrow \infty$) values of these low-energy parameters, considered as large-$s$ asymptotic expansions.

The ground-state (GS) phase diagram of the frustrated spin-$s$ $J_{1}$--$J_{2}$--$J_{3}$ Heisenberg antiferromagnet on the honeycomb lattice is studied using the coupled cluster method, for spin quantum numbers $s=1,\,\frac{3}{2},\,2\,,\frac{5}{2}$. We study the case $J_{3}=J_{2}=\kappa J_{1}$, in the range $0 \leq \kappa \leq 1$, which includes the point of maximum classical ($s \to \infty$) frustration, viz., the classical critical point at $\kappa_{{\rm cl}}=\frac{1}{2}$, separating the N\'{e}el phase for $\kappa < \kappa_{{\rm cl}}$ and the collinear striped AFM phase for $\kappa > \kappa_{{\rm cl}}$. Results are presented for the GS energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. For all spins $s \geq \frac{3}{2}$ we find a quantum phase diagram very similar to the classical one, with a direct first-order transition between the two collinear AFM states at a value $\kappa_{c}(s)$ which is slightly greater than $\kappa_{{\rm cl}}$ [e.g., $\kappa_{c}(\frac{3}{2}) \approx 0.53(1)$] and which approaches it monotonically as $s \to \infty$. By contrast, for the case $s=1$ the transition is split into two such that the stable GS phases are ones with N\'{e}el AFM order for $\kappa < \kappa_{c_{1}} = 0.485(5)$ and with striped AFM order for $\kappa > \kappa_{c_{2}} = 0.528(5)$, just as in the case $s=\frac{1}{2}$ (for which $\kappa_{c_{1}} \approx 0.47$ and $\kappa_{c_{2}} \approx 0.60$). For both the $s=\frac{1}{2}$ and $s=1$ models the transition at $\kappa_{c_{2}}$ appears to be of first-order type, while that at $\kappa_{c_{1}}$ appears to be continuous. However, whereas in the $s=\frac{1}{2}$ case the intermediate phase appears to have PVBC order over the entire range $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$, in the $s=1$ case PVBC ordering either exists only over a very small part of the region or, more likely, is absent everywhere.

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 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 use the coupled cluster method implemented to high orders of approximation to investigate the frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{3}$ antiferromagnet on the honeycomb lattice with isotropic Heisenberg interactions of strength $J_{1} > 0$ between nearest-neighbor pairs, $J_{2}>0$ between next-nearest-neighbor pairs, and $J_{3}>0$ between next-next-neareast-neighbor pairs of spins. In particular, we study both the ground-state (GS) and lowest-lying triplet excited-state properties in the case $J_{3}=J_{2} \equiv \kappa J_{1}$, in the window $0 \leq \kappa \leq 1$ of the frustration parameter, which includes the (tricritical) point of maximum classical frustration at $\kappa_{{\rm cl}} = \frac{1}{2}$. We present GS results for the spin stiffness, $\rho_{s}$, and the zero-field uniform magnetic susceptibility, $\chi$, which complement our earlier results for the GS energy per spin, $E/N$, and staggered magnetization, $M$, to yield a complete set of accurate low-energy parameters for the model. Our results all point towards a phase diagram containing two quasiclassical antiferromagnetic phases, one with N\'eel order for $\kappa < \kappa_{c_{1}}$, and the other with collinear striped order for $\kappa > \kappa_{c_{2}}$. The results for both $\chi$ and the spin gap $\Delta$ provide compelling evidence for a quantum paramagnetic phase that is gapped over a considerable portion of the intermediate region $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$, especially close to the two quantum critical points at $\kappa_{c_{1}}$ and $\kappa_{c_{2}}$. Each of our fully independent sets of results for the low-energy parameters is consistent with the values $\kappa_{c_{1}} = 0.45 \pm 0.02$ and $\kappa_{c_{2}} = 0.60 \pm 0.02$, and with the transition at $\kappa_{c_{1}}$ being of continuous (and probably of the deconfined) type and that at $\kappa_{c_{2}}$ being of first-order type.

We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-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 $0 \leq \kappa < 1$. The classical version of the model has a single GS phase transition at $\kappa^{cl}=$1/8 in this window from a phase with 3-sublattice antiferromagnetic (AFM) 120$^{\circ}$ N\'{e}el order for $\kappa < \kappa^{cl}$ to an infinitely degenerate family of 4-sublattice AFM N\'{e}el phases for $\kappa > \kappa^{cl}$. 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-1/2 model for the GS energy per spin, $E/N=-0.5521(2)J_{1}$, and the GS magnetic order parameter, $M=0.198(5)$, which are among the best available. For the spin-1/2 $J_{1}$--$J_{2}$ model we find that the classical transition at $\kappa=\kappa^{cl}$ 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 $\kappa < \kappa^{c}_{1}$ and $\kappa > \kappa^{c}_{2}$, respectively, while in the intermediate regimes $\kappa^{c}_{1} < \kappa < \kappa^{c}_{2}$ the stable GS phase has no evident long-range magnetic order.

We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spin-lattice models of interest in quantum magnetism, including their quantum phase transitions. The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (so-called LSUB$m$) approximations, by the use of computer-algebraic techniques. The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the "raw" LSUB$m$ results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, $m \rightarrow \infty$, of the truncation index $m$, which denotes the {\it only} approximation made. All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half $J^{XXZ}_{1}$--$J^{XXZ}_{2}$ model on a 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, where both interactions are of the same anisotropic $XXZ$ type. We show how the method can be used to determine the entire zero-temperature ground-state phase diagram of the model in the range $0 \leq \kappa \leq 1$ of the frustration parameter and $0 \leq \Delta \leq 1$ of the spin-space anisotropy parameter. In particular, we identify a candidate quantum spin-liquid region in the phase space.

We study the zero-temperature ($T=0$) ground-state (GS) properties of a frustrated spin-half $J^{XXZ}_{1}$--$J^{XXZ}_{2}$ 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.

We study the zero-temperature ground-state (GS) phase diagram of a spin-half $J_{1}$--$J_{2}$ $XY$ model on the honeycomb lattice with nearest-neighbor exchange coupling $J_{1}>0$ 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 $0 \leq \kappa \leq 1$. In this range we find phases exhibiting, respectively, N\'{e}el $xy$ planar [N(p)], N\'{e}el $z$-aligned [N($z$)], SDVBC, and N\'{e}el-II $xy$ planar [N-II(p)] orderings which break the lattice rotational symmetry. 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 range $\kappa_{c_{2}} < \kappa < \kappa_{c_{3}} \approx 0.52(3)$ 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.

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 $J_{2} \equiv \kappa J_{1}>0$. 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 $\kappa_{c_{1}} \approx 1.00 \pm 0.01$ (as does the classical model at $\kappa_{{\rm cl}}=1$) between two antiferromagnetic phases, viz., a quasiclassical N\'{e}el phase (for $\kappa < \kappa_{c_{1}}$) and one of the infinitely degenerate family of quasiclassical phases (for $\kappa > \kappa_{c_{1}}$) that exists in the classical model for $\kappa > \kappa_{{\rm cl}}$, which is now chosen by the {\it order by disorder} mechanism as (probably) the "doubled N\'{e}el" (or N\'{e}el$^{\ast}$) 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 $\kappa_{c_{2}} \approx 2.0 \pm 0.5$ in the spin-1 model, such that for $\kappa > \kappa_{c_{2}}$ the quasiclassical (N\'{e}el$^{\ast}$) 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 N\'{e}el 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 $\kappa > \kappa_{c_{2}}$.

We investigate ground states of $s$=1/2 Heisenberg antiferromagnets on the eleven two-dimensional (2D) Archimedian 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 Archimedian 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., Comment: 2 figures and 2 tables

Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half ($s=1/2$) $J_{1}$-$J_{2}$ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has 4 nearest-neighbour exchange bonds of strength $J_{1}$ and 2 next-nearest-neighbour (diagonal) bonds of strength $J_{2}$. The $J_{2}$ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no $J_{2}$ bonds included. The classical ($s \rightarrow \infty$) version of the model has 4 collinear phases when $J_{1}$ and $J_{2}$ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'{e}el, double N\'{e}el and double columnar striped order respectively, while the fourth is ferromagnetic. For the quantum $s=1/2$ model we use the 3 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 5 phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the 5 quantum critical points are determined with high accuracy. While all 4 phase transitions in the classical model are first order, we find strong evidence that 3 of the 5 quantum phase transitions in the $s=1/2$ model are of continuous deconfined type.

The coupled cluster method (CCM) is used to study the zero-temperature properties of a frustrated spin-half ($s={1}{2}$) $J_{1}$--$J_{2}$ Heisenberg antiferromagnet (HAF) on a 2D chevron-square lattice. Each site on an underlying square lattice has 4 nearest-neighbor exchange bonds of strength $J_{1}>0$ and 2 next-nearest-neighbor (diagonal) bonds of strength $J_{2} \equiv x J_{1}>0$, with each square plaquette having only one diagonal bond. The diagonal bonds form a chevron pattern, and the model thus interpolates smoothly between 2D HAFs on the square ($x=0$) and triangular ($x=1$) lattices, and also extrapolates to disconnected 1D HAF chains ($x \to \infty$). The classical ($s \to \infty$) version of the model has N\'{e}el order for $0 < x < x_{{\rm cl}}$ and a form of spiral order for $x_{{\rm cl}} < x < \infty$, where $x_{{\rm cl}} = {1}{2}$. For the $s={1}{2}$ model we use both these classical states, as well as other collinear states not realized as classical ground-state (GS) phases, as CCM reference states, on top of which the multispin-flip configurations resulting from quantum fluctuations are incorporated in a systematic truncation scheme, which we carry out to high orders and extrapolate to the physical limit. We calculate the GS energy, GS magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order, including plaquette and two different dimer forms. We find that the $s={1}{2}$ model has two quantum critical points, at $x_{c_{1}} \approx 0.72(1)$ and $x_{c_{2}} \approx 1.5(1)$, with N\'{e}el order for $0 < x < x_{c_{1}}$, a form of spiral order for $x_{c_{1}} < x < x_{c_{2}}$ that includes the correct three-sublattice $120^{\circ}$ spin ordering for the triangular-lattice HAF at $x=1$, and parallel-dimer VBC order for $x_{c_{2}} < x < \infty$.

Using the coupled cluster method we study the phase diagram of the spin-1/2 Heisenberg antiferromagnet on a honeycomb lattice with nearest-neighbor exchange coupling $J_{1}>0$ and frustrating next-nearest-neighbor coupling $J_{2} \equiv xJ_{1}>0$. In the range $0

We use the coupled cluster method to study the zero-temperature properties of an extended two-dimensional Heisenberg antiferromagnet formed from spin-1/2 moments on an infinite spatially anisotropic kagome lattice of corner-sharing isosceles triangles, with nearest-neighbor bonds only. The bonds have exchange constants $J_{1}>0$ along two of the three lattice directions and $J_{2} \equiv \kappa J_{1} > 0$ along the third. In the classical limit the ground-state (GS) phase for $\kappa < 1/2$ has collinear ferrimagnetic (N\'{e}el$'$) order where the $J_2$-coupled chain spins are ferromagnetically ordered in one direction with the remaining spins aligned in the opposite direction, while for $\kappa > 1/2$ there exists an infinite GS family of canted ferrimagnetic spin states, which are energetically degenerate. For the spin-1/2 case we find that quantum analogs of both these classical states continue to exist as stable GS phases in some regions of the anisotropy parameter $\kappa$, namely for $0<\kappa<\kappa_{c_1}$ for the N\'{e}el$'$ state and for (at least part of) the region $\kappa>\kappa_{c_2}$ for the canted phase. However, they are now separated by a paramagnetic phase without either sort of magnetic order in the region $\kappa_{c_1} < \kappa < \kappa_{c_2}$, which includes the isotropic kagome point $\kappa = 1$ where the stable GS phase is now believed to be a topological ($\mathbb{Z}_2$) spin liquid. Our best numerical estimates are $\kappa_{c_1} = 0.515 \pm 0.015$ and $\kappa_{c_2} = 1.82 \pm 0.03$.

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 $J_{3}<0$. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at: (a) $(J_{2}^{c_1},J_{3}^{c_1})=(0.51 \pm 0.01,0.69 \pm 0.01)$ between the N, S, and QP phases; (b) $(J_{2}^{c_2},J_{3}^{c_2})=(0.65 \pm 0.02,0.55 \pm 0.01)$ between the S, spiral, and QP phases; and (c) $(J_{2}^{c_3},J_{3}^{c_3})=(0.69 \pm 0.01,0.12 \pm 0.01)$ between the spiral, aN, and QP phases.

We use the coupled cluster method to investigate the ground-state (GS) properties of the frustrated spin-1/2 $J_{1}$-$J_{2}$-$J_{3}$ model on the honeycomb lattice, with nearest-neighbor exchange coupling $J_1$ plus next-nearest-neighbor ($J_2$) and next-next-nearest-neighbor ($J_3$) 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 $J_{2}/J_{1} = -1.17 \pm 0.01$ when $J_{1}>0$, 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., Comment: 4 pages, 4 figures

We study the ground-state (gs) phases of the spin-half anisotropic planar pyrochlore (or crossed chain) model using the coupled cluster method (CCM). The model is a frustrated antiferromagnetic (AFM) $J_{1}$--$J_{2}$ system on the checkerboard lattice, with nearest-neighbor exchange bonds $J_{1}>0$ and next-nearest-neighbor bonds $J_{2} \equiv \kappa J_{1} > 0$. Using various AFM classical ground states as CCM model states we present results for their gs energy, average on-site magnetization, and susceptibilities to plaquette valence-bond crystal (PVBC) and crossed-dimer valence-bond crystal (CDVBC) ordering. We show that the state with Neel ordering is the gs phase for $\kappa < \kappa_{c_1} \approx 0.80 \pm 0.01$, but that none of the fourfold set of AFM states selected by quantum fluctuations at $O(1/s)$ in a large-$s$ analysis (where $s$ is the spin quantum number) from the infinitely degenerate set of AFM states that form the gs phase for the classical version of the model (for $\kappa>1$) survives the quantum fluctuations to form a stable magnetically-ordered gs phase for the spin-half case. The Neel state becomes susceptible to PVBC ordering at or very near to $\kappa = \kappa_{c_1}$, and the fourfold AFM states become infinitely susceptible to PVBC ordering at $\kappa = \kappa_{c_2} \approx 1.22 \pm 0.02$. In turn, we find that these states become infinitely susceptible to CDVBC ordering for all values of $\kappa$ above a certain critical value at or very near to $\kappa = \kappa_{c_2}$. We thus find a Neel-ordered gs phase for $\kappa<\kappa_{c_1}$, a PVBC-ordered phase for $\kappa_{c_1} < \kappa < \kappa_{c_2}$, and a CDVBC-ordered phase for $\kappa > \kappa_{c_2}$. Both transitions are probably 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., Comment: 12 pages, 6 figures

We study the ground-state (gs) phase diagram of the frustrated spin-1/2 $J_{1}$--$J_{2}$ antiferromagnet with $J_{2}=\kappa J_1>0$ ($J_{1}>0$) on the honeycomb lattice, using the coupled-cluster method. We present results for the ground-state energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. We find a paramagnetic PVBC phase for $\kappa_{c_1}<\kappa<\kappa_{c_2}$, where $\kappa_{c_1} \approx 0.207 \pm 0.003$ and $\kappa_{c_2} \approx 0.385 \pm 0.010$. The transition at $\kappa_{c_1}$ to the N\'{e}el phase seems to be a continuous deconfined transition (although we cannot exclude a very narrow intermediate phase in the range $0.21 \lesssim \kappa \lesssim 0.24$), while that at $\kappa_{c_2}$ is of first-order type to another quasiclassical antiferromagnetic phase that occurs in the classical version of the model only at the isolated and highly degenerate critical point $\kappa = 1/2$. The spiral phases that are present classically for all values $\kappa > 1/6$ are absent for all $\kappa \lesssim 1$., Comment: 6 pages, 5 figures

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))

Starting with the sqrt{3} x sqrt{3} and the q=0 states as reference states we use the coupled cluster method to high orders of approximation to investigate the ground state of the Heisenberg antiferromagnet on the kagome lattice for spin quantum numbers s=1/2,1,3/2,2,5/2, and 3. Our data for the ground-state energy for s=1/2 are in good agreement with recent large-scale density-matrix renormalization group and exact diagonalization data. We find that the ground-state selection depends on the spin quantum number s. While for the extreme quantum case, s=1/2, the q=0 state is energetically favored by quantum fluctuations, for any s>1/2 the sqrt{3} x sqrt{3} state is selected. For both the sqrt{3} x sqrt{3} and the q=0 states the magnetic order is strongly suppressed by quantum fluctuations. Within our coupled cluster method we get vanishing values for the order parameter (sublattice magnetization) M for s=1/2 and s=1, but (small) nonzero values for M for s>1. Using the data for the ground-state energy and the order parameter for s=3/2,2,5/2, and 3 we also estimate the leading quantum corrections to the classical values., Comment: 7 pages, 6 figures

We study the ground-state (gs) properties of the frustrated spin-1/2 $J_{1}$--$J_{2}$--$J_{3}$ Heisenberg model on a honeycomb lattice with ferromagnetic (FM) nearest-neighbor ($J_{1}=-1$) exchange and frustrating antiferromagnetic (AFM) next-nearest-neighbor ($J_{2}>0$) and next-next-nearest-neighbor ($J_{3}>0$) exchanges, for the case $J_{3}=J_{2}$. We use the coupled cluster method in high orders of approximation, complemented by the exact diagonalization of a lattice with 32 sites, and calculate the gs energy, magnetic order parameter, and spin-spin correlation functions. We find a quantum phase transition between regions characterized by FM order and a form of AFM ("striped") collinear order at $J^{c}_{2} \approx 0.1095 \pm 0.0005$. We compare results for the FM case (with $J_{1}=-1$) to previous results for the corresponding AFM case (with $J_{1}=+1$). While the magnetic order parameters behave similarly for the FM and the AFM models for large values of the frustration parameter $J_{2}$, there are considerable differences between them for $J_{2}/|J_{1}| \lesssim 0.6$. For example, the quasiclassical collinear magnetic long-range order for the AFM model (with $J_{1}=+1$) breaks down at $J^{c_{2}}_{2} \approx 0.60$, whereas the "equivalent" point for the FM model (with $J_{1}=-1$) occurs at $J^{c}_{2} \approx 0.11$. Unlike in the AFM model (with $J_{1}=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical AFM phases (with N\'eel and striped order) for $J^{c_{1}}_{2} < J_{2} < J^{c_{2}}_{2}$, with $J^{c_{1}}_{2} \approx 0.47$, we find no clear indications in the FM model for an intermediate magnetically disordered phase between the phases exhibiting FM and striped order. Instead, the evidence points strongly to a direct first-order transition between the two ordered phases of the FM model., Comment: 21 pages, 6 figures (a & b)

We study the ground-state (gs) phase diagram of the frustrated spin-1/2 $J_{1}$-$J_{2}$-$J_{3}$ antiferromagnet with $J_{2} = J_{3} =\kappa J_1$ on the honeycomb lattice, using coupled-cluster theory and exact diagonalization methods. We present results for the gs energy, magnetic order parameter, spin-spin correlation function, and plaquette valence-bond crystal (PVBC) susceptibility. We find a N\'eel antiferromagnetic (AFM) phase for $\kappa < \kappa_{c_{1}} \approx 0.47$, a collinear striped AFM phase for $\kappa > \kappa_{c_{2}} \approx 0.60$, and a paramagnetic PVBC phase for $\kappa_{c_{1}} \lesssim \kappa \lesssim \kappa_{c_{2}}$. The transition at $\kappa_{c_{2}}$ appears to be of first-order type, while that at $\kappa_{c_{1}}$ is continuous. Since the N\'eel and PVBC phases break different symmetries our results favor the deconfinement scenario for the transition at $\kappa_{c_{1}}$.

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: 12

A new approximation hierarchy, called the LPSUB$m$ scheme, is described for the coupled cluster method (CCM). It is applicable to systems defined on a regular spatial lattice. We then apply it to two well-studied prototypical (spin-1/2 Heisenberg antiferromagnetic) spin-lattice models, namely: the XXZ and the XY models on the square lattice in two dimensions. Results are obtained in each case for the ground-state energy, the ground-state sublattice magnetization and the quantum critical point. They are all in good agreement with those from such alternative methods as spin-wave theory, series expansions, quantum Monte Carlo methods and the CCM using the alternative LSUB$m$ and DSUB$m$ schemes. Each of the three CCM schemes (LSUB$m$, DSUB$m$ and LPSUB$m$) for use with systems defined on a regular spatial lattice is shown to have its own advantages in particular applications.

The coupled cluster method is applied to a spin-half model at zero temperature ($T=0$), which interpolates between Heisenberg antiferromagnets (HAF's) 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 $J_{1} \geq 0$ along two of the equivalent directions and $J_{2} \geq 0$ along the third. Sites connected by $J_{2}$ bonds are themselves connected to the missing NN non-kagome sites of the triangular lattice by bonds of strength $J_{1}' \geq 0$. When $J_{1}'=J_{1}$ and $J_{2}=0$ 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., Comment: 21 pages, 8 figures

We use the coupled cluster method (CCM) to study the zero-temperature phase diagram of a 2D frustrated spin-half antiferromagnet, the so-called Union Jack model. It is defined on a square lattice such that all nearest-neighbor bonds are present with a strength $J_{1} > 0$, but only half the next-nearest-neighbor bonds are present with a 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. We find strong evidence for a first phase transition between a N\'{e}el phase and a canted ferrimagnetic phase at a critical coupling $\kappa_{c_{1}} = 0.66 \pm 0.02$. At the transition the energy and its first derivative seem continuous, thus providing a typical scenario of a second-order transition, although a weakly first-order transition cannot be excluded. By contrast, the average on-site magnetization $M$ approaches a nonzero value $M_{c_{1}}=0.195 \pm 0.005$ on both sides of the transition, which is more typical of a first-order transition. The slope $dM/d\kappa$ also appears to be continuous, or very nearly so, at the critical point $\kappa_{c_{1}}$. We find strong evidence that the canted phase becomes unstable at large values of $\kappa$, and hence we have also used the CCM with a model collinear semi-stripe-ordered ferrimagnetic state in which alternating rows (and columns) are ferromagnetically and antiferromagnetically ordered. We find tentative evidence, based on the relative energies of the two states, for a second (first-order) phase transition between the canted and semi-stripe-ordered states at a large value of the coupling parameter around $\kappa_{c_{2}} \approx 125 \pm 5$. This prediction, however, is based on an extrapolation of the CCM results for the canted state into regimes where the CCM equations at any level of approximation beyond the lowest have no solutions., Comment: 31 pages, 8 figures

A new approximate scheme, DSUB$m$, is described for the coupled cluster method. We then apply it to two well-studied (spin-1/2 Heisenberg antiferromagnet) spin-lattice models, namely: the $XXZ$ and the $XY$ models on the square lattice in two dimensions. Results are obtained in each case for the ground-state energy, the sublattice magnetization and the quantum critical point. They are in good agreement with those from such alternative methods as spin-wave theory, series expansions, quantum Monte Carlo methods and those from the CCM using the LSUB$m$ scheme., Comment: 18 pages, 10 figures