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arXiv:cond-mat/0506161 · Physical Review B

Diagrammatic approach in the variational coupled-cluster method

Yang Xian

Published Dec 30, 2005• 9 citations
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Abstract

We present an alternative approach based on a diagrammatic technique to the recently proposed variational coupled-cluster method. We apply this method to quantum antiferromagnetic bipartite lattices. In our analysis, infinite resummations of diagrams can be done in a straightforward manner. One such resummation reproduces spin-wave theory (SWT). Approximations beyond SWT can also be easily made. Interestingly, one such approximation produces a convergent, precise number for the order parameter of the one-dimensional isotropic model, in contrast to the well-known divergence of SWT; it also produces improved results for the two-dimensional square-lattice system. We also discuss relations and a possible combination of our approach with other established many-body theories.

DOI
10.1103/physrevb.72.224438

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We present an alternative approach based on a diagrammatic technique to the recently proposed variational coupled-cluster method. We apply this method to quantum antiferromagnetic bipartite lattices. In our analysis, infinite resummations of diagrams can be done in a straightforward manner. One such resummation reproduces spin-wave theory (SWT). Approximations beyond SWT can also be easily made. Interestingly, one such approximation produces a convergent, precise number for the order parameter of the one-dimensional isotropic model, in contrast to the well-known divergence of SWT; it also produces improved results for the two-dimensional square-lattice system. We also discuss relations and a possible combination of our approach with other established many-body theories.
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 spinlattice 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 (socalled LSUBm) 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" LSUBm results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, m → ∞, of the truncation index m, which denotes the 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 nearestneighbor and next-nearest-neighbor interactions with exchange couplings J 1 > 0 and J 2 ≡ κJ 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 zerotemperature ground-state phase diagram of the model in the range 0 ≤ κ ≤ 1 of the frustration parameter and 0 ≤ Δ ≤ 1 of the spin-space anisotropy parameter.In particular, we identify a candidate quantum spin-liquid region in the phase space.
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.
We develop a general quantum many-body theory in configuration space by extending the traditional coupled-cluster method to a variational formalism. A set of distribution functions are introduced to evaluate the expectation value of the Hamiltonian. An algebraic technique for calculations of these distribution functions via a set of self-consistent set of equations is given. As a demonstration, we apply this method to a quantum antiferromagnetic spin model. It is shown that one of the lowest-order approximations within this new technique agrees with the traditional coupled-cluster method. Comparison with the method of correlated basis functions is also made.
A coupled-cluster formalism similar to that used in electronic-structure theory is developed for quantum-mechanical systems localized to sites on a lattice. It is shown how diagrammatic methods may be used to represent the cluster contributions in a convenient and systematic manner. The formalism is applied to the square S=1/2 Heisenberg antiferromagnet, yielding with minor computational effort results of quality comparable to those from computationally intensive quantum Monte Carlo calculations.
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-mmm and analytically for SUB2. It is found that SUB2-mmm results converge rapidly to the full SUB2 solution, including new results for the SUB2-mmm 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.
We extend the recently proposed variational coupled-cluster method to describe excitation states of quantum many-body interacting systems. We discuss, in general terms, both quasiparticle excitations and quasiparticle-density-wave excitations (collective modes). In application to quantum antiferromagnets, we reproduce the well-known spin-wave excitations, i.e. quasiparticle magnons of spin ± 1. In addition, we obtain new, spin-zero magnon-density-wave excitations, which have been missing in Anderson's spin-wave theory. Implications of these new collective modes are discussed.