There are efficient many-body methods, such as the (symmetry-restored) generator coordinate method in nuclear physics, that formulate the A-body Schrödinger equation within a set of nonorthogonal many-body states. Solving the corresponding secular equation requires the evaluation of the norm matrix and thus the capacity to compute its entries consistently and without any phase ambiguity. This is not always a trivial task, e.g. it remained a long-standing problem for methods based on general Bogoliubov product states. While a solution to this problem was found recently in Ref. [L. M. Robledo, Phys. Rev. C79, 021302 (2009)], the present work introduces an alternative method that can be generically applied to other classes of states of interest in many-body physics. The method is presently exemplified in the case of Bogoliubov states and numerically illustrated on the basis of a toy model.
There are efficient many-body methods, such as the (symmetry-restored) generator coordinate method in nuclear physics, that formulate the A-body Schrödinger equation within a set of nonorthogonal many-body states. Solving the corresponding secular equation requires the evaluation of the norm matrix and thus the capacity to compute its entries consistently and without any phase ambiguity. This is not always a trivial task, e.g. it remained a long-standing problem for methods based on general Bogoliubov product states. While a solution to this problem was found recently in Ref. [L. M. Robledo, Phys. Rev. C79, 021302 (2009)], the present work introduces an alternative method that can be generically applied to other classes of states of interest in many-body physics. The method is presently exemplified in the case of Bogoliubov states and numerically illustrated on the basis of a toy model.
Background: State-of-the-art multi-reference energy density functional calculations require the computation of norm overlaps between different Bogoliubov quasiparticle many-body states. It is only recently that the efficient and unambiguous calculation of such norm kernels has become available under the form of Pfaffians [L. M. Robledo, Phys. Rev. C 79, 021302 (2009)]. Recently developed particle-number-restored Bogoliubov coupled-cluster (PNR-BCC) and particle-number-restored Bogoliubov many-body perturbation (PNR-BMBPT) ab initio theories [T. Duguet and A. Signoracci, J. Phys. G 44, 015103 (2017)] make use of generalized norm kernels incorporating explicit many-body correlations. In PNR-BCC and PNR-BMBPT, the Bogoliubov states involved in the norm kernels differ specifically via a global gauge rotation.Purpose: The goal of this work is threefold. We wish (i) to propose and implement an alternative to the Pfaffian method to compute unambiguously the norm overlap between arbitrary Bogoliubov quasiparticle states, (ii) to extend the first point to explicitly correlated norm kernels, and (iii) to scrutinize the analytical content of the correlated norm kernels employed in PNR-BMBPT. Point (i) constitutes the purpose of the present paper while points (ii) and (iii) are addressed in a forthcoming paper.Methods: We generalize the method used in another work [T. Duguet and A. Signoracci, J. Phys. G 44, 015103 (2017)] in such a way that it is applicable to kernels involving arbitrary pairs of Bogoliubov states. The formalism is presently explicated in detail in the case of the uncorrelated overlap between arbitrary Bogoliubov states. The power of the method is numerically illustrated and benchmarked against known results on the basis of toy models of increasing complexity.Results: The norm overlap between arbitrary Bogoliubov product states is obtained under a closed-form expression allowing its computation without any phase ambiguity. The formula is physically intuitive, accurate, and versatile. It equally applies to norm overlaps between Bogoliubov states of even or odd number parity. Numerical applications illustrate these features and provide a transparent representation of the content of the norm overlaps.Conclusions: The complex norm overlap between arbitrary Bogoliubov states is computed, without any phase ambiguity, via elementary linear algebra operations. The method can be used in any configuration mixing of orthogonal and non-orthogonal product states. Furthermore, the closed-form expression extends naturally to correlated overlaps at play in PNR-BCC and PNR-BMBPT. As such, the straight overlap between Bogoliubov states is the zero-order reduction of more involved norm kernels to be studied in a forthcoming paper.
Given elementary inter-nucleon interactions,the resolution of the A-body Schrödinger equation gives access to the properties of the nuclear eigenstates. Many-body perturbation theory (MBPT) based on a Hartree-Fock mean field allows one to treat the dynamical correlations at play in doubly-closed-shell nuclei. Beyond shell closures,breaking U(1) symmetry associated with the conservation of neutron and proton numbers further permits to capture static correlations at the meanfield level and to formulate the well-behaved Bogoliubov many-body perturbation theory (BMBPT).Nevertheless, BMBPT results show contaminations associated with the broken symmetry that is only emergent on finite quantum system such as theatomic nucleus. Thus, the restoration of U(1) symmetry beyond the mean field is necessary for acorrect description and gives rise to the projected BMBPT method (PBMBPT). The goal is to implement PBMBPT in order to perform ab initio calculations of singly-open-shell mid-mass nuclei.The present work provides systematic solutions to the formal and technical problems arising during the implementation of PBMBPT: an automatic and safe method to generate PBMBPT diagrams and associated expressions, a framework to calculate other observables than the energy, an extension of the normal ordering approximation to symmetry breaking methods as well as an automatic and safe tool for spherical reduction of tensor networks.
The solution of Schrödinger’s equation for systems composed of a large number of strongly interacting particles is an unsolved problem. Indeed, even the classical three-body problem cannot be solved in general form. Fortunately, the problem of obtaining the Ψ-function of such a system is not only insoluble but unnecessary; such a detailed description of the many-body problem is not required by any experiment that is possible in real life. Any real experimental set-up contains a comparatively small number of particle indicators, and hence, in cases where a large number of particles take part in a given process, can give only their averaged characteristics. Any attempt to determine the coordinates of every particle of such a system would lead (because of the uncertainty relation) to a complicated excited state and would indeed change the properties of the system. Thus, the description of a macroscopic system by means of a wave function is an inadequate approach * : we need 262 instead methods of incomplete description, which determine only the relations between averaged quantities. An example of such a description is hydrodynamics, the equations of which determine only the average velocity of the particles at each point (i. e. the velocity field). A second example, closer to our problem, is the kinetic equation, which allows us to find the distribution of particles with respect to velocity and position. A knowledge of the one-particle distribution function f ( r ˜ , p ˜ , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1304.tif"/> , which depends on the coordinate r ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1305.tif"/> and velocity p ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1306.tif"/> of the particles, allows us to calculate the mean values of additive quantities such as the density n ( r ˜ ) = ∑ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1307.tif"/> or the momentum per unit volume j ˜ ( r ˜ ) = ∑ i p ˜ i δ ( r ˜ - r ˜ i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1308.tif"/> . The two-particle distribution function f ( r ˜ 1 , r ˜ 2 ; p ˜ 1 , p ˜ 2 ; t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1309.tif"/> allows us to find the correlations between the coordinates and the velocities of two particles and to determine the averages of quantities which depend on the coordinates of two particles, for instance the mean value of the pairwise interaction energy V = ∑ ik V ( r ˜ i - r ˜ k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429497940/5adca591-a306-49c7-affc-2d249cfef85c/content/eq1310.tif"/> . The periodic or weakly damped solutions of the equation for the distribution function give the eigenfrequencies of the system.
The finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation often breaks symmetries of the underlying many-body Hamiltonian. Restricting the calculation of the HFB partition function to a subspace with good quantum numbers through projection after variation restores some of the correlations lost in breaking these symmetries, although effects of the broken symmetries such as sharp kinks at phase transitions remain. However, the most general projection after variation formula in the finite-temperature HFB approximation is limited by a sign ambiguity. Here, I extend the Pfaffian formula for the many-body traces of HFB density operators introduced by Robledo [L. M. Robledo, Phys. Rev. C. 79, 021302(R) (2009)] to eliminate this sign ambiguity and evaluate the more complicated many-body traces required in projection after variation in the most general HFB case. The method is validated through a proof-of-principle calculation of the particle-number-projected HFB thermal energy in a simple model.
Perturbative and non-perturbative expansion methods already constitute a tool of choice to perform ab initio calculations over a significant part of the nuclear chart. In this context, the categories of accessible nuclei directly reflect the class of unperturbed state employed in the formulation of the expansion. The present work generalizes to the nuclear many-body context the versatile method of Ref. \cite{burton20a} by formulating a perturbative expansion on top of a multi-reference unperturbed state mixing deformed non-orthogonal Bogoliubov vacua, i.e. a state obtained from the projected generator coordinate method (PGCM). Particular attention is paid to the part of the mixing taking care of the symmetry restoration, showing that it can be exactly contracted throughout the expansion, thus reducing significantly the dimensionality of the linear problem to be solved to extract perturbative corrections. While the novel expansion method, coined as PGCM-PT, reduces to the PGCM at lowest order, it reduces to single-reference perturbation theories in appropriate limits. Based on a PGCM unperturbed state capturing (strong) static correlations in a versatile and efficient fashion, PGCM-PT is indistinctly applicable to doubly closed-shell, singly open-shell and doubly open-shell nuclei. The remaining (weak) dynamical correlations are brought consistently through perturbative corrections. This symmetry-conserving multi-reference perturbation theory is state-specific and applies to both ground and excited PGCM unperturbed states, thus correcting each state belonging to the low-lying spectrum of the system under study.
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 Schrodinger 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 so-called 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 Schrodinger 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-orderedk-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. The PNOkB approximation of an arbitrary N-body operator is formulated. Based on this systematic approach, it is demonstrated that naive extensions of the normal-ordered two-body (NO2B) approximation employed so far on the basis of symmetry-conserving reference states lead to particle non-conserving operators. Alternatively, the PNOkB procedure is now available to generate particle-number-conserving approximate operators. The formal analysis is validated numerically. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed. The future formulation of an angular-momentum-conserving normal-ordered k-body approximation based on deformed Slater determinant or Bogoliubov reference states is envisioned.