PHYSICAL REVIEW C 96, 014305 (2017) Particle-number projection in the finite-temperature mean-field approximation P. Fanto,1 Y. Alhassid,1 and G. F. Bertsch2 1Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520, USA 2Department of Physics and Institute for Nuclear Theory, Box 351560, University of Washington, Seattle, Washington 98915, USA (Received 9 January 2017; published 7 July 2017) 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, and 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. DOI: 10.1103/PhysRevC.96.014305 I. INTRODUCTION Here, we study the restoration of particle-number con- servation in the HFB by means of an exact particle-number Finite-temperature mean-field approximations, in partic- projection [11,12]. In particular, we use a projection after ular the finite-temperature Hartree–Fock (HF) and Hartree– Fock–Bogoliubov (HFB) approximations [1,2], are commonly variation (PAV) method, in which the projection operator is used in the calculation of statistical properties of nuclei such as applied to the grand-canonical mean-field solution. We refer level densities [3]. These approximations are computationally to this particle-number PAV method as the particle-number efficient and therefore suitable for global studies of nuclear projection (PNP). In this procedure, we determine the approx- properties. imate canonical partition function by taking a trace of the The appropriate ensemble to describe the nucleus is grand-canonical mean-field density operator over a complete the canonical ensemble with fixed numbers of protons and basis of many-particle states with the correct particle number. neutrons. However, the finite-temperature HF and HFB ap- In contrast, in the DG approximations, the grand-canonical proximations are formulated in the grand-canonical ensem- partition function is multiplied by an approximate correction ble. It is therefore necessary to carry out a reduction to factor. A general formalism for PNP was presented in Ref. [12], the canonical ensemble to restore the correct proton and and an alternate but equivalent formalism was derived in neutron numbers. This reduction is usually carried out in Ref. [13]. However, in the HFB case the application of the the continuous saddle-point approximation, which treats the general formula is limited by a sign ambiguity. Moreover, particle number as a continuous variable [4–6]. The accuracy there has been no systematic assessment of the accuracy of of finite-temperature mean-field approximations was recently PNP in finite-temperature mean-field theories. benchmarked [7] against shell-model Monte Carlo (SMMC) In this article, we derive a general expression for PNP [8–10] results, which are accurate up to statistical errors, in the HFB approximation, assuming only that the HFB and significant problems were identified with this continuous Hamiltonian is invariant under time reversal. We then show saddle-point approach. In particular, the continuous saddle- how this expression reduces to known formulas for the special point approximation breaks down when the particle-number cases of the HF and the Bardeen–Cooper–Schrieffer (BCS) fluctuations are small. These problems were addressed in approximations. We apply this method to three even-even Ref. [7] by the introduction of the discrete Gaussian (DG) nuclei in the rare-earth region: (i) a strongly deformed nucleus approximations. Specifically, in Ref. [7] two DG approxi- with weak pairing (162Dy), (ii) a spherical nucleus with strong mations were introduced, which we label DG1 and DG2 pairing (148Sm), and (iii) a transitional deformed nucleus with and discuss in detail in Sec. III. In these approximations, non-negligible pairing (150Sm). The results from the PNP are the saddle-point correction to the grand-canonical partition compared with results from the DG approximations and the function is given by a sum over discrete Gaussians (in particle SMMC method. number). This overcomes a divergence in the continuous The outline of this article is as follows: In Sec. II, we derive saddle-point approximation at low temperatures in the HF a general expression for the particle-number projected HFB approximation, but still gives an unphysical negative value partition function in the case where the HFB Hamiltonian of the canonical entropy in the low-temperature limit of the is time-reversal invariant and show how this expression HFB. simplifies in the HF and BCS limits. In Sec. III, we discuss 2469-9985/2017/96(1)/014305(10) 014305-1 ©2017 American Physical Society P. FANTO, Y. ALHASSID, AND G. F. BERTSCH PHYSICAL REVIEW C 96, 014305 (2017) the DG approximations and describe how we calculate the Using Ek = Ek¯, the HFB Hamiltonian in Eq. (2) can be approximate canonical thermal energy, entropy, and state rewritten as density from the particle-number projected partition function. † † 1 In Sec. IV, we apply the PNP formula to three heavy nuclei and Hˆ − μNˆ = Ek(αkαk − αk¯α ) + tr (h − μ). (4) HFB k¯ 2 compare the results with those from the DG approximations k>0 and with SMMC results. Finally, in Sec. V, we summarize our In a more compact notation findings and provide an outlook for future work. The computer Hˆ − μNˆ = ξ †Eξ + 1 h − μ , codes for the canonical reductions and the data files to generate HFB 2 tr ( ) (5) the results described here are provided in the Supplemental † † † where ξ = (α ,...,α ,αk¯ ,...,αk¯ ) and Material depository for this article [14]. k1 kNs /2 1 Ns /2 E 0 II. PARTICLE-NUMBER PROJECTION IN E = . (6) 0 −E HARTREE–FOCK–BOGOLIUBOV APPROXIMATION E We assume a nuclear Hamiltonian in Fock space spanned by The matrix is the diagonal matrix of the HFB quasiparticle Ek k = ,...,Ns / a set of Ns single-particle orbitals k (k = 1,...,Ns ) with a one- energies ( 1 2). Similarly, the number operator Nˆ body part described by the matrix t and an antisymmetrized can be written as two-body interactionv ¯, † † † † Ns Nˆ = (a ak + a ak¯) = (a ak − ak¯a ) + k k¯ k k¯ 2 † 1 † † k>0 k>0 Hˆ = tij ai aj + v¯ij kl ai aj alak, (1) ij 4 ij kl Ns = ξ †(W†NW)ξ + , (7) † 2 where a and ak are, respectively, creation and annihilation k where operators of the single-particle states k. For nuclei with strong pairing condensates, the appropriate mean-field theory is the 10 N = , (8) HFB approximation. The HFB Hamiltonian can be written in 0 −1 matrix notation as [11] and where we have used the transformation in Eq. (3). a 1 † h − μ The particle-number-projected HFB partition function is HˆHFB − μNˆ = (a a) ∗ ∗ − −(h − μ) a† HFB −β(Hˆ −Vˆ ) 2 defined by ZN = Tr[PˆN e HFB ], where PˆN is the 1 operator that projects any many-particle state in Fock space + tr (h − μ), (2) N Vˆ 2 onto the Hilbert space of -particle states, and is the two-body interaction of the Hamiltonian (1). For a finite where μ is the chemical potential, h = t + v¯ is the single-particle model space of dimension Ns , PˆN can be density-dependent single-particle Hamiltonian, and ij = expressed as a discrete Fourier sum over Ns quadrature angles ij kl v¯ij kl κkl/2 is the pairing field, with being the one-body ϕn = 2πn/Ns density and κ the pairing tensor. The 2Ns × 2Ns matrix in Eq. (2) can be diagonalized by a unitary transformation to the e−βμN Ns † ZHFB = e−iϕnN ζ HFB, quasiparticle basis αk,αk. N n (9) Ns We assume that the HFB Hamiltonian in Eq. (2) is invariant n=1 under time reversal, and thus its quasiparticle states come where in time-reversed pairs |k and |k¯ with degenerate energies βVˆ iϕ Nˆ −β Hˆ −μNˆ 1 HFB n ( HFB ) Ek = Ek¯. The Bogoliubov transformation that defines the ζn = e Tr[e e ], (10) quasiparticle basis can then be fully expressed by μ and is the chemical potential determined in the grand- αk ak U −V canonical ensemble. The subtraction of Vˆ =tr (v¯ )/2 + = W† , W = , (3) κ†vκ / α† a† VU tr ( ¯ ) 4 accounts for the double counting of the interaction k¯ k¯ terms and ensures the thermodynamic consistency of the HFB where k runs over half the number of single-particle states in the grand-canonical ensemble. Because of the nonzero Hˆ Nˆ from 1 to Ns /2. In the following, we denote these states by pairing gap in Eq. (2), HFB and do not commute. k>0. W is an Ns × Ns unitary matrix, in contrast with the Using Eqs.