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Physical review. B, Condensed matter

Combining coupled-cluster and perturbative expansions through intermediate Hamiltonians: Theory and application to 1D and 2D spin lattices

Beatriz Miguel, Jean‐Paul Malrieu

Published Jul 15, 1996• 7 citations
Publisher

Abstract

Using the perturbative expansion of the wave function from a single determinantal reference, it is possible to define a genealogical hierarchy of model spaces. The corresponding configuration-interaction matrices are dressed in the frame of intermediate Hamiltonian theory from estimates of the coefficients of the outer space determinants, which may take a coupled cluster (CC) or a perturbative form. The deviations from the additivity of transition energies are taken into account as a modification of coupled-cluster amplitudes, and higher-order exclusion-principle-violating corrections are introduced. The resulting equations relative to the first and second generations are applied to anisotropic Heisenberg Hamiltonians for one-dimensional and square two-dimensional spin lattices. The results show the efficiency of the method, which gives as accurate results with adjacent two-body operators as the traditional CC expansion with local four-body operators. With local four-body operators we obtain results comparable to those obtained with local six-body operators in normal CC treatments. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.1652

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Using the perturbative expansion of the wave function from a single determinantal reference, it is possible to define a genealogical hierarchy of model spaces. The corresponding configuration-interaction matrices are dressed in the frame of intermediate Hamiltonian theory from estimates of the coefficients of the outer space determinants, which may take a coupled cluster (CC) or a perturbative form. The deviations from the additivity of transition energies are taken into account as a modification of coupled-cluster amplitudes, and higher-order exclusion-principle-violating corrections are introduced. The resulting equations relative to the first and second generations are applied to anisotropic Heisenberg Hamiltonians for one-dimensional and square two-dimensional spin lattices. The results show the efficiency of the method, which gives as accurate results with adjacent two-body operators as the traditional CC expansion with local four-body operators. With local four-body operators we obtain results comparable to those obtained with local six-body operators in normal CC treatments. \textcopyright{} 1996 The American Physical Society.
The coupled-cluster method (CCM) is applied to anisotropic-quantum-antiferromagnetic models in one and two dimensions (1D and 2D). Several hierarchical approximation schemes of the CCM are defined, which are specialized for the spin-lattice models. Good results are obtained for both ground-state and excited-state energies. The spin-spin correlation functions and the staggered magnetizations are also calculated. We have found that the CCM gives a qualitatively correct description of the entire Heisenberg-Ising phase, including the critical point where a phase transition occurs in 1D. Some interesting differences between 1D and 2D models near the critical point are discussed.
We introduce perturbation and coupled-cluster theories based on a cluster mean-field reference for describing the ground state of strongly correlated spin systems. In cluster mean-field, the ground state wave function is written as a simple tensor product of optimized cluster states. The cluster language and the mean-field nature of the ansatz allow for a straightforward improvement which uses perturbation theory and coupled-cluster to account for intercluster correlations. We present benchmark calculations on the 1D chain and 2D square J 1 – J 2 Heisenberg model, using cluster mean-field, perturbation theory, and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies accurately. Our results indicate that, with sufficiently large clusters, the correlated methods (cPT2, cPT4, and cCCSD) can provide a relatively accurate description of the Heisenberg model in the regimes considered, which suggests that the methods presented can be used for other strongly correlated systems. Some ways to improve upon the methods presented in this work are discussed.
Strongly interacting quantum spin–lattice models exhibit a wide variety of phases with diverse and subtle magnetic ordering properties. Their detailed description within a unified microscopic framework poses a real challenge for the many-body theorist. By specific application to the spin-½ anisotropic Heisenberg model on a square lattice, we show how the ab initio coupled cluster method, which has already been very successfully applied to a wide variety of quantum many-body and field-theoretic systems, may be very efficiently and systematically implemented for spin–lattice models. Results for such local properties as the ground-state energy and sublattice magnetization are thereby obtained which are on a par with those from the best of the available alternative methods. Furthermore, we demonstrate explicitly how the coupled cluster method now also provides an effective and fully microscopic tool to yield systematic and accurate estimates of the zero-temperature quantum phase transition boundaries between states of different quantum order, as well as of the critical behaviour of the system in the vicinity of the transition points.
The Coupled-Cluster (CC) method is one of the most popular and efficient correlation methods in quantum chemistry. \nEspecially the CCSD(T) approximation, which includes single and double excitations by means of the application of the cluster operator \nto a reference determinant and triple excitations via a perturbative treatment, has become a standard tool \nin quantum chemical applications. However, the method is restricted to relatively small system sizes due to its unfavourable scaling (N^7, where N is the number of basis functions applied). \nFor the full treatment of triple excitations the scaling advances to N^8 and every further excitation level increases the exponent by two. \n \nThe goal of this work is to reduce the calculation time for closed shell systems at least by a factor growing with the excitation level for arbitrary truncation levels. \nThis is done by restricting the spatial parts of the spin orbitals and thus treat pairs of spin orbitals on the same footing. The restrictions can be easily constructed \nfor the CCSD model by spin integration. \nThe derivation of the restrictions arising in higher excited case will be done employing \nthe spatial orbital excitation operators E. \n \nIn the first part of this work an algorithm is derived that is capable of the derivation of the energy and amplitude equations for arbitrary excitation levels. In the second part \nan implementation of this algorithm is presented.
A modified coupled cluster method is applied to the research of the ground-state energy of periodic systems described by model Hamiltonians. The reference function is always a strongly localized function. The method is applied first to Heisenberg Hamiltonians and spin-frustrated one-dimensional (1-D) chain and square lattices, starting from Neel functions or from products of bond singlets. The same method is then applied to Hubbard Hamiltonian for 1-D chain and two-dimensional (2-D) square lattices starting from Neel function or products of bond molecular orbitals (MOs). In both cases the wave operators involve a very limited number of local operators. Despite its simplicity, the method is able to treat quite satisfactorily highly degenerate situations, approaching correctly the highly delocalized regime from the Neel function or the highly correlated regime from a product of bond MOs. However, the method is not precise enough to treat the subtle phenomenon of bond alternation of polyacetylene. The coupled cluster method from strongly localized reference functions represents an elegant and quite efficient exploratory tool, but its accuracy is limited by the poor treatment of collective effects. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 115–132, 1998
In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC (DUCC) formalism, the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects—used to define the effective Hamiltonian—from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of CAS representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing. Given growing interest in quantum computing, we provide energies for H2 and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.