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arXiv:cond-mat/9909317 · Journal of Physics Condensed Matter

Construction of size-consistent effective Hamiltonians for systems with arbitrary Hilbert space

A. Hübsch, Matthias Vojta C, K. W. Becker

Published Oct 15, 1999• 4 citations
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Abstract

Effective Hamiltonians are usually constructed by using canonical transformations or projection techniques. In contrast to this, we present a method for systems with arbitrary Hilbert space based on the introduction of cumulants. Cumulants guarantee size consistency, a property that is not always evident in other treatments. As a non-trivial example of use the derived method is applied to the strong-coupling limit of the half-filled Hubbard model on a general lattice in arbitrary spatial dimension for which the fourth-order expansion in t / U of the effective Hamiltonian is derived.

DOI
10.1088/0953-8984/11/43/316

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Effective Hamiltonians are usually constructed by using canonical transformations or projection techniques. In contrast to this, we present a method for systems with arbitrary Hilbert space based on the introduction of cumulants. Cumulants guarantee size consistency, a property that is not always evident in other treatments. As a non-trivial example of use the derived method is applied to the strong-coupling limit of the half-filled Hubbard model on a general lattice in arbitrary spatial dimension for which the fourth-order expansion in t / U of the effective Hamiltonian is derived.
For a system consisting of two interacting subsystems, effective Hamiltonians for one subsystem are usually constructed by integrating out the degrees of freedom of the second subsystem. In contrast to usual treatments, our method for deriving effective Hamiltonians is based on the introduction of cumulants. Cumulants guarantee size consistency, a property that is not always evident in other approaches. The formalism applies for any temperature. Our method is based on a matrix formulation in the framework of a recently introduced cumulant-projection method. This method allows the treatment of strongly correlated systems, where conventional diagrammatic techniques are difficult to apply. The relation with perturbation theory is also shown. As a simple application we discuss the small polaron problem. Evaluating the one-electron dispersion relation in the strong-coupling case shows good agreement with recent results from exact Lanczos diagonalization. Our method has substantial advantages over perturbation theory.
By applying projection techniques a unified description of weakly and strongly correlated electronic systems can be given. For that purpose the theory is formulated in Liouville instead of Hilbert space. By introducing an appropriate metric in terms of cumulants it is ensured that all energies are size consistent. An energy expression is derived which is a generalization of Goldstone’s linked cluster theorem for arbitrary Hamiltonians. It is suitable for applying the projection method of Zwanzig and Mori. The equations for the correlation energy within the Local Ansatz and within coupled electron pair approximation CEPA-2 follow straightforwardly from the present theory.
Cumulants represent a natural language for expressing macroscopic properties of many-body systems. The most important property of cumulants is that of size consistency, i.e. a cumulant expression for an extensive variable scales with the size of the system, independent of possible further approximations used in the evaluation procedure. Cumulants can be considered as a generalization of linked diagrams known from diagrammatic technique of many-body theory. In this paper we outline a recently introduced method based on cumulants in order to derive expressions for zero-temperature properties of many-particle systems, i.e. the ground-state energy, static expectation values and dynamical correlation functions. This cumulant formalism allows one to describe weakly and strongly correlated systems along the same lines. We show that the coupled-cluster method known from quantum chemistry can be derived from our cumulant approach. Finally, we demonstrate the usefulness of the cumulant method by applying it to examples from solid-state physics and quantum chemistry.
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
Three well-known perturbative approaches to deriving low-energy effective theories, the degenerate Brillouin-Wigner perturbation theory (projection method), the canonical transformation, and the resolvent methods, are compared. We use the Hubbard model as an example to show how, to fourth order in hopping ttt, all methods lead to the same effective theory, namely the $t\text{\ensuremath{-}}J$ model with ring exchange and various correlated hoppings. We emphasize subtle technical difficulties that make such a derivation less trivial to carry out for orders higher than second. We also show that in higher orders, different approaches can lead to seemingly different forms for the low-energy Hamiltonian. All of these forms are equivalent since they are connected by an additional unitary transformation whose generator is given explicitly. The importance of transforming the operators is emphasized and the equivalence of their transformed structure within the different approaches is also demonstrated.
Starting from the full many-body Hamiltonian of interacting electrons the effective self-energy acting on electrons residing in a subspace of the full Hilbert space is derived. This subspace may correspond to, for example, partially filled narrow bands, which often characterize strongly correlated materials. The formalism delivers naturally the frequency-dependent effective interaction (the Hubbard U) and provides a general framework for constructing theoretical models based on the Green's function language. It also furnishes a general scheme for first-principles calculations of complex systems in which the main correlation effects are concentrated on a small subspace of the full Hilbert space.