The coupled cluster method (CCM) is a powerful and widely applied technique of modern-day quantum many-body theory. It has been used with great success in order to understand the properties of quantum magnets at zero temperature. This is largely due to the application of computational techniques that allow the method to be applied to high orders of approximation using a localized scheme known as the LSUBm scheme. A hitherto unreported aspect of this scheme is that results for LSUBm expectation values behave in distinctly different ways with odd and even values of m. Here, we consider the behavior of ground-state expectation values of odd and even orders of the CCM LSUBm approximation for unfrustrated spin-half Heisenberg antiferromagnets on the square and honeycomb lattice and the frustrated spin-half Heisenberg antiferromagnet on the triangular lattice. We demonstrate that results for odd and even orders of approximation show qualitatively different behavior for both the ground-state energy and the sublattice magnetization. Indeed, the odd series consistently forms an upper branch of results, and the even series a lower branch with respect to both ground-state energy and sublattice magnetization, for all of the models considered here.
The coupled cluster method (CCM) is a powerful and widely applied technique of modern-day quantum many-body theory. It has been used with great success in order to understand the properties of quantum magnets at zero temperature. This is largely due to the application of computational techniques that allow the method to be applied to high orders of approximation using a localized scheme known as the LSUBm scheme. A hitherto unreported aspect of this scheme is that results for LSUBm expectation values behave in distinctly different ways with odd and even values of m. Here, we consider the behavior of ground-state expectation values of odd and even orders of the CCM LSUBm approximation for unfrustrated spin-half Heisenberg antiferromagnets on the square and honeycomb lattice and the frustrated spin-half Heisenberg antiferromagnet on the triangular lattice. We demonstrate that results for odd and even orders of approximation show qualitatively different behavior for both the ground-state energy and the sublattice magnetization. Indeed, the odd series consistently forms an upper branch of results, and the even series a lower branch with respect to both ground-state energy and sublattice magnetization, for all of the models considered here.
High-Order Coupled Cluster Method (CCM) Formalism 1: Ground- and Excited-State Properties of Lattice Quantum Spin Systems with s>=1/2
The coupled cluster method (CCM) is a powerful and widely applied technique of modern-day quantum many-body theory. It has been used with great success in order to understand the properties of quantum magnets at zero temperature. This is due largely to the application of computational techniques that allow the method to be applied to high orders of approximation using localised approximation schemes, e.g., such as the LSUBm scheme. In this article, the high-order CCM formalism for the ground and excited states of quantum magnetic systems are extended to those with spin quantum number s≥21. Solution strategies for the ket- and bra-state equations are also considered. Aspects of extrapolation of CCM expectation values are discussed and future topics regarding extrapolations are presented.
Recent developments of high-order CCM have been to extend existing formalism and codes to s≥21 for both the ground and excited states. In this article, we describe how "generalised" expectation values for a wide range of one- and two-body spin operators may also be determined using existing the CCM code for the ground state. We present new results for the spin-spin correlation functions of the spin-half square- and triangular-lattice antiferromagnets by using the LSUBm approximation. We show that the absolute values of the spin-spin correlation functions ∣<s(0).s(r)>∣ converge with increasing approximation level for both lattices. We believe that the LSUBm approximation provides reasonable results for the correlation functions for lattice separations roughly of order r≈m for the square lattice. We compare qualitatively our results for the square lattice to those results of quantum Monte Carlo (QMC) and we see that good correspondence is observed. Indeed as seen by QMC, the spin-spin correlation function initially decays strongly with ∣r∣ before becoming constant for larger values of ∣r∣. CCM results are also compared to results of exact diagonalisations for both lattices. ED results demonstrate a strong finite-size effects at lattice separations r=L/2 (where N=L×L) for both lattices. The behaviour of the correlation function for the triangular lattice is qualitatively similar to that of the square lattice, namely, that it decays strongly at first before becoming constant. This is in keeping with the behaviour of both models, which are believed strongly to be Néel-ordered from approximate studies.
In this article, we present new results of high-order coupled cluster method (CCM) calculations, based on a Néel model state with spins aligned in the z-direction, for both the ground- and excited-state properties of the spin-half {\it XXZ} model on the linear chain, the square lattice, and the simple cubic lattice. In particular, the high-order CCM formalism is extended to treat the excited states of lattice quantum spin systems for the first time. Completely new results for the excitation energy gap of the spin-half {\it XXZ} model for these lattices are thus determined. These high-order calculations are based on a localised approximation scheme called the LSUBm scheme in which we retain all k-body correlations defined on all possible locales of m adjacent lattice sites (k≤m). The ``raw'' CCM LSUBm results are seen to provide very good results for the ground-state energy, sublattice magnetisation, and the value of the lowest-lying excitation energy for each of these systems. However, in order to obtain even better results, two types of extrapolation scheme of the LSUBm results to the limit m→∞ (i.e., the exact solution in the thermodynamic limit) are presented. The extrapolated results provide extremely accurate results for the ground- and excited-state properties of these systems across a wide range of values of the anisotropy parameter.
A high-order coupled cluster method (CCM) for the zero-temperature ground states of quantum lattice spin systems of general spin quantum number (s>1/2) is described and illustrated via the spin-1 anisotropic Heisenberg antiferromagnet on the square lattice. Results for the ground-state energy and sublattice magnetization are seen to be in excellent agreement with the best of other approximate methods. Furthermore, we are able to follow the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter $\ensuremath{\Delta}$ from the trivial Ising limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty})$ down to a critical value ${\ensuremath{\Delta}}_{c},$ at which point the physical solution terminates. The results for the ground-state energy, sublattice magnetization, and phase transition point thus support our choice of this model as a good nontrivial test case of our CCM formalism.
A new approximate scheme, DSUBm, is described for the coupled cluster method.We apply it to two wellstudied (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, exact diagonalization techniques, quantum Monte Carlo methods and those from the CCM using the LSUBm scheme.
An approximation hierarchy, called the lattice-path-based subsystem (LPSUBm) approximation 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-21 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 lattice-animal-based subsystem (LSUBm) and the distance-based subsystem (DSUBm) schemes. Each of the three CCM schemes (LSUBm, DSUBm, and LPSUBm) for use with systems defined on a regular spatial lattice is shown to have its own advantages in particular applications.