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R. F. Bishop

Publications and source records attributed to R. F. Bishop.

At least 55 records · Page 3Linked to original sources
arXiv2009

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

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

cond-mat.str-el↗
arXiv2009

Magnetic order in a spin-half interpolating square-triangle Heisenberg antiferromagnet

Using the coupled cluster method we study the zero-temperature phase diagram of a spin-half Heisenberg antiferromagnet (HAF), the so-called $J_{1}$--$J_{2}'$ model, defined on an anisotropic 2D lattice. With respect to an underlying square-lattice geometry the model contains antiferromagnetic ($J_{1} > 0$) bonds between nearest neighbors and competing ($J_{2}'>0$) 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 $J_{2}' \equiv κJ_{1}$ bonds along parallel chains and $J_{1}$ bonds providing an interchain coupling. 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 ($κ\to \infty$), with the spin-half HAF on the triangular lattice in between at $κ= 1$. We find strong evidence that quantum fluctuations favor a first-order transition from quasiclassical Néel order to a quantum helical state at a first critical point at $κ_{c_{1}} = 0.80 \pm 0.01$, by contrast with the corresponding second-order transition between the equivalent classical states at $κ_{\rm cl} = 0.5$. We also find strong evidence for a second critical point at $κ_{c_{2}} = 1.8 \pm 0.4$ where another first-order transition occurs, this time from the quantum helical phase to a collinear stripe-ordered phase. This latter result provides quantitative verification of a recent qualitative prediction of Starykh and Balents [Phys.\ Rev. Lett. {\bf 98}, 077205 (2007)] for the $J_{1}$--$J_{2}'$ model that did not, however, evaluate the corresponding critical point.

cond-mat.str-el↗
arXiv2009

The DSUB$m$ Approximation Scheme for the Coupled Cluster Method and Applications to Quantum Magnets

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.

cond-mat.str-el↗
arXiv2008

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

Using the coupled-cluster method for infinite lattices and the exact diagonalization method for finite lattices, we study the influence of an exchange anisotropy Delta on the ground-state phase diagram of the spin-1/2 frustrated J1-J2 XXZ antiferromagnet on the square lattice. We find that increasing Delta>1 (i.e. an Ising type easy-axis anisotropy) as well as decreasing Delta<1 (i.e. an XY type easy-plane anisotropy) both lead to a monotonic shrinking of the parameter region of the magnetically disordered quantum phase. Finally, at Delta~1.9 this quantum phase disappears, whereas in pure XY limit (Delta=0) there is still a narrow region around J2 =0.5J1 where the quantum paramagnetic ground-state phase exists.

cond-mat.str-el↗
arXiv2008

The quantum $J_{1}$--$J_{1}'$--$J_{2}$ spin-1 Heisenberg model: Influence of the interchain coupling on the ground-state magnetic ordering in 2D

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 Néel and stripe states, as compared with all previous results for the corresponding spin-1/2 case. However, we find a quantum tricritical point at $J_{1}'/J_{1} \approx0.66 \pm 0.03$, $J_{2}/J_{1} \approx0.35\pm0.02$, where a line of second-order phase transitions between the quasi-classical Néel and stripe-ordered phases (for $J_{1}'/J_{1} \lesssim 0.66$) meets a line of first-order phase transitions between the same two states (for $J_{1}'/J_{1} \gtrsim 0.66$)

cond-mat.str-el↗
arXiv2008

Effect of anisotropy on the ground-state magnetic ordering of the spin-half quantum $J_1^{XXZ}$--$J_2^{XXZ}$ model on the square lattice

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

cond-mat.str-el↗
arXiv2008

The quantum $J_1$-$J_1'$-$J_2$ spin-1/2 Heisenberg model: Influence of the interchain coupling on the ground-state magnetic ordering in 2D

We study the phase diagram of the 2D $J_1$-$J_1'$-$J_2$ spin-1/2 Heisenberg model by means of the coupled cluster method. The effect of the coupling $J_1'$ on the Neel and stripe states is investigated. We find that the quantum critical points for the Neel and stripe phases increase as the coupling strength $J_1'$ is increased, and an intermediate phase emerges above the region at $J_1' \approx 0.6$ when $J_1=1$. We find indications for a quantum triple point at $J_1' \approx 0.60 \pm 0.03$, $J_2 \approx 0.33 \pm 0.02$ for $J_1=1$.

cond-mat.str-el↗
arXiv2006

Frustrated Quantum Antiferromagnets: Application of High-Order Coupled Cluster Method

We report on recent results for strongly frustrated quantum $J_1$-$J_2$ antiferromagnets in dimensionality d=1,2,3 obtained by the coupled cluster method (CCM). We demonstrate that the CCM in high orders of approximation allows us to investigate quantum phase transitions driven by frustration and to discuss novel quantum ground states. In detail we consider the ground-state properties of (i) the Heisenberg spin-1/2 antiferromagnet on the cubic lattice in d=1,2,3, and use the results for the energy, the sublattice magnetization and the spin stiffness as a benchmark test for the precision of the method; (ii) coupled frustrated spin chains (the quasi-one-dimensional $J_1$--$J_2$ model) and discuss the influence of the quantum fluctuations and the interchain coupling on the incommensurate spiral state present in the classical model; (iii) the Shastry-Sutherland antiferromagnet on the square lattice; and (iv) a stacked frustrated square-lattice Heisenberg antiferromagnet (the quasi-two-dimensional $J_1$--$J_2$ model), and discuss the influence of the interlayer coupling on the quantum paramagnetic ground-state phase that is present for the strictly two-dimensional model.

cond-mat.str-el↗
arXiv2006

The Coupled Cluster Method Applied to the {\it XXZ} Model on the Square Lattice

A review of the coupled cluster method (CCM) applied to lattice quantum spin systems is presented here. The CCM formalism is explained and an application to the spin-half {\it XXZ} model on the square lattice is presented. Low orders of approximation are carried out analytically and a high-order CCM formulation is presented. Results for the SUB2 approximation are carried out numerically for SUB2-$m$ and analytically for SUB2. It is found that SUB2-$m$ results converge rapidly to the full SUB2 solution, including new results for the SUB2-$m$ limiting points compared to the SUB2 critical point. Results for the ground-state energy and the sublattice magnetisation are presented. A study of the excitation spectrum of this model at the SUB2 critical point is given. Indeed, the shape of the excitation spectrum at the SUB2 critical point is identical to that predicted by spin-wave theory for the isotropic model, albeit with a multiplicative factor of 1.1672. This result compares very well to results of cumulant series expansions and Monte Carlo simulation that again predict a similar shape for the excitation spectrum, but with multiplicative factors of 1.18 and 1.21$\pm$0.03, respectively. Results for the isotropic Heisenberg model on the square lattice for the spin-one antiferromagnet and the spin-one/spin-half ferrimagnet are also given.

cond-mat.str-el↗
arXiv2003

High-Order Coupled Cluster Method (CCM) Calculations Via Parallel Processing: An Application To The Kagome Antiferromagnet

A simple ``brute-force'' parallelisation procedure for the computational implementation of high-order coupled cluster method (CCM) calculations is presented here. This approach is investigated and illustrated by an application of high-order CCM to the Heisenberg antiferromagnet on the Kagomé lattice with nearest- and next-nearest-neighbour bonds. Ferromagnetic next-nearest-neighbour bonds are used to stabilise a model state which contains three sublattices in which the spins make angles of 120$^{\circ}$ to each other. Ground-state results for up to approximately 10000 fundamental clusters are presented, and our best estimate for the ground-state energy per spin of the spin-half Kagomé lattice antiferromagnet with only nearest-neighbour bonds is $E_g/N = -0.43104$. We believe that further increases (of at least another order of magnitude) in the number of fundamental clusters might be possible in future by using parallel processing techniques. The extension of high-order CCM calculations in order to consider non-Néel (e.g., dimer solid) model states, simulation of excitation spectra, lattice boson and fermion models, and finite-sized systems is very briefly considered.

cond-mat.str-el↗
arXiv2003

Coupled Cluster Method Calculations Of Quantum Magnets With Spins Of General Spin Quantum Number

We present a new high-order coupled cluster method (CCM) formalism for the ground states of lattice quantum spin systems for general spin quantum number, $s$. This new ``general-$s$'' formalism is found to be highly suitable for a computational implementation, and the technical details of this implementation are given. To illustrate our new formalism we perform high-order CCM calculations for the one-dimensional spin-half and spin-one antiferromagnetic {\it XXZ} models and for the one-dimensional spin-half/spin-one ferrimagnetic {\it XXZ} model. The results for the ground-state properties of the isotropic points of these systems are seen to be in excellent quantitative agreement with exact results for the special case of the spin-half antiferromagnet and results of density matrix renormalisation group (DMRG) calculations for the other systems. Extrapolated CCM results for the sublattice magnetisation of the spin-half antiferromagnet closely follow the exact Bethe Ansatz solution, which contains an infinite-order phase transition at $Δ=1$. By contrast, extrapolated CCM results for the sublattice magnetisation of the spin-one antiferromagnet using this same scheme are seen to go to zero at $Δ\approx 1.2$, which is in excellent agreement with the value for the onset of the Haldane phase for this model. Results for sublattice magnetisations of the ferrimagnet for both the spin-half and spin-one spins are non-zero and finite across a wide range of $Δ$, up to and including the Heisenberg point at $Δ=1$.

cond-mat.str-el↗
arXiv2003

The unreasonable accuracy of the Jastrow approach in many-body physics

We investigate in a simple model whether a Jastrow-based approach for a many-body system containing two-body interactions can be exact. By comparison with recent assertions to the contrary, we find that in general the exact wave function is \emph{not} purely two-body correlated. Nonetheless, we show that the energy estimate obtained from the calculations is as accurate as it can possibly be, suggesting why Jastrow wave functions are such a good choice.

physics.chem-ph↗
arXiv2002

Exact isolated solutions for the two-photon Rabi Hamiltonian

The two-photon Rabi Hamiltonian is a simple model describing the interaction of light with matter, with the interaction being mediated by the exchange of two photons. Although this model is exactly soluble in the rotating-wave approximation, we work with the full Hamiltonian, maintaining the non-integrability of the model. We demonstrate that, despite this non-integrability, there exist isolated, exact solutions for this model analogous to the so-called Juddian solutions found for the single-photon Rabi Hamiltonian. In so doing we use a Bogoliubov transformation of the field mode, as described by the present authors in an earlier publication.

quant-ph↗
arXiv2002

Time evolution of the Rabi Hamiltonian from the unexcited vacuum

The Rabi Hamiltonian describes a single mode of electromagnetic radiation interacting with a two-level atom. Using the coupled cluster method, we investigate the time evolution of this system from an initially empty field mode and an unexcited atom. We give results for the atomic inversion and field occupation, and find that the virtual processes cause the field to be squeezed. No anti-bunching occurs.

quant-ph↗
arXiv2001

Quantum Phase Transitions in Spin Systems

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.

cond-mat.str-el↗
arXiv2001

Phase Transition Of The Spin-One Square-Lattice Anisotropic Heisenberg Model

The coupled cluster method (CCM) is applied to the spin-one anisotropic Heisenberg antiferromagnet (HAF) on the square lattice at zero temperature using a new high-order CCM ground-state formalism for general quantum spin number ($s \ge 1/2$). The results presented constitute the first such application of this new formalism, and they are shown to be among the most accurate results for the ground-state energy and sublattice magnetisation of this model as yet determined. We ``track'' the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter, $Δ$, from the trivial Ising limit ($Δ\to \infty$) down to a critical value $Δ_c$, at which point this solution terminates. This behaviour is associated with a phase transition in the system, and hence a primary result of these high-order CCM calculations is that they provide an accurate and unbiased (i.e., {\it ab initio}) estimation of the position of the quantum phase transition point as a function of the anisotropy parameter. Our result is, namely, that this point occurs at (or slightly below) the isotropic Heisenberg point at $Δ=1$, for this model.

cond-mat.str-el↗
arXiv2000

Ab Initio Treatments of the Ising Model in a Transverse Field

In this article, new results are presented for the zero-temperature ground-state properties of the spin-half transverse Ising model on various lattices using three different approximate techniques. These are, respectively, the coupled cluster method, the correlated basis function method, and the variational quantum Monte Carlo method. The methods, at different levels of approximation, are used to study the ground-state properties of these systems, and the results are found to be in excellent agreement both with each other and with results of exact calculations for the linear chain and results of exact cumulant series expansions for lattices of higher spatial dimension. The different techniques used are compared and contrasted in the light of these results, and the constructions of the approximate ground-state wave functions are especially discussed.

cond-mat.str-el↗