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arXiv:1908.03780 · The European Physical Journal Plus

Non-Hermitian coupled cluster method for non-stationary systems and its interaction-picture reinterpretation

R. F. Bishop, Miloslav Znojil

Published Apr 1, 2020• 20 citations
arXivHTMLPDF

Abstract

Abstract The interaction picture in a non-Hermitian realization is discussed in detail and considered for its practical use in many-body quantum physics. The resulting non-Hermitian interaction-picture (NHIP) description of dynamics, in which both the wave functions and operators belonging to physical observables cease to remain constant in time, is a non-Hermitian generalization of the traditional Dirac picture of standard quantum mechanics, which itself is widely used in quantum field theory calculations. Particular attention is paid here to the variational (or, better, bivariational) and dynamical (i.e., non-stationary) aspects that are characteristic of the coupled cluster method (CCM) techniques that nowadays form one of the most versatile and most accurate of all available formulations of quantum many-body theory. In so doing we expose and exploit multiple parallels between the NHIP and the CCM in its time-dependent versions.

DOI
10.1140/epjp/s13360-020-00374-z

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Abstract The interaction picture in a non-Hermitian realization is discussed in detail and considered for its practical use in many-body quantum physics. The resulting non-Hermitian interaction-picture (NHIP) description of dynamics, in which both the wave functions and operators belonging to physical observables cease to remain constant in time, is a non-Hermitian generalization of the traditional Dirac picture of standard quantum mechanics, which itself is widely used in quantum field theory calculations. Particular attention is paid here to the variational (or, better, bivariational) and dynamical (i.e., non-stationary) aspects that are characteristic of the coupled cluster method (CCM) techniques that nowadays form one of the most versatile and most accurate of all available formulations of quantum many-body theory. In so doing we expose and exploit multiple parallels between the NHIP and the CCM in its time-dependent versions.
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.
In the framework of quantum mechanics using quasi-Hermitian operators the standard unitary evolution of a non-stationary but still closed quantum system is only properly described in the non-Hermitian interaction picture (NIP). In this formulation of the theory both the states and the observables vary with time. A few aspects of implementation of this picture are illustrated via the "wrong-sign" quartic oscillators. It is shown that in contrast to the widespread belief, both of the related Schrödinger-equation generators G(t) and the Heisenberg-equation generators Σ(t) are just auxiliary concepts. Their spectra are phenomenologically irrelevant and, in general, complex. It is argued that only the sum H(t)=G(t)+Σ(t) of the latter operators retains the standard physical meaning of the instantaneous energy of the unitary quantum system in question.
We present the diagrammatic theory of the irreducible self-energy and Bethe-Salpeter kernel that naturally arises within the Green’s function formalism for a general N -body non-Hermitian interaction. In this work, we focus specifically on the coupled-cluster (CC) self-energy generated by the similarity transformation of the electronic structure Hamiltonian. The main result of this work is a Green’s function formalism for non-Hermitian many-body quantum systems. We develop the biorthogonal quantum theory to construct dynamical correlation functions where the time dependence of operators is governed by a non-Hermitian Hamiltonian. We extend the Gell-Mann and Low theorem to include non-Hermitian interactions and to generate perturbative expansions of many-body Green’s functions. We introduce the single-particle coupled-cluster Green’s function and derive the perturbative diagrammatic expansion for the non-Hermitian coupled-cluster self-energy in terms of the “noninteracting” reference Green’s function, Σ ̃ [ G 0 ] . From the exact equation of motion of the single-particle coupled-cluster Green’s function, we derive the self-consistent renormalized coupled-cluster self-energy, Σ ̃ [ G ̃ ] , and demonstrate its relationship to the perturbative expansion of the self-energy, Σ ̃ [ G 0 ] . Subsequently, we show that the usual electronic self-energy can be recovered from the coupled-cluster self-energy by neglecting the effects of the similarity transformation. We show how the coupled-cluster ground-state energy can be obtained from the coupled-cluster self-energy and provide an overview of the relationship between approximations for the coupled-cluster self-energy, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the G 0 W 0 approximation. As a result, we introduce the CC- G 0 W 0 self-energy by leveraging the connections between Green’s function and coupled-cluster theory. Finally, we derive the diagrammatic expansion of the coupled-cluster Bethe-Salpeter kernel.
The coupled cluster method (CCM) is one of the most powerful and most successful fully microscopic, ab initio formulations available for quantum N-body theory, with N finite or infinite. It has probably been applied to more systems in quantum chemistry, nuclear, condensed matter and other areas of physics, and quantum field theory than any other competing method. In nearly all such cases the numerical results are either the best or among the best available. The CCM can deal with ground- and excited-state energies of closed- and open-shell systems, density matrices and hence other properties, sum rules, and the sub-sum-rules that follow from imbedding linear response theory within it. Extensions exist to deal with systems at nonzero temperature and out of equilibrium. At the formal level it provides an exact mapping of the quantum-mechanical problem onto a classical Hamiltonian phase space where the multiconfigurational canonical classical coordinates have specific cluster and locality properties. In this way it can provide exact hierarchical generalizations of mean-field theory and the random phase approximation. We discuss here both the formalism itself and a selection of its applications.
This work presents a time-dependent (TD) unitary coupled-cluster (UCC) formulation for electronic quantum dynamics, including the propagation of excited states and their superpositions, in both single- and multi-reference regimes. Standard TD coupled-cluster techniques offer size-extensivity, but they rely on non-Hermitian bivariational action functionals that break time-reversibility, giving transition matrix elements and amplitude estimators that are asymmetric (though accurate and systematically improvable). Here we use the time-evolution operator as an exponential map driven by TD anti-Hermitian cluster operators and first-order generators. Applying the Dirac-Frenkel action principle, we extract equations of motion governed by Heisenberg-picture-like commutators. This approach connects to our previous non-Hermitian formulations, where observables are expressed in terms of regular and extended cluster operators. From that connection we obtain a generator cluster operator whose unperturbed TD limit leads to the UCC eigenvalue problem. Even though this problem can be exact, the time dependence of the generator holds only at short propagation times, so we use the generator to prepare the initial state and then propagate that state with the formal TD UCC equation of motion. The theory is tested on an extended hard-core Bose-Hubbard ring with connections to neutral atom chains, and we discuss the theory's present limitations and possible extensions.