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arXiv:2006.02871 · Physical Review C

Zero-pairing limit of Hartree-Fock-Bogoliubov reference states

T. Duguet, B. Bally, A. Tichai

Published Nov 13, 2020• 21 citations
arXivHTMLPDF

Abstract

Background: The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. In this context, pairing correlations may be driven towards zero either on purpose via HFB calculations constrained on, e.g., the particle-number variance or simply because internucleon interactions cannot sustain pairing correlations in the first place in, e.g., closed-shell systems. While taking this limit constitutes a text-book problem when the system is of closed-shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system.Purpose: The present work demonstrates that the zero-pairing limit of an HFB state is well-defined independently of the average particle number $\mathrm{A}$ it is constrained to. Still, the nature of the limit state is shown to depend on the regime, i.e., on whether the nucleus characterizes as a closed-shell or an open-shell system when taking the limit. Finally, the consequences of the zero-pairing limit on Bogoliubov many-body perturbation theory (BMBPT) calculations performed on top of the HFB reference state are illustrated.Methods: The zero-pairing limit of a HFB state constrained to carry an arbitrary (integer) number of particles $\mathrm{A}$ on average is worked out analytically and realized numerically using a two-nucleon interaction derived within the frame of chiral effective field theory.Results: The zero-pairing limit of the HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number $\mathrm{A}$ of interest. First, the textbook situation is recovered for closed-shell systems, i.e., the limit state is reached for a finite value of the pairing strength (the well-known BardeenCooperSchrieffer (BCS) collapse) and takes the form of a single Slater determinant displaying (i) zero pairing energy, (ii) nondegenerate elementary excitations, and (iii) zero particle-number variance. Contrarily, a nonstandard situation is obtained for open-shell systems, i.e., the limit state is only reached for a zero value of the pairing interaction (no BCS collapse) and takes the form of a specific finite linear combination of Slater determinants displaying (a) a nonzero pairing energy, (b) degenerate elementary excitations, and (c) a nonzero particle-number variance for which an analytical formula is derived. This nonzero particle-number variance acts as a lower bound that depends in a specific way on the number of valence nucleons and on the degeneracy of the valence shell. All these findings are confirmed and illustrated numerically. Last but not least, BMBPT calculations of closed-shell (open-shell) nuclei are shown to be well-defined (ill-defined) in the zero-pairing limit.Conclusions: While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. In the present paper, the zero-pairing limit of a HFB state carrying an arbitrary number of particles has been worked out and shown to lead to drastic differences and consequences depending on the closed-shell or open-shell nature of the system in that limit. From a general perspective, the present analysis demonstrates that HFB theory does not reduce to HF theory when the pairing field is driven to zero in the HFB Hamiltonian matrix.

DOI
10.1103/physrevc.102.054320

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Zero-pairing and zero-temperature limits of finite-temperature Hartree-Fock-Bogoliubov theory

T. Duguet, Wouter Ryssens

Background: Recently, variational Hartree-Fock-Bogoliubov (HFB) mean-field equations were shown to possess a mathematically well-defined zero-pairing limit, independently of the closed- or open-shell character of the system under consideration. This limit is nontrivial for open-shell systems such that HFB theory does not reduce to the Hartree-Fock (HF) formalism in all cases.Purpose: The present work extends this analysis to finite-temperature HFB (FTHFB) theory by investigating the behavior of this more general formalism in the combined zero-temperature and zero-pairing limits.Methods: The zero-pairing and zero-temperature limits of the FTHFB statistical density operator constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction.Results: While the FTHFB density operator reduces to the projector corresponding to a pure HF Slater determinant for closed-shell nuclei, the FTHFB formalism does not reduce to the HF theory in all cases in the zero-temperature and zero-pairing limits, i.e., for open-shell nuclei. However, the fact that a nucleus can be of open-shell character in these joint limits is necessarily the result of some symmetry restrictions. Whenever it is the case, the nontrivial description obtained for open-shell systems is shown to depend on the order with which both limits are taken, i.e., the two limits do not commute for these systems. When the zero-temperature limit is performed first, the FTHFB density operator is demoted to a projector corresponding to a pure state made out of a linear combination of a finite number of Slater determinants with different (even) numbers of particles. When the zero-pairing limit is performed first, the FTHFB density operator remains a statistical mixture of a finite number of Slater determinants with both even and odd particle numbers. While the entropy (pairing density) is zero in the first (second) case, it does not vanish in the second (first) case in spite of the temperature (pairing) tending towards zero. The difference between both limits can have striking consequences for the (thermal) expectation values of observables. For instance, the particle-number variance does not vanish in either case and has limiting values that differ by a factor of two in both cases.Conclusions: While in the textbook situation associated with closed-shell nuclei Hartree-Fock-Bogoliubov (finite-temperature Hartree-Fock) theory reduces to Hartree-Fock theory in the zero-pairing (zero-temperature) limit, the present analysis demonstrates that a nontrivial and unexpected limit is obtained for this formalism in open-shell systems. This result sheds a new light on certain aspects of this otherwise very well-studied many-body formalism.

arXiv:2008.01859•Physical Review C•Oct 27, 2020•7 citations

Bogoliubov many-body perturbation theory for open-shell nuclei

A. Tichai, P. Arthuis, T. Duguet +3 more

A Rayleigh–Schrödinger many-body perturbation theory (MBPT) approach is introduced by making use of a particle-number-breaking Bogoliubov reference state to tackle (near-)degenerate open-shell fermionic systems. By choosing a reference state that solves the Hartree–Fock–Bogoliubov variational problem, the approach reduces to the well-tested Møller–Plesset, i.e., Hartree–Fock based, MBPT when applied to closed-shell systems. Due to its algorithmic simplicity, the newly developed framework provides a computationally simple yet accurate alternative to state-of-the-art non-perturbative many-body approaches. At the price of working in the quasi-particle basis associated with a single-particle basis of sufficient size, the computational scaling of the method is independent of the particle number. This paper presents the first realistic applications of the method ranging from the oxygen to the nickel isotopic chains on the basis of a modern nuclear Hamiltonian derived from chiral effective field theory.

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A. Signoracci, T. Duguet, G. Hagen +1 more

Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in $m$ scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an ${N}_{\text{max}}=6$ spherical harmonic oscillator basis for $^{16,18}\mathrm{O}$ and $^{18}\mathrm{Ne}$ in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while $^{20}\mathrm{O}$ and $^{20}\mathrm{Mg}$ display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.

arXiv:1412.2696•Physical Review C•Jun 29, 2015•96 citations

Projected Bogoliubov Many-Body Perturbation Theory : Overcoming formal and technical challenges

Julien Ripoche

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.

•Sep 24, 2019•2 citations

Many-Body Perturbation Theory for Ab Initio Nuclear Structure

A. Tichai

The solution of the quantum many-body problem for medium-mass nuclei using realistic nuclear interactions poses a superbe challenge for nuclear structure research. Because an exact solution can only be provided for the lightest nuclei, one has to rely on approximate solutions when proceeding to heavier systems. Over the past years, tremendous progress has been made in the development and application of systematically improvable expansion methods and an accurate description of nuclear observables has become viable up to mass number $A \approx 100$. While closed-shell systems are consistently described via a plethora of different many-body methods, the extension to genuine open-shell systems still remains a major challenge and up to now there is no \textit{ab initio} many-body method which applies equally well to systems with even and odd mass numbers. The goal of this thesis is the development and implementation of innovative perturbative approaches with genuine open-shell capabilities. This requires the extension of well-known single-reference approaches to more general vacua. In this work we choose two complementary routes for the usage of generalized reference states. First, we derive a new \emph{ab initio} approach based on multi-configurational reference states that are conveniently derived from a prior no-core shell model calculation. Perturbative corrections are derived via second-order many-body perturbation theory, thus, merging configuration interaction and many-body perturbation theory. The generality of this ansatz enables for a treatment of medium-mass systems with arbitrary mass number, as well as the extension to low-lying excited states such that ground and excited states are treated on an equal footing. In a complementary approach, we use reference states that break a symmetry of the underlying Hamiltonian. In the simplest case this corresponds to the expansion around a particle-number-broken Hartree-Fock-Bogoliubov vacuum which is obtained from a mean-field calculation. Pairing correlations are already absorbed in the reference state. The mild scaling allows for investigations up to tin isotopic chains. All benchmarks were performed using state-of-the-art chiral two- plus three-nucleon interactions thus allowing for a universal description of nuclear observables in the medium-mass regime.

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Particle-number projection in the finite-temperature mean-field approximation

P. Fanto, Y. Alhassid, G. F. Bertsch

Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei $^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}$, and $^{150}\mathrm{Sm}$. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.

arXiv:1610.08954•Physical Review C•Jul 7, 2017•18 citations
Background: The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. In this context, pairing correlations may be driven towards zero either on purpose via HFB calculations constrained on, e.g., the particle-number variance or simply because internucleon interactions cannot sustain pairing correlations in the first place in, e.g., closed-shell systems. While taking this limit constitutes a text-book problem when the system is of closed-shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system.Purpose: The present work demonstrates that the zero-pairing limit of an HFB state is well-defined independently of the average particle number A\mathrm{A}A it is constrained to. Still, the nature of the limit state is shown to depend on the regime, i.e., on whether the nucleus characterizes as a closed-shell or an open-shell system when taking the limit. Finally, the consequences of the zero-pairing limit on Bogoliubov many-body perturbation theory (BMBPT) calculations performed on top of the HFB reference state are illustrated.Methods: The zero-pairing limit of a HFB state constrained to carry an arbitrary (integer) number of particles A\mathrm{A}A on average is worked out analytically and realized numerically using a two-nucleon interaction derived within the frame of chiral effective field theory.Results: The zero-pairing limit of the HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number A\mathrm{A}A of interest. First, the textbook situation is recovered for closed-shell systems, i.e., the limit state is reached for a finite value of the pairing strength (the well-known BardeenCooperSchrieffer (BCS) collapse) and takes the form of a single Slater determinant displaying (i) zero pairing energy, (ii) nondegenerate elementary excitations, and (iii) zero particle-number variance. Contrarily, a nonstandard situation is obtained for open-shell systems, i.e., the limit state is only reached for a zero value of the pairing interaction (no BCS collapse) and takes the form of a specific finite linear combination of Slater determinants displaying (a) a nonzero pairing energy, (b) degenerate elementary excitations, and (c) a nonzero particle-number variance for which an analytical formula is derived. This nonzero particle-number variance acts as a lower bound that depends in a specific way on the number of valence nucleons and on the degeneracy of the valence shell. All these findings are confirmed and illustrated numerically. Last but not least, BMBPT calculations of closed-shell (open-shell) nuclei are shown to be well-defined (ill-defined) in the zero-pairing limit.Conclusions: While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. In the present paper, the zero-pairing limit of a HFB state carrying an arbitrary number of particles has been worked out and shown to lead to drastic differences and consequences depending on the closed-shell or open-shell nature of the system in that limit. From a general perspective, the present analysis demonstrates that HFB theory does not reduce to HF theory when the pairing field is driven to zero in the HFB Hamiltonian matrix.
Background: Recently, variational Hartree-Fock-Bogoliubov (HFB) mean-field equations were shown to possess a mathematically well-defined zero-pairing limit, independently of the closed- or open-shell character of the system under consideration. This limit is nontrivial for open-shell systems such that HFB theory does not reduce to the Hartree-Fock (HF) formalism in all cases.Purpose: The present work extends this analysis to finite-temperature HFB (FTHFB) theory by investigating the behavior of this more general formalism in the combined zero-temperature and zero-pairing limits.Methods: The zero-pairing and zero-temperature limits of the FTHFB statistical density operator constrained to carry an arbitrary (integer) number of particles A on average is worked out analytically and realized numerically using a two-nucleon interaction.Results: While the FTHFB density operator reduces to the projector corresponding to a pure HF Slater determinant for closed-shell nuclei, the FTHFB formalism does not reduce to the HF theory in all cases in the zero-temperature and zero-pairing limits, i.e., for open-shell nuclei. However, the fact that a nucleus can be of open-shell character in these joint limits is necessarily the result of some symmetry restrictions. Whenever it is the case, the nontrivial description obtained for open-shell systems is shown to depend on the order with which both limits are taken, i.e., the two limits do not commute for these systems. When the zero-temperature limit is performed first, the FTHFB density operator is demoted to a projector corresponding to a pure state made out of a linear combination of a finite number of Slater determinants with different (even) numbers of particles. When the zero-pairing limit is performed first, the FTHFB density operator remains a statistical mixture of a finite number of Slater determinants with both even and odd particle numbers. While the entropy (pairing density) is zero in the first (second) case, it does not vanish in the second (first) case in spite of the temperature (pairing) tending towards zero. The difference between both limits can have striking consequences for the (thermal) expectation values of observables. For instance, the particle-number variance does not vanish in either case and has limiting values that differ by a factor of two in both cases.Conclusions: While in the textbook situation associated with closed-shell nuclei Hartree-Fock-Bogoliubov (finite-temperature Hartree-Fock) theory reduces to Hartree-Fock theory in the zero-pairing (zero-temperature) limit, the present analysis demonstrates that a nontrivial and unexpected limit is obtained for this formalism in open-shell systems. This result sheds a new light on certain aspects of this otherwise very well-studied many-body formalism.
A Rayleigh–Schrödinger many-body perturbation theory (MBPT) approach is introduced by making use of a particle-number-breaking Bogoliubov reference state to tackle (near-)degenerate open-shell fermionic systems. By choosing a reference state that solves the Hartree–Fock–Bogoliubov variational problem, the approach reduces to the well-tested Møller–Plesset, i.e., Hartree–Fock based, MBPT when applied to closed-shell systems. Due to its algorithmic simplicity, the newly developed framework provides a computationally simple yet accurate alternative to state-of-the-art non-perturbative many-body approaches. At the price of working in the quasi-particle basis associated with a single-particle basis of sufficient size, the computational scaling of the method is independent of the particle number. This paper presents the first realistic applications of the method ranging from the oxygen to the nickel isotopic chains on the basis of a modern nuclear Hamiltonian derived from chiral effective field theory.
Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in mmm scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an Nmax=6{N}_{\text{max}}=6Nmax​=6 spherical harmonic oscillator basis for 16,18O^{16,18}\mathrm{O}16,18O and 18Ne^{18}\mathrm{Ne}18Ne in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while 20O^{20}\mathrm{O}20O and 20Mg^{20}\mathrm{Mg}20Mg display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.
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 the quantum many-body problem for medium-mass nuclei using realistic nuclear interactions poses a superbe challenge for nuclear structure research. Because an exact solution can only be provided for the lightest nuclei, one has to rely on approximate solutions when proceeding to heavier systems. Over the past years, tremendous progress has been made in the development and application of systematically improvable expansion methods and an accurate description of nuclear observables has become viable up to mass number A≈100A \approx 100A≈100. While closed-shell systems are consistently described via a plethora of different many-body methods, the extension to genuine open-shell systems still remains a major challenge and up to now there is no \textit{ab initio} many-body method which applies equally well to systems with even and odd mass numbers. The goal of this thesis is the development and implementation of innovative perturbative approaches with genuine open-shell capabilities. This requires the extension of well-known single-reference approaches to more general vacua. In this work we choose two complementary routes for the usage of generalized reference states. First, we derive a new \emph{ab initio} approach based on multi-configurational reference states that are conveniently derived from a prior no-core shell model calculation. Perturbative corrections are derived via second-order many-body perturbation theory, thus, merging configuration interaction and many-body perturbation theory. The generality of this ansatz enables for a treatment of medium-mass systems with arbitrary mass number, as well as the extension to low-lying excited states such that ground and excited states are treated on an equal footing. In a complementary approach, we use reference states that break a symmetry of the underlying Hamiltonian. In the simplest case this corresponds to the expansion around a particle-number-broken Hartree-Fock-Bogoliubov vacuum which is obtained from a mean-field calculation. Pairing correlations are already absorbed in the reference state. The mild scaling allows for investigations up to tin isotopic chains. All benchmarks were performed using state-of-the-art chiral two- plus three-nucleon interactions thus allowing for a universal description of nuclear observables in the medium-mass regime.
Finite-temperature mean-field theories, such as the Hartree--Fock (HF) and Hartree--Fock--Bogoliubov (HFB) theories, are formulated in the grand-canonical ensemble, and their applications to the calculation of statistical properties of nuclei such as level densities require a reduction to the canonical ensemble. In a previous publication [Y. Alhassid et al., Phys. Rev. C 93, 044320 (2016)], it was found that ensemble-reduction methods based on the saddle-point approximation are not reliable in cases in which rotational symmetry or particle-number conservation is broken. In particular, the calculated HFB canonical entropy can be unphysical as a result of the inherent violation of particle-number conservation. In this work, we derive a general formula for exact particle-number projection after variation in the HFB approximation, assuming that the HFB Hamiltonian preserves time-reversal symmetry. This formula reduces to simpler known expressions in the HF and Bardeen--Cooper--Schrieffer (BCS) limits of the HFB. We apply this formula to calculate the thermodynamic quantities needed for level densities in the heavy nuclei 162Dy,148Sm^{162}\mathrm{Dy}, ^{148}\mathrm{Sm}162Dy,148Sm, and 150Sm^{150}\mathrm{Sm}150Sm. We find that the exact particle-number projection gives better physical results and is significantly more computationally efficient than the saddle-point methods. However, the fundamental limitations caused by broken symmetries in the mean-field approximation are still present.