Many-body Formalisms Research Articles

Automatic derivation of many-body theories based on general Fermi vacua.

This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.

Natural Orbital Branching Scheme for Time-Dependent Density Functional Theory Nonadiabatic Simulations.

Real time-time-dependent density functional theory (rt-TDDFT) has now been used to study a wide range of problems, from optical excitation to charge transfer, to ion collision, and to ultrafast phase transition. However, conventional rt-TDDFT Ehrenfest dynamics for nuclear movement lacks a few critical features to describe many problems: the detail balance between state transitions, decoherence for the wave function evolution, and stochastic branching of the nuclear trajectory. There are many-body formalisms to describe such nonadiabatic molecular dynamics, especially the ones based on mixed quantum/classical simulations, like the surface hopping and wave function collapsing schemes. However, there are still challenges to implement such many-body formalisms to the rt-TDDFT simulations, especially for large systems where the excited state electronic structure configuration space is large. Here we introduce two new algorithms for nonadiabatic rt-TDDFT simulations: the first is a Boltzmann factor algorithm which introduces decoherence and detailed balance in the carrier dynamics but uses mean field theory for nuclear trajectory. The second is a natural orbital branching (NOB) formalism, which uses a time-dependent density matrix for electron evolution and a natural orbital set to collapse the wave function upon. It provides the features of decoherence, detailed balance, and trajectory branching. We have tested these methods for a molecule radiolysis decay problem. We found that these methods can be used to study such radiolysis problems in which the molecule is broken into many fragments following complex electronic structure transition paths. The computational time of NOB is similar to that of the original plain rt-TDDFT simulations.

ADG: Automated generation and evaluation of many-body diagrams II. Particle-number projected Bogoliubov many-body perturbation theory

We describe the second version (v2.0.0) of the code ADG that automatically (1) generates all valid off-diagonal Bogoliubov many-body perturbation theory diagrams at play in particle-number projected Bogoliubov many-body perturbation theory (PNP-BMBPT) and (2) evaluates their algebraic expression to be implemented for numerical applications. This is achieved at any perturbative order p for a Hamiltonian containing both two-body (four-legs) and three-body (six-legs) interactions (vertices). All valid off-diagonal BMBPT diagrams of order p are systematically generated from the set of diagonal, i.e.,unprojected, BMBPT diagrams. The production of the latter were described at length in Arthuis et al. (2019) dealing with the first version of ADG. The automated evaluation of off-diagonal BMBPT diagrams relies both on the application of algebraic Feynman’s rules and on the identification of a powerful diagrammatic rule providing the result of the remaining p-tuple time integral. The new diagrammatic rule generalizes the one already identified in Arthuis et al. (2019) to evaluate diagonal BMBPT diagrams independently of their perturbative order and topology. The code ADG is written in Python3 and uses the graph manipulation package NetworkX. The code is kept flexible enough to be further expanded throughout the years to tackle the diagrammatics at play in various many-body formalisms that already exist or are yet to be formulated. Program summaryProgram Title:ADGCPC Library link to program files:https://doi.org/10.17632/6h4xrydwfb.2Licensing provisions: GPLv3Programming language: Python3Journal reference of previous version: P. Arthuis, T. Duguet, A. Tichai, R.-D. Lasseri and J.-P. Ebran, ”ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory”, Computer Physics Communications 240 (2019), pp. 202-227.Does the new version supersede the previous version?: Yes.Reasons for the new version: Incorporation of a new formalism into the program.Summary of revisions: Addition of off-diagonal BMBPT to the formalisms for which diagrams can be generated, fix of a wrong symmetry factor, move of the codebase from Python2 to Python3 while maintaining support for Python2, various optimizations to reduce the time and memory necessary to the program.Nature of problem: As formal and numerical developments in many-body-perturbation-theory-based ab initio methods make higher orders reachable, manually producing and evaluating all the diagrams becomes rapidly untractable as both their number and complexity grow quickly, making it prone to mistakes and oversights.Solution method: Diagonal BMBPT diagrams are encoded as square matrices known as oriented adjacency matrices in graph theory, and then turned into graph objects using the NetworkX package. Off-diagonal BMBPT diagrams can then be generated from those graphs. Checks on the diagrams and evaluation of their time-integrated expression are eventually done on a purely diagrammatic basis. HF-MBPT diagrams are produced and evaluated as well using the same principle.

Normal-ordered k-body approximation in particle-number-breaking theories

The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schr\"odinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schr\"odinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle number conservation is broken (and potentially restored), a particle-number-conserving normal-ordered k-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed.

ADG: Automated generation and evaluation of many-body diagrams I. Bogoliubov many-body perturbation theory

We describe the first version (v1.0.0) of the code ADG that automatically (1) generates all valid Bogoliubov many-body perturbation theory (BMBPT) diagrams and (2) evaluates their algebraic expression to be implemented for numerical applications. This is achieved at any perturbative order p for a Hamiltonian containing both two-body (four-legs) and three-body (six-legs) interactions (vertices). The automated generation of BMBPT diagrams of order p relies on elements of graph theory, i.e., it is achieved by producing all oriented adjacency matrices of size (p+1)×(p+1) satisfying topological Feynman’s rules. The automated evaluation of BMBPT diagrams of order p relies both on the application of algebraic Feynman’s rules and on the identification of a powerful diagrammatic rule providing the result of the remaining p-tuple time integral. The diagrammatic rule in question constitutes a novel finding allowing for the straight summation of large classes of time-ordered diagrams at play in the time-independent formulation of BMBPT. Correspondingly, the traditional resolvent rule employed to compute time-ordered diagrams happens to be a particular case of the general rule presently identified. The code ADG is written in Python2.7 and uses the graph manipulation package NetworkX. The code is also able to generate and evaluate Hartree–Fock-MBPT (HF-MBPT) diagrams and is made flexible enough to be expanded throughout the years to tackle the diagrammatics used in various many-body formalisms that already exist or are yet to be formulated. Program summaryProgram Title:ADGProgram Files doi:http://dx.doi.org/10.17632/6h4xrydwfb.1Licensing provisions: GPLv3Programming language: Python2.7Nature of problem: As formal and numerical developments in many-body-perturbation-theory-based ab initio methods make higher orders reachable, producing and evaluating all the diagrams become rapidly undoable on a handmade basis as both their number and complexity grow quickly, making it prone to mistakes and oversights.Solution method: BMBPT diagrams are encoded as square matrices known as oriented adjacency matrices in graph theory, and then turned into graph objects using the NetworkX package. Checks on the diagrams and evaluation of their time-integrated expression are then done on a purely diagrammatic basis. HF-MBPT diagrams are produced and evaluated as well using the same principle.

Emergence of Clusters: Halos, Efimov States, and Experimental Signals.

We investigate the emergence of halos and Efimov states in nuclei by use of a newly designed model that combines self-consistent mean-field and three-body descriptions. Recent interest in neutron heavy calcium isotopes makes ^{72}Ca (^{70}Ca+n+n) an ideal realistic candidate on the neutron dripline, and we use it as a representative example that illustrates our broadly applicable conclusions. By smooth variation of the interactions we simulate the crossover from well-bound systems to structures beyond the threshold of binding, and find that halo configurations emerge from the mean-field structure for three-body binding energy less than ∼100 keV. Strong evidence is provided that Efimov states cannot exist in nuclei. The structure that bears the most resemblance to an Efimov state is a giant halo extending beyond the neutron-core scattering length. We show that the observable large-distance decay properties of the wave function can differ substantially from the bulk part at short distances, and that this evolution can be traced with our combination of few- and many-body formalisms. This connection is vital for interpretation of measurements such as those where an initial state is populated in a reaction or by a beta decay.

Electron spectra in forbiddenβdecays and the quenching of the weak axial-vector coupling constantgA

Evolution of the electron spectra with the effective value of the weak axial-vector coupling constant ${g}_{\mathrm{A}}$ was followed for 26 first-, second-, third-, fourth- and fifth-forbidden ${\ensuremath{\beta}}^{\ensuremath{-}}$ decays of odd-$A$ nuclei by calculating the involved nuclear matrix elements (NMEs) in the framework of the microscopic quasiparticle-phonon model (MQPM). The next-to-leading-order terms were included in the $\ensuremath{\beta}$-decay shape factor of the electron spectra. The spectrum shapes of third- and fourth-forbidden nonunique decays were found to depend strongly on the value of ${g}_{\mathrm{A}}$, while first- and second-forbidden decays were mostly unaffected by the tuning of ${g}_{\mathrm{A}}$. The ${g}_{\mathrm{A}}$-driven evolution of the normalized $\ensuremath{\beta}$ spectra was found to be quite universal, largely insensitive to the small changes in the nuclear mean field and the adopted residual many-body Hamiltonian producing the excitation spectra of the MQPM. This makes the comparison of experimental and theoretical electron spectra, coined ``the spectrum-shape method'' (SSM), a robust tool for extracting information on the effective values of the weak coupling constants. In this exploratory work two new experimentally interesting decays for the SSM treatment were discovered: the ground-state-to-ground-state decays of $^{99}\mathrm{Tc}$ and $^{87}\mathrm{Rb}$. Comparing the experimental and theoretical spectra of these decays could shed light on the effective values of ${g}_{\mathrm{A}}$ and ${g}_{\mathrm{V}}$ for second- and third-forbidden nonunique decays. The measurable decay transitions of $^{135}\mathrm{Cs}$ and $^{137}\mathrm{Cs}$, in turn, can be used to test the SSM in different many-body formalisms. The present work can also be considered as a (modest) step towards solving the ${g}_{\mathrm{A}}$ problem of the neutrinoless double beta decay.

Inactive excitations in Mukherjee's state-specific multireference coupled cluster theory treated with internal contraction: Development and applications

One generic difficulty of most state-specific many-body formalisms using the Jeziorski-Monkhorst ansatz: ψ = Σ(μ)exp(T(μ))|φ(μ)>c(μ) for the wave-operators is the large number of redundant cluster amplitudes. The number of cluster amplitudes up to a given rank is many more in number compared to the dimension of the Hilbert Space spanned by the virtual functions of up to the same rank of excitations. At the same time, all inactive excitations--though linearly independent--are far too numerous. It is well known from the success of the contracted multi-reference configuration interaction (MRCI(SD)) that, at least for the inactive double excitations, their model space dependence (μ-dependence) is weak. Considerable simplifications can thus be obtained by using a partially internally contracted description, which uses the physically appealing approximation of taking the inactive excitations T(i) to be independent of the model space labels (μ-independent). We propose and implement in this paper such a formalism with internal contractions for inactive excitations (ICI) within Mukherjee's state-specific multi-reference coupled cluster theory (SS-MRCC) framework (referred to from now on as the ICI-SS-MRCC). To the extent the μ-independence of T(i) is valid, we expect the ICI-SS-MRCC to retain the conceptual advantages of size-extensivity yet using a drastically reduced number of cluster amplitudes without sacrificing accuracy. Moreover, greater coupling is achieved between the virtual functions reached by inactive excitations as a result of the internal contraction while retaining the original coupling term for the μ-dependent excitations akin to the parent theory. Another major advantage of the ICI-SS-MRCC, unlike the other analogous internally contracted theories, such as IC-MRCISD, CASPT2, or MRMP2, is that it can use relaxed coefficients for the model functions. However, at the same time it employs projection manifolds for the virtuals obtained from inactive n hole-n particle (nh-np) excitations on the entire reference function containing relaxed model space coefficients. The performance of the method has been assessed by applying it to compute the potential energy surfaces of the prototypical H(4); to the torsional potential energy barrier for the cis-trans isomerism in C(2)H(4) as well as that of N(2)H(2), automerization of cyclobutadiene, single point energy calculation of CH(2), SiH(2), and comparing them against the SS-MRCC results, benchmark full CI results, wherever available and those from the allied MR formalisms. Our findings are very much reminiscent of the experience gained from the IC-MRCISD method.

On the extensivity of the roots of effective Hamiltonians in many-body formalisms employing incomplete model spaces

We analyze the current controversies regarding the extensivity of energies computed from an effective Hamiltonian defined over an incomplete model space (IMS). We show that the recently developed formalism in Fock space, using a size-extensive normalization for a valence universal operator Ω, generates both a connected H eff and size-extensive energies. In contrast, the corresponding Hilbert space formalisms, with intermediate normalization for Ω, produce size-inextensive energies. It is emphasized that the extensivity of energies for the Fock space theory stems not just from the connectivity of H eff but also due to the existence of certain special null matrix-elements in the matrix of H eff demanded by the decoupling conditions defined in Fock space.

Connected-diagram expansions of effective Hamiltonians in incomplete model spaces. II. The general incomplete model space

We generalize here the formalism of the preceeding paper to encompass the case of the general incomplete model space. The classification of operators as diagonal or nondiagonal depends in this case upon the specific m-valence model space. It is stressed that even then one has to work in Fock space in order to get connected-diagram expansions, since connectedness is a Fock space property. Two choices of separable normalization of the wave operator W leading to a connected Heff are discussed. It is shown that the intermediate normalization is not separable in general and hence not compatible with a connected-diagram expansion. We also discuss how to generate ‘‘subduced’’ incomplete model spaces of lower particle rank such that Heff remains a valid effective Hamiltonian for these subduced model spaces as well. We discuss the nature of the various disconnected diagrams encountered in many-body formalisms and point out which of these are really worth worrying about. We finally comment on the question of the separability of the wave function into proper fragments.

Coupled channel T-operator equations with connected kernels : (II). N coupled two-body channels

A time-independent theory of rearrangement collisions involving transitions between two-body states is presented. It is assumed that the system of interest consists of particles that may be partitioned into two-body systems in N ways, including interchanges of particle labels without changing the kind of channel. An infinite family of sets of N coupled T-operator equations is derived by use of the channel coupling array, as in previous work on the three-body problem. Specialization to the channel-permuting arrays guaranteeing connected ( N−1)th iterates of the kernel of the coupled equations is made in the N-channel case ( N > 3) and the nature of the solutions to the coupled equations is discussed. Various approximation schemes to be used with numerical calculations are suggested. Since the transition operators for all rearrangement channels are coupled together, no problems concerning non-orthogonality of the eigenstates of different channel Hamiltonians are encountered; also the presence of the outgoing wave boundary condition in all channels is made explicit. The close resemblance of the equations in matrix form to those of one-channel scattering is exploited by introducing Møller wave operators and associated channel scattering states, an optical potential formalism that leads to rearrangement channel optical potential operators, and a variational formulation of the coupled equations using a Schwinger-like variational principle. A brief comparison with other many-body formalisms is also given.

Electroabsorption in Semiconductors: The Excitonic Absorption Edge

Numerical calculations of the optical-absorption coefficient for direct, excitonic transitions in a uniform applied electric field are presented. The electron-hole scattering is treated within the effective-mass approximation and leads to an absorption coefficient which differs markedly in size and shape from the Franz-Keldysh absorption spectrum. A detailed numerical study of the shape of the absorption-edge spectrum at photon energies somewhat below the zero-field absorption threshold suggests that for small field strengths the dominant asymptotic form of the absorption coefficient is $\mathrm{exp}(\ensuremath{-}\frac{{C}_{0}|E\ensuremath{-}{{E}_{0}}^{\ensuremath{'}}|}{f})$, where $f=\frac{|e|\mathrm{Fa}}{R}$ is the electric field strength in units of exciton Rydbergs per electron-exciton Bohr radius. This result contradicts the existing belief that the electron-hole interaction does not alter the asymptotic form of the Franz-Keldysh shape: $\mathrm{exp}(\ensuremath{-}\frac{{{C}_{0}}^{\ensuremath{'}}{|E\ensuremath{-}{{E}_{0}}^{\ensuremath{'}}|}^{\frac{3}{2}}}{f})$. Physical arguments are presented to show why the exciton effects should be important. A discussion is presented of the interrelationships among the present treatment of electro-absorption and various one-electron, exciton, and many-body formalisms.

Realistic Nuclear Single-Particle Hamiltonians and the Proton Shell 114

A simple self-consistent single-particle equation is investigated and compared with similar attempts. The proposed model is designed to be particularly suitable for the calculation of (adiabatic) fission processes. The kernel of this integrodifferential equation has a structure that allows satisfactory reproduction, with one constant set of five physical parameters, of (1) charge density distributions, including isotope shifts, (2) $1s$ proton levels as measured in ($e,{e}^{\ensuremath{'}}p$) scattering, (3) total binding energies or nuclear mass defects, and (4) the shell-model spin assignments and mass structure throughout the periodic table. Hence, it seems that all future work in this direction has to confirm quantitatively the essential features determined here, particularly the nonlocality and rearrangement effects. Rearrangement energies appear explicitly, since the present model, like self-consistent fields of appropriate many-body formalisms, yields different eigenvalue spectra and mass defects for different occupation functions. The partial derivative $\frac{\ensuremath{\partial}E}{\ensuremath{\partial}Z}$ of the total binding energy (mass) changes considerably at the proton number $Z=114$ when the present Hamiltonian is used for superheavy nuclei. This confirms an earlier suggestion made by this author on the basis of a gap in the proton eigenvalue spectrum at $Z=114$. The present calculations show that this shell effect becomes insignificant for isotopes too far from the extrapolated $\ensuremath{\beta}$ stability line, in particular for $N\ensuremath{\lesssim}172$.