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Physical Review A

Constrained Hamiltonian approach to the phase space of the coupled cluster method

J. Arponen

Published Apr 1, 1997• 26 citations
Publisher

Abstract

The coupled cluster method (CCM) introduces a linked-cluster parametrization for the many-body wave function. The method can be formulated through an action principle, which introduces a symplectic structure into the CCM phase space. We study the phase spaces of the normal and extended coupled cluster methods (NCCM and ECCM, respectively) with a special emphasis on the ensuing characteristic non-Hermiticity of the action and the average-value functionals. By introducing the complex conjugates of the usual CCM cluster amplitudes the phase space is enlarged into a genuine complex manifold. The extra degrees of freedom are then eliminated by the Dirac bracket method, leaving as independent coordinates a minimal set of complex conjugate (eventually, real) coordinate functions. The constraint functions performing this elimination are shown to be of second class. The induced symplectic structure in the physical submanifold (or the constraint surface) is derived, and the physical shell itself found to be a K\"ahler manifold. Furthermore, the significance of the CCM star product in generating the average-value functionals of compound operators is emphasized. It is proven that the star product can also be defined on the constraint surface in both the NCCM and ECCM cases. The results suggest that Hermiticity-preserving truncation schemes based on the on-shell star product may become feasible.

DOI
10.1103/physreva.55.2686

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Physical Review A•Feb 1, 1988•23 citations
The coupled cluster method (CCM) introduces a linked-cluster parametrization for the many-body wave function. The method can be formulated through an action principle, which introduces a symplectic structure into the CCM phase space. We study the phase spaces of the normal and extended coupled cluster methods (NCCM and ECCM, respectively) with a special emphasis on the ensuing characteristic non-Hermiticity of the action and the average-value functionals. By introducing the complex conjugates of the usual CCM cluster amplitudes the phase space is enlarged into a genuine complex manifold. The extra degrees of freedom are then eliminated by the Dirac bracket method, leaving as independent coordinates a minimal set of complex conjugate (eventually, real) coordinate functions. The constraint functions performing this elimination are shown to be of second class. The induced symplectic structure in the physical submanifold (or the constraint surface) is derived, and the physical shell itself found to be a K"ahler manifold. Furthermore, the significance of the CCM star product in generating the average-value functionals of compound operators is emphasized. It is proven that the star product can also be defined on the constraint surface in both the NCCM and ECCM cases. The results suggest that Hermiticity-preserving truncation schemes based on the on-shell star product may become feasible.
The quantum-mechanical many-body problem is studied using the extended coupled-cluster method (ECCM) as a convenient parametrization of the Hilbert space. A systematic development of the formalism is given, and the quantum-mechanical problem is cast into the form of a classical Hamiltonian field theory in a complex symplectic phase space. The equations of motion of the basic ECCM amplitudes are derived both from a dynamical action principle using the ECCM average-value functional, and directly from the Schr\"odinger equation using the ECCM double similarity transformation. Rules are given for the construction of the Hamiltonian and other observables as well as their products. In particular, commutators are shown to be mapped into generalized classical Poisson brackets. The description is interpreted as an exact bosonization of the quantum theory in which the concept of bosonization is carried to the logical extreme, namely, the resulting generalized coherent bosons are identifiable with classical fields. The quantum-mechanical states of the system are points in the ECCM phase space, and their time evolution, or trajectories, are controlled by a classical Hamiltonian. The bosonization has a perturbation-theoretical basis in terms of maximally linked generalized tree diagrams. The increased degree of locality of the basic amplitudes allows applications to topological excitations and cases with spontaneous symmetry breaking.
This article gives a discussion of the application of the extended coupled-cluster method (ECCM) to the excited states of a general quantum many-body system. The direct eigenvalue equations for the excitation amplitudes of both the ket and the bra eigenstates are derived in the biorthogonal basis obtained by a double similarity transformation. The equations correspond to the diagonalization of a matrix involving second-order functional derivatives of the average-value functional for the Hamiltonian with respect to the basic ECCM amplitudes. The same excitation spectrum is obtained by considering small oscillations around the equilibrium. The problem with its associated effective Hamiltonian has the structure of a generalized random-phase approximation. By diagonalizing the effective Hamiltonian we perform a canonical or symplectomorphic coordinate transformation into normal coordinates in the symplectic ECCM phase space. In this coordinate system the exact average-value functional for the Hamiltonian has a structure analogous to that of classical lattice dynamics or the phenomenological Ginzburg-Landau theory. At all stages the method satisfies the property of quantum locality, which in real space shows up as a definite quasilocality. Due to this property the method allows, for example, the treatment of mesonlike excitations in the presence of topological objects or in other symmetry-broken equilibrium states.
The quantum many-body bound-state problem in its computationally successful coupled cluster method (CCM) representation is reconsidered. In conventional practice one factorizes the groundstate wave functions |Ψ) = eS |Φ) which live in the “physical” Hilbert space H(P) using an elementary ansatz for |Φi plus a formal expansion of S in an operator basis of multi-configurational creation operators C+. In our paper a reinterpretation of the method is proposed. Using parallels between the CCM and the so called quasi-Hermitian, alias three-Hilbert-space (THS), quantum mechanics, the CCM transition from the known microscopic Hamiltonian (denoted by usual symbol H), which is self-adjoint in H(P), to its effective lower-case isospectral avatar h = e−SHeS, is assigned a THS interpretation. In the opposite direction, a THS-prescribed, non-CCM, innovative reinstallation of Hermiticity is shown to be possible for the CCM effective Hamiltonian h, which only appears manifestly non-Hermitian in its own (“friendly”) Hilbert space H(F). This goal is achieved via an ad hoc amendment of the inner product in H(F), thereby yielding the third (“standard”) Hilbert space H(S). Due to the resulting exact unitary equivalence between the first and third spaces, H(F) ∼ H(S), the indistinguishability of predictions calculated in these alternative physical frameworks is guaranteed.
The widely used coupled cluster method (CCM) in quantum many-body theory has recently provided very accurate descriptions of a large number of extended systems. Although its earlier applications to closed-shell and neighboring finite nuclei were also very successful, they have been shrouded in algebraic and technical complexity. Furthermore, they are difficult to compare with more traditional calculations of generalized shell-model theory since, at least at the important level of two-body correlations, they have been largely implemented in relative-coordinate space rather than the more usual oscillator configuration space. The CCM is reviewed here in the precise context of applications to simple finite systems. Special attention is paid to formulate it in such a way that comparison may be made with generalized shell-model or configuration-interaction (CI) theories. Particular regard is paid to an exact incorporation of translational invariance, so that any spuriosity associated with the center-of-mass motion is always avoided. An important side benefit is that the number of many-body configurations in the usual oscillator basis is dramatically reduced. We are thereby able to present both CI and CCM calculations on 4He^{4}\mathrm{He}4He up to the essentially unprecedented level of 60\ensuremath{\Elzxh}\ensuremath{\omega} in oscillator excitation energy, for two popular and quasirealistic choices of the nucleon-nucleon interaction for which exact Monte Carlo results are available for this nucleus. Although even our simplest approximations attain about 95% of the total binding energy, the convergence in the oscillator configuration space is shown to be both very slow and of a complicated nonuniform nature. Strong implications are drawn for standard implementations of generalized shell-model techniques for heavier nuclei.
We review the basic principles of the various coupled cluster (CC) methods based on an exponential form for the many-body wavefunction, and contrast them with the configuration–interaction (CI) method. Particular emphasis is placed on their applicability to problems in quantum chemistry. We prove that in all cases we can construct an energy functional which variationally determines both the ground–state wavefunction and the dynamic equations of motion for nonstationary states. As a result the equations of motion assume the familiar classical canonical Hamiltonian form in some well-defined (multibody) configuration space. We also thereby construct the expectation-value functional for an arbitrary operator in such a way that the Feynman–Hellmann theorem is preserved at all natural levels of truncation of the appropriate configuration space. We show in detail that only in the case of the recently introduced extended CC method (ECCM) is the expectation-value functional expressed fully in terms of linked (multilocal) amplitudes. The ECCM is thereby capable of describing such global phenomena as shape transitions and other stereochemical properties, and the large-scale behavior of the molecular energy surfaces. We illustrate our methodology on the one-body density matrix, which is now much more easily discussed than by conventional methods in quantum chemistry.
The extended coupled-cluster method (ECCM) of quantum many-body theory, which has been studied and developed in earlier papers in this series, is now applied to condensed Bose systems. The formalism is seen to provide a concise and convenient description of quite general nonstatic states of such systems with arbitrary spatial inhomogeneity. The entire ECCM description is based on the equations of motion for the set of linked-cluster amplitudes which, we have shown, completely characterize an arbitrary quantum system with a Schr\"odinger dynamics. Since all such amplitudes obey the cluster property, and hence may be regarded as a set of quasilocal, many-body, classical order parameters, the formalism is in principle perfectly capable of describing phase transitions and states of topological excitation or deformation and broken symmetry. At the lowest (mean-field) level of truncation, the formalism degenerates to the well-known Gross-Pitaevskii description of the condensate wave function or one-body order parameter. The treatment is developed in a fully gauge-invariant fashion, and is thereby shown to provide a complete hydrodynamical description, valid in the zero-temperature limit. In particular, by studying the off-diagonal one- and two-body density matrices in terms of the basic ECCM amplitudes, we derive balance (or local conservation) equations for such local observables as the number density, momentum density, and energy density. These are shown to be obeyed not only by the exact untruncated formalism but also by most practical truncation schemes in the ECCM configuration space.