Particle-Number Projection in the Finite-Temperature Mean-Field

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 conservation 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. (5) and (7), we can rewrite (10)as general Bogoliubov transformation matrix which is a 2Ns × −βU iϕ N / iϕ ξ † W†NW ξ −βξ†Eξ ζ HFB = e 0 e n s 2Tr[e n ( ) e ], (11) 2Ns -dimensional matrix [11]. n

where U0 = tr (h − μ)/2 −Vˆ . To evaluate the trace in Eq. (11), we use the group property of the exponentials of 1If the condensate is also axially symmetric, the angular mo- one-body fermion operators written in quadratic form. This mentum m about the symmetry axis is conserved. If this is the property states that the product of two such group elements is case, we choose the k states to have positive signature, i.e., m = another group element 1/2,−3/2,5/2,−7/2,...,andk¯ states have negative signature m = ξ †Aξ ξ †Bξ ξ †Cξ −1/2,3/2,−5/2,7/2,.... e e = e , (12)

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C where the matrix is determined from the single-particle Equation (19) can be derived directly using [HˆHF,Nˆ ] = representation of the group 0 even when time-reversal symmetry is broken. A general eAeB = eC. (13) derivation is given in Ref. [12]. Applying this property to Eq. (11), we can rewrite it in the form B. The Bardeen–Cooper–Schrieffer limit † HFB n −βU0 ξ Cn(β)ξ Equation (17) also simplifies in the BCS limit, in which ζn = (−) e Tr[e ], (14) the quasiparticle representation mixes particle state k with where the matrix Cn(β) is determined from only its time-reversed counterpart k¯. In this case, in the † Cn β iϕnW NW −βE † iϕnN −βE Bogoliubov transformation matrix given in Eq. (3), U is e ( ) = e e = W e We . (15) diagonal and V antidiagonal. Consequently, the transformation Using the formula for the trace of the exponential of a one-body can be decomposed into a set of Ns /2 transformations fermionic operator (see Appendix A), we find αk ak HFB n −βU0 Cn(β) uk −vk ζn = (−) e det(1 + e ). (16) = (20) α† vk uk a† Combining Eq. (16) with Eq. (15), we obtain the final k¯ k¯ expression for each pair {k,k¯} of time-reversed states. Here, uk and vk are HFB n −βU0 † iϕnN −βE u2 + v2 = ζn = (−) e det(1 + W e We ), (17) real numbers satisfying k k 1. Equation (17) can then be rewritten as a product of 2 × 2 block determinants. Using W, E N where the matrices , and are given, respectively, by Eqs. (3), (6), and (8). 2 iϕn 2 −iϕn iϕn −iϕn † u e + v e ukvk(e − e ) iϕnW NW k k Equation (17) is a general formula that applies when the e |k = u v eiϕn − e−iϕn v2eiϕn + u2e−iϕn HFB Hamiltonian is time-reversal invariant and thus the k k( ) k k quasiparticle energies come in degenerate time-reversed pairs (21) Ek = Ek¯. A formula valid for the most general case can be † † ξ † = α α α α e−iϕn = eiϕnNs /2 = − n derived in a similar fashion by using ( k k k¯ k¯) for each block and k>0 ( ) , we find the (where k = 1,...,Ns /2) and making the dimension of final expression the relevant matrices 2Ns × 2Ns [12]. However, the final HFB BCS −βU0 βEk 2 2iϕn 2 −βEk +iϕn expression for ζn , given in Eq. (3.46) of Ref. [12], involves ζn = e e uk + e vk + 2e a square root of a determinant. This square root leads to a sign k> 0 ambiguity that is difficult to resolve. The method discussed − βE iϕ + e 2 k v2 + e2 n u2 . (22) here eliminates this sign ambiguity completely for the case k k in which the HFB Hamiltonian is time-reversal invariant by The result in Eq. (22) can also be derived by writing explic- working with matrices of reduced dimension Ns × Ns . eiϕnNˆ Equation (17) becomes numerically unstable at large β.The itly the matrix elements of in the subspace spanned by the | = u + v a†a† | α†| α†| reason for this instability can be seen in Eq. (15). At large β,the four many-body states k ( k k k k¯ ) , k k, k¯ k, and −βE † † iϕ Nˆ −β Hˆ −μNˆ −Vˆ e α α | e n e ( HFB ) diverging scales in the diagonal matrix will dominate the k k¯ k, and evaluating the traces of in smaller scales in the matrix product W†eiϕnN W. We stabilize each of these subspaces. the calculation by the method discussed in Appendix B. The result in Eq. (22) is given by Eqs. (24) and (25) of Ref. [15] but is obtained here as the special limit of a more A. The Hartree–Fock limit general formula. A formula for the BCS limit as a product × In the limit in which the pairing condensate vanishes, of the corresponding 2 2 determinants is also presented in i.e., → 0, the HFB approximation reduces to the HF Ref. [16]. approximation. The matrix in Eq. (2) becomes diagonal, and An important case of the BCS limit is that of a spherical the mean-field Hamiltonian can be rewritten as condensate, in which the angular-momentum quantum number j and the magnetic quantum number m are preserved by Hˆ = h a†a . HF ij i j (18) the Bogoliubov transformation. In this case, the quasiparticle ij energies for a given j are independent of m. If the single- In this limit, the Bogoliubov transformation reduces to a particle model space does not include more than one shell unitary transformation among the particle basis operators, and with the same j, the Bogoliubov transformation is then of the V vanishes. Because the particle-number operator is diagonal form (20), where |k=|jm and |k¯=±|j − m, with m = in a particle basis, W†eiϕnN W = eiϕnN . Equation (17) can then 1/2,−3/2,5/2,−7/2,... being the positive-signature states. be written as The parameters uk = uj ,vk = vj are independent of m, and βVˆ −βh+ βμ+iϕ Eq. (22) simplifies to ζ HF = e det[1 + e ( n)] n 1 Ns HFB −βU0 β(j+ )Ej 2 2iϕn 2 −βEj +iϕn ζn = e e 2 uj + e vj + 2e βVˆ −β( k −μ)+iϕn = e [1 + e ], (19) j k=1 − βE iϕ j+ 1 + e 2 j v2 + e2 n u2 2 . (23) where k are the HF single-particle energies. j j

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III. DISCRETE GAUSSIAN APPROXIMATIONS where i, j = p, n. In the DG approximation, ζ is guaranteed AND STATISTICAL QUANTITIES to be at least unity, so this approximation will not break down when the particle-number fluctuation is small. When the A. Saddle-point and discrete Gaussian approximations particle-number fluctuation is large, the DG correction factor In this section, we describe briefly the discrete Gaussian in Eq. (28) agrees with the continuous saddle-point correction (DG) approximations introduced in Ref. [7]. For more details, factor in Eq. (26). we refer the reader to Sec. II of Ref. [7]. The traditional method In Ref. [7], two versions of the DG approximation were for calculating level densities from a grand-canonical finite- introduced. In the first, which we call DG1, the matrix ∂N/∂α temperature mean-field theory is by a three-dimensional (3D) is determined numerically. In the second, which we call DG2, saddle-point approximation. Specifically, the state density is the matrix ∂N/∂α is replaced by a diagonal matrix whose given by diagonal elements are given by the particle-number variances 2 −1/2 (Ni ) calculated in the grand-canonical mean-field theory. 1 ∂(E,Np,Nn) S ρ E,N ,N ≈ e gc , p n / (24) Because it neglects the potentially nonzero off-diagonal terms, (2π)3 2 ∂(β,αp,αn) DG2 is expected to be less accurate than DG1 at low S where gc is the grand-canonical entropy calculated in the temperatures. mean-field approximation and the energy and particle numbers Z −β are set equal to the derivatives of ln gc with respect to B. Canonical energy, entropy, and state density and αp,n, respectively. Here αp,n = βμp,n, where μp,n are the proton and neutron chemical potentials. We summarize here the formulas used to find the statistical Two refinements to this standard procedure were introduced quantities calculated in Sec. IV. Given the canonical partition Z ρ E E in Ref. [7]. The first refinement is to separate the αp,n and function c, the average state density ( ) at energy is eval- β integrations in the 3D saddle-point approximation. The uated by a one-dimensional (1D) saddle-point approximation α as p,n integrations are carried out first, giving the approximate canonical partition function ∂E −1/2 ρ E ≈ π eSc(β), ( ) 2 ∂β (29) ln Zc ≈ ln Zgc − αi Ni − ln ζ, (25) i=p,n where β is determined as a function of the energy E by where the correction factor ζ (obtained in the two-dimensional the saddle-point condition E =−∂ ln Zc/∂β = Ec and Sc is saddle-point approximation) is given by the canonical entropy Sc = ln Zc + βEc. When Zc(β)isthe Z β = Pˆ e−βHˆ ∂ N ,N 1/2 exact canonical partition function c( ) Tr ( N ) (here ζ = π ( p n) . Pˆ = Pˆ Pˆ N N 2 (26) N Np Nn , with p and n being the numbers of protons ∂(αp,αn) −βHˆ and neutrons, respectively), Ec = Tr (PˆN e Hˆ )/Zc(β) and In a second step, the state density is obtained from the ap- Sc =−Tr (Dˆ N ln Dˆ N ) are, respectively, the exact canonical proximate canonical partition function in Eq. (25) by carrying energy and entropy of the correlated density matrix Dˆ N = out the β integration in the saddle-point approximation. The Pˆ e−βHˆ /Z β S N c( ). canonical entropy obtained in this procedure differs from gc In the particle-number-projected HFB approximation, not only through the inclusion of ζ but also through the HFB HFB HFB HFB Zc = ZN ZN , where ZN is given by Eqs. (9) and (17). ζ β p n p(n) dependence of on .Itisgivenby HFB The approximate canonical HFB energy Ec is determined Sc ≈ Sgc − ln ζ − βδE, (27) by the saddle-point condition where δE =−d ln ζ/dE. The approximate state density is ∂ ln ZHFB E =− c = EHFB, given by an expression similar to the one given in Eq. (29) ∂β c (30) below. Equation (26) is derived by treating the particle numbers as continuous variables, and we therefore refer to this method and the approximate canonical HFB entropy is HFB HFB HFB as the continuous saddle-point approximation. Sc = ln Zc + βEc . (31) The next refinement to the standard 3D saddle-point approximation originates from the observation that Np and The average state density in the HFB approximation is then Nn are discrete integers and should not be treated as con- given by Eq. (29) with the canonical energy and entropy tinuous variables when the particle-number fluctuation is replaced by the their HFB expressions (30) and (31). EHFB small. Specifically, the continuous saddle-point approximation We note that the above c differs from the expectation 2 Hˆ breaks down when 2π(Ni ) 1fori = p, n. This prob- value of in the particle-number projected HFB density op- HFB −β(Hˆ −Vˆ ) HFB HFB lem is dealt with in Ref. [7] by the introduction of the discrete erator Dˆ N = PˆN e HFB /ZN , and similarly Sc = HFB HFB Gaussian (DG) approximation, in which the correction factor −Tr (Dˆ N ln Dˆ N ). The reason for these differences is the ζ is not given by Eq. (26) but instead by explicit dependence of Hˆ on the grand-canonical one-body ⎛ ⎞ HFB density and pairing tensor. ∂N −1 ⎝ 1 ⎠ In the DG approximations, ln Zc isgivenbyEq.(25), where ζ = exp − (Ni − Ni )(Nj − Nj ) , ∂α ζ is given by Eq. (28). The energy is given by the saddle-point N ,N 2 i,j ij i j HFB condition (30), and the entropy by Eq. (31), where Zc is (28) replaced by the DG1 or DG2 partition function.

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IV. RESULTS approximation accounts only for the intrinsic K states and not for the rotational bands that are built on top of these intrinsic Here we present results for the particle-number-projected states. finite-temperature mean-field theories in three heavy nuclei. The PNP, DG1, and DG2 give nearly identical results for In this section, we refer to the particle-number projection as −1 162 β<4MeV . In DG1 and DG2, however, the entropy exhibits the PNP method. First, we discuss Dy, a typical strongly − unphysical oscillatory behavior for 4 β 10 MeV 1.In deformed nucleus for which the appropriate mean-field theory contrast, the entropy asymptotes monotonically to zero at large is the HF approximation. Next, we present results for 148Sm, β in the PNP, as would be physically expected. a typical spherical nucleus with a strong pairing condensate The PNP canonical excitation energy and state density for which the BCS limit of the HFB is appropriate. Finally, closely resemble the corresponding results for the DG ap- we discuss a transitional nucleus 150Sm, in which the pairing proximations, which are shown in Figs. 6 and 10 of Ref. [7]. condensate is deformed and the general projection formula The deviation between the PNP entropy and the DG entropy (17) is required. The results from the PNP are compared with observed at low temperatures does not lead to any significant the discrete Gaussian approximations of Ref. [7] and with the difference in the state densities. Figures showing these observ- SMMC results [17,18]. ables are included in the Supplemental Material [14]. The Hamiltonian for these calculations is taken from Refs. [17,18], where the original calculations of the SMMC level density were carried out. It is given in a shell-model basis B. Particle-number-projected Hartree–Fock–Bogoliubov having Ns = 40 proton orbitals and Ns = 66 neutron orbitals. for spherical condensate: 148Sm To test our formulas for the particle-number-projected HFB A. Particle-number-projected Hartree–Fock for strongly partition function in the BCS limit, given by Eqs. (9) and (23), deformed nucleus: 162Dy we apply the particle-number-projected HFB approximation We applied the particle-number-projected HF approxima- to 148Sm, for which the pairing condensate is spherical. The 162 tion to the strongly deformed nucleus Dy, in which the canonical entropies for the PNP, the DG approximations, and pairing is weak, by using Eqs. (9) and (19) to calculate the SMMC are shown in Fig. 2. The kinks in the HFB results the particle-number-projected partition function. In Fig. 1, in the region 2 β 3MeV−1 are due to the proton and we compare the approximate canonical entropy (31) from neutron pairing transitions. For β values below the first pairing the PNP with those from the DG approximations and with transition, there is good agreement between the HFB results −1 the SMMC entropy. The kink at β ≈ 0.83 MeV in the HF and the SMMC results. The kinks that indicate the pairing results is due to the sharp shape transition that occurs in the transitions are more pronounced in the PNP and DG2 than in grand-canonical HF approximation. At β values below the DG1. shape transition (i.e., in the spherical phase), the HF results In the paired phase, the approximate canonical entropies are in good agreement with those from the SMMC. However, for the PNP and the DG approximations decrease rapidly, at β values above the shape transition, the entropies from the dropping below zero around β ≈ 4MeV−1. A negative entropy PNP and the DG approximations are noticeably lower than the is unphysical because the entropy of a nondegenerate ground SMMC entropy. The reason for this discrepancy is that the HF state is zero. This negative entropy originates in the intrinsic violation of particle-number conservation in the grand-canonical HFB approximation and will be explained for the PNP in 70 6 60 4

S 2 60 50 8 0 40 50 4 4 6 8 10 12 14 16 18 20 S S 30 β (MeV -1) 40 0 -4 20 30 4 8 12 16 20 24 28 S 10 β (MeV -1) 20 0 0 0.5 1 1.5 2 2.5 3 3.5 4 10 -1 β (MeV ) 0 0 0.5 1 1.5 2 2.5 3 162 β FIG. 1. Canonical entropy of Dy vs in the HF approximation. -1 The approximate PNP canonical entropy (31) (solid black line) is β (MeV ) compared with the approximate canonical entropy from DG1 (dashed red line) and from DG2 (dashed-dotted blue line). The open circles FIG. 2. Canonical entropies of 148Sm vs β in the BCS limit of the represent the SMMC entropy. The inset shows the various entropies HFB approximation. Lines and symbols are as in Fig. 1. The inset at higher β values. shows the entropies for higher β values.

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Sec. IV D. An explanation for the DG approximations was 120 given in Ref. [7]. 6 β 100 However, for large values, the PNP exhibits qualitative 4 and quantitative improvements over the DG approximations. (MeV) 2 80 x

The entropy from the PNP asymptotes smoothly to a value E 0 of Sc ≈−2.30. The entropy from DG1 reaches a minimum around β ≈ 8MeV−1 and subsequently increases with increas- 60 2 3 4 5 6 7 8 9 10

(MeV) -1 ing β. Such an increase is unphysical because, for these values x β (MeV ) E 40 of β, the system is already in its ground state. The entropy from DG2 asymptotes smoothly to a negative value of Sc ≈−3.68. 20 The absolute error in the estimate for the ground-state entropy is thus larger in DG2 than in the PNP by more than a unit. 0 At small and intermediate values of β, the PNP excitation 0 0.5 1 1.5 2 energy for 148Sm closely resembles the DG1 excitation energy β (MeV-1) showninFig.13ofRef.[7]. The PNP state density is similar to the DG2 state density shown by the dotted line in Fig. 16 150 β of Ref. [7] while the DG1 state density is somewhat enhanced FIG. 3. Excitation energy of Sm vs in the HFB approxi- at low excitation energies. Figures showing these observables mation. The approximate canonical energy calculated from the PNP (solid black line) is compared with the approximate canonical energy are included in the Supplemental Material [14]. from DG2 (dashed-dotted blue line). DG1 becomes unstable for this nucleus and is not shown. The open circles are the SMMC excitation C. Particle-number-projected Hartree–Fock–Bogoliubov energies. The inset shows higher β values. for deformed condensate: 150Sm To calculate the particle-number-projected HFB partition The entropy from the PNP is very close to that from DG2 in function for 150Sm, which has a deformed pairing condensate, the unpaired phase but shows a quantitative improvement over we must use the general PNP HFB formalism of Sec. II, i.e., DG2 in the paired phase. The entropy from the PNP asymptotes Eqs. (9) and (17). In this case, the advantages of the PNP to Sc ≈−1.20, while the entropy from DG2 asymptotes to 148 over the DG approximations are signiﬁcant for the excitation Sc ≈−2.52. As with Sm, the absolute error of the ground- energy, canonical entropy, and state density. In particular, DG1, state entropy in the PNP is lower by more than a unit of entropy the more accurate of the two DG methods used in Ref. [7], than the error in DG2. Furthermore, DG2 shows a large spike becomes numerically unstable for temperatures below the near the neutron pairing transition, which is not present in the shape transition. Because of this instability, we omit DG1 PNP. from the ﬁgures in this section. In contrast, the PNP remains stable for all temperatures. 3. State density The behavior of the state density, which is shown for the 1. Excitation energy PNP, DG2 and SMMC in Fig. 5, closely resembles that of the In Fig. 3, we show the excitation energy as a function canonical entropy. At energies above the shape transition, the of β for the PNP and DG2 in comparison with the SMMC PNP and DG2 results agree well with the SMMC results. The energy. The kink at β ≈ 1.5MeV−1 is the shape transition, and the kinks at β ≈ 3MeV−1 and β ≈ 6MeV−1 are the 60 proton and neutron pairing transitions, respectively. Except 8 around the phase transitions, there is good agreement between 50 4 the PNP and DG2 results and the SMMC results. The DG2 S approximation shows a larger discrepancy from the SMMC 40 0 around the pairing transitions than does the PNP. -4 4 8 12 16 S 30 -1 2. Canonical entropy β (MeV ) 20 The canonical entropies from the PNP, DG2 and SMMC are shown in Fig. 4.Atβ values below the shape transition, there 10 is close agreement between the SMMC and the HFB results. Between the shape transition at β ≈ 1.5MeV−1 and the proton 0 − 0 0.5 1 1.5 2 2.5 3 3.5 4 pairing transition at β ≈ 3MeV 1, the PNP and DG2 entropies are lower than the SMMC entropy because of the contributions β (MeV-1) of rotational bands to the SMMC. For β values above the proton pairing transition, the HFB entropies are reduced even FIG. 4. Canonical entropy of 150Sm vs β in the HFB approxi- − further, falling below zero above β ≈ 4MeV 1. As in the case mation. Lines and symbols for the PNP, DG2, and SMMC are as in of 148Sm, this negative entropy originates in the violation of Fig. 3. The inset shows an expanded scale at large values of β. particle-number conservation in the HFB approximation.

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1020 related explanation for the DG approximations is given in Ref. [7]. This negative entropy is an inherent limitation of 1016 the particle-number projection after variation method in the grand-canonical HFB theory.

) 12

-1 10

) 108 8 -1 E. Computational efﬁciency 10 4

(MeV 10

ρ An important advantage of the PNP over all saddle-

4 (MeV 0 10 ρ 10 point methods in which the partial derivatives are calculated 0 4 8 explicitly is computational efficiency. Both the PNP and E (MeV) 100 x these saddle-point methods require finding the self-consistent 0 5 10 15 20 25 30 35 40 mean-field solutions for a set of β values. The additional cost 2 of the PNP HF approximation scales as Ns , since calculating E (MeV) HF x ζn in Eq. (19) takes Ns operations and must be done for each of the Ns quadrature points in the Fourier sum. The PNP HFB 150 E 4 FIG. 5. State density of Sm vs excitation energy x in the HFB approximation scales as Ns because the matrix decomposition approximation. Lines and symbols are as in Fig. 3. The inset shows in the stabilization method (discussed in Appendix B) requires 3 the low-excitation-energy results. Ns operations for each of the Ns quadrature points. In DG1 it is necessary to calculate numerically the HFB results are reduced at energies below the shape transition derivatives of the proton and neutron numbers with respect and reduced further at energies below the pairing transitions. to the chemical potentials, which requires finding additional The discontinuities in both the PNP and in DG2 around Ex ≈ mean-field solutions. The cost of calculating these derivatives 10 MeV are due to the sharp kink in the excitation energy at accurately can be very large, especially in the vicinity of the shape transition. The discrepancy between the PNP and the phase transitions where the mean-field solution can take DG2 entropies in the high-β limit is not noticeable in the state many iterations to converge. These timing considerations do density because this discrepancy becomes significant only for not apply to DG2, since in that approximation the relevant very low excitation energies. derivatives are replaced by the particle-number variances calculated as mean-field observables. However, as discussed in Ref. [7], DG2 is significantly less accurate than DG1 in D. Approximate canonical Hartree–Fock–Bogoliubov the paired phase. In contrast, the PNP is as accurate or more entropy in limit T → 0 accurate than DG1 in each of the nuclei considered here. We found in Secs. IV B and IV C that the approximate In the Supplemental Material [14], we include timing data canonical entropy of 148Sm and 150Sm asymptotes at low comparing the efficiency of the PNP to that of the DG1 temperatures to a negative number. Here, we show how approximation for the spherical nucleus 148Sm. this unphysical result arises from the inherent violation of particle-number conservation in the grand-canonical HFB approximation. We consider the limit of sufficiently large β, V. CONCLUSION AND OUTLOOK in which we can neglect the contribution of excited states to the partition function. Assuming for simplicity one type of We have derived a general formula for exact particle- particle, we have in this limit number projection after variation in the finite-temperature HFB approximation, assuming only that the HFB Hamiltonian HFB −βE0 ZN → e |PˆN |, (32) is invariant under time reversal. This general formula reduces to simpler known expressions in the HF and BCS limits. where E is the ground-state energy and | is the HFB ground 0 We have assessed the accuracy of the PNP, using the state. This state can be written as a linear superposition of SMMC as a benchmark. In addition, we have compared states with even particle numbers, |= αN |ψ , N=0,2,4,... N the performance of this method to the DG approximations where |ψ is an N-particle state. Equation (32) can then be N formulated in Ref. [7], which were introduced to improve on expressed as the saddle-point approximation. Our results show that, like HFB −βE0 2 ZN → e |αN | . (33) the DG approximations, the PNP is in agreement with the SMMC for temperatures above the shape or pairing phase 2 where |αN | is the probability that the HFB ground-state con- transitions. In general, we find that the PNP provides both densate contains N particles. Using Eq. (31) and generalizing quantitative and qualitative improvements over the DG and to the case of both protons and neutrons, we find in the limit saddle-point approximations at low temperatures. In the HF of zero temperature case, the PNP suppresses an instability that develops in the SHFB → |α |2 + |α |2, canonical entropy calculated by the DG approximations at c ln Np ln Nn (34) large β. In the paired phase of the HFB, the PNP entropy where Np and Nn are the numbers of protons and neu- shows the correct qualitative behavior, i.e., it is monotonically trons, respectively. Because particle-number is not conserved, decreasing with increasing β, unlike the DG1 approximation. |α |2, |α |2 < Np Nn 1, and the entropy (34) is negative. A closely Furthermore, in the paired phase, the PNP reduces the error

014305-7 P. FANTO, Y. ALHASSID, AND G. F. BERTSCH PHYSICAL REVIEW C 96, 014305 (2017) in the DG2 approximation by more than a unit of entropy. ACKNOWLEDGMENTS This reduction is significant, since the errors in both cases This work was supported in part by the U.S. DOE Grants are of order unity. Finally, the PNP is significantly more No. DE-FG02-91ER40608 and No. DE-FG02-00ER411132. computationally efficient than saddle-point methods in which derivatives have to be calculated explicitly, such as the DG1 approximation. APPENDIX A: TRACE OF EXPONENTIAL OF However, the PAV method is inherently limited by the FERMIONIC ONE-BODY OPERATOR broken symmetries in the grand-canonical mean-field theory to which it is applied. In a deformed nucleus, the projected Here, we derive the formula for the grand-canonical trace of mean-field theory cannot describe the rotational enhancement the exponential of a quadratic fermionic operator of the form that is observed in the SMMC below the shape transition of Eq. (6). Assuming a diagonalizable Ns × Ns matrix C, temperature. In a nucleus with a strong pairing condensate, † the intrinsic violation of particle-number conservation below Tr eξ Cξ = det(1 + eC), (A1) the pairing transition temperature leads to negative values of the PNP canonical entropy. It is therefore desirable to explore where ξ is defined in Sec. II. improvements to finite-temperature mean-field theories that Since C is diagonalizable, there is a similarity transforma- can address these issues. tion S that brings C to a diagonal form, One avenue for improvement is the variation after projection (VAP) method. VAP conserves particle number during ⎛ ⎞ λk the variation to determine the mean-field solution and is ⎜ 1 ⎟ ⎜ . ⎟ therefore expected to smooth the sharp phase transitions ⎜ .. ⎟ of the grand-canonical mean-field approximations and cor- ⎜ λ ⎟ ⎜ kNs /2 ⎟ −1 ⎜ ⎟ rect the unphysical negative entropy in the paired phase. SCS = ⎜ λk¯ ⎟. (A2) ⎜ 1 ⎟ This method has been formulated for the zero-temperature ⎜ .. ⎟ ⎜ . ⎟ HFB approximation [19] and applied to even-even nuclei ⎝ ⎠ λk¯ [20,21]. VAP has also been formulated at finite temperature Ns /2 in the BCS approximation to calculate the pairing gaps in small model spaces [22,23]. However, a general VAP † method for finite-temperature calculations in large model If S is unitary, it can be used to transform αk and αk to new spaces would involve a considerable computational cost. † fermionic operators dk and d such that In particular, the entropy of the projected HFB density k S =− Dˆ Dˆ N Tr ( N ln N ), which is required for the calculation † † † ξ Cξ (λk d dk +λk¯ dk¯ d ) of the free energy at finite temperature, is difficult to calculate Tr e = Tr e k>0 k k¯ † † (λk d dk −λk¯ d dk¯ +λk¯ ) in the paired phase because the HFB Hamiltonian does not = Tr e k>0 k k¯ . (A3) commute with the particle-number operator. Nevertheless, given the potential advantages of this method, it would be Since dk are fermionic operators, the grand-canonical trace in worth investigating the possibility of developing a general VAP Eq. (A3) can be evaluated as usual to give method for the finite-temperature HF and HFB approximations in large model spaces. † ξ Cξ λk −λ λ Another direction, perhaps more tractable, would be to use Tr e = [(1 + e )(1 + e k¯ )e k¯ ] k>0 the PNP in the static-path approximation (SPA) [24–27]. The SPA takes into account the static fluctuations of the mean field = [(1 + eλk )(1 + eλk¯ )] = det(1 + eC). (A4) beyond its self-consistent solution. As with the mean-field k>0 approximation, the particle-number projection can be carried out either before or after the SPA integration. It would be useful However, in general, S is not unitary. In this case, we make use to conduct a systematic assessment of the particle-number- of a nonunitary Bogoliubov transformation [30]. We define the projected SPA. operators {d,d˜} by the canonical transformation Finally, the formula derived here for the particle-number projected HFB partition function assumes that the HFB † − η = Sξ, η˜ = ξ S 1, (A5) Hamiltonian is time-reversal invariant and therefore is not completely general. A general formula for the projected HFB T where η = (dk ,...,dk ,d˜k¯ ,...,d˜k¯ ) andη ˜ = partition function was derived in Ref. [12], but its application 1 Ns /2 1 Ns /2 d˜k ,...,d˜k ,d ,...,d {d,d˜} is limited by a sign ambiguity. For the grand-canonical traces ( 1 Ns /2 k¯1 k¯Ns /2 ). It can be shown that have of statistical density operators, a similar sign ambiguity was the same anticommutation relations as {α,α†} [30]. However, resolved by relating the trace over Fock space to a Pfaffian this transformation does not preserve the Hermiticity relation [28]. An extension of the Pfaffian approach to symmetry of the operators. Therefore we must treat the kets and restoration projection at finite temperature for a general HFB bras related to these operators differently. Because the Hamiltonian, and specifically for particle-number projection, anticommutation relations are preserved, the usual creation will be presented in Ref. [29]. and annihilation formulas apply to the left and right bases

014305-8 PARTICLE-NUMBER PROJECTION IN THE FINITE- . . . PHYSICAL REVIEW C 96, 014305 (2017) separately. We deﬁne the left and right vacuums by = eλk nk +λk¯ (1−nk¯ ) k> n = , | = d d | , ¯|= | d˜ d˜ , 0 k 0 1 0 d k k¯ 0 d 0 0 k k¯ (A6) nk¯ =0,1 k>0 k>0 = (1 + eλk )(1 + eλk¯ ) = det(1 + eC). (A11) and left and right states by k>0 nk nk¯ This completes the proof of Eq. (A1). |φ= (d˜k) (d˜k¯) |0d , k>0 APPENDIX B: STABILIZATION OF nk nk¯ φ¯|=d 0¯| (dk) (dk¯) . (A7) HARTREE–FOCK–BOGOLIUBOV k>0 PARTICLE-NUMBER PROJECTION The anticommutation relations ensure that Equation (17) becomes numerically unstable at large values β of . We rewrite φ¯|φ =δφ,φ , φ¯|d˜kdk|φ=nk, (A8) det(1 + W†eiϕnN We−βE )

n k † iϕnN † −iϕnN −βE where k is the occupation number of state . Furthermore, = det(W ) det(e ) det(W) det(W e W + e ). as discussed in Ref. [30], these left and right states form a bi-orthogonal basis for the Fock space and therefore satisfy (B1) the completeness relation, Using det(eiϕnN ) = 1 and det W† = [det W]−1, we find † iϕ N −βE † −iϕ N −βE |φφ¯|= . det(1 + W e n We ) = det(W e n W + e ). 1 (A9) (B2) φ The determinant on the right-hand side can be computed stably † † −iϕ N −βE We can rewrite ξ Cξ in this bi-orthogonal basis, [31] by decomposing the matrix An ≡ W e n W + e in the form † −1 ξ Cξ = ηS˜ CS η = λkd˜kdk − λk¯d˜k¯dk¯ + λk¯. (A10) An = QnDnRn, (B3) k>0 Q R We can now compute the trace as follows (|ψ below is an where n is an orthogonal matrix, n is an upper triangular D arbitrary state in the α, α† basis): matrix in which each diagonal entry is 1, and n is a diagonal matrix. Qn and Rn are well-conditioned matrices, † † Tr eξ Cξ = ψ|eξ Cξ |ψ while Dn contains the scales of the problem. Consequently, ψ the eigenvalues of Qn and Rn can be computed stably. We use these eigenvalues, together with the diagonal entries of Dn, k> (λk d˜k dk −λk¯ d˜k¯ dk¯ +λk¯ ) = ψ|φφ¯|e 0 |φ φ¯ |ψ to find det An. In practice, to avoid numerical overflow, we ψ,φ,φ compute the quantity λ d˜ d −λ d˜ d +λ = φ¯| e( k k k k¯ k¯ k¯ k¯ )|φ ln det An = ln det Qn + ln det Dn + ln det Rn. (B4) φ k>0

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