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Quantum number projection at finite temperature via thermo field dynamics K. Tanabe∗ Department of Physics, Faculty of Science, Saitama University, Sakura, Saitama 338-8570, Japan H. Nakada† Department of Physics, Faculty of Science, 5 0 Chiba University, Inage, Chiba 263-8522, Japan 0 2 (Dated: February 8, 2008) n Abstract a J Applying the thermo field dynamics, we reformulate exact quantum number projection in the 2 1 finite-temperature Hartree-Fock-Bogoliubov theory. Explicit formulae are derived for the simulta- 2 neous projection of particle number and angular momentum, in parallel to the zero-temperature v case. We also propose a practical method for the variation-after-projection calculation, by approx- 4 5 imating entropy without conflict with the Peierls inequality. The quantum number projection in 0 0 the finite-temperature mean-field theory will be useful to study effects of quantum fluctuations 1 associated with the conservation laws on thermal properties of nuclei. 4 0 / h PACS numbers: 21.60.Jz,21.10.-k, 21.90.+f,05.30.Fk t - l c u n : v i X r a ∗Electronic address: [email protected] †Electronic address: [email protected] 1 I. INTRODUCTION Transitions among different phases have been observed as temperature increases or de- creases, in wide variety of physical systems from elementary particles to condensed matters. They have attracted great interest both experimentally and theoretically, and profound physics content has been revealed particularly in infinite systems. Despite an isolated system, statistical properties of atomic nuclei have been investigated in terms of the so-called nuclear temperature [1], providing us with a ground to study phase transitions in nuclei. However, it is not straightforward to identify phase transitions in finite systems like nuclei, because the quantum fluctuations associated with finiteness of the system obscure the transitions [2, 3] so that even definition of ‘phases’ should depend on theoretical models to a considerable extent. It will be natural, from theoretical points of view, to define phases in nuclei via the mean- field picture, since ‘phases’ in physical systems are more or less a semi-classical concept. Indeed, we discuss spherical/deformed and normal-fluid/superfluid phases in nuclei based on the mean-field approximation. Phase transitions are often connected to the spontaneous symmetry breaking, leading to violation of conservation laws. In the mean-field picture of nuclei, the rotational symmetry is broken in the deformed phase, and the particle number conservation is violatedin thesuperfluid phase. However, the symmetry breaking innuclei is a hypothetical effect due to the mean-field approximation. In practice, the conservation laws are restored via a sort of quantum fluctuations arising from correlations beyond the mean field. In studying statistical properties of finite systems, it will be significant to investigate effects of the symmetry restoration. As the most general mean-field theory, we shall primarily consider the Hartree-Fock- Bogoliubov (HFB) approximation. To restore the symmetries, projection with respect to the quantum numbers should be applied [4]. The particle number projection as well as the angular momentum projection methods have been developed for zero-temperature cases; i.e. for the ground-state wave functions [5, 6], and also for the quantum-number-constrained HFB (CHFB) solutions along the yrast line [7, 8, 9]. For finite-temperature problems, projections with respect to the discrete symmetries such as theparity and the number-parity were explored more than two decades ago [10]. However, there is a certain complication in extending them to the continuous symmetries, arising from the non-commutability between the projection operator and the Boltzmann-Gibbs operator in the mean-field approximation. A way to overcome these problems has been found in Ref. [11] for the particle number projection within the Bardeen-Cooper-Schrieffer (BCS) approximation. General projection formalism in the variation-before-projection (VBP) scheme is developed in Ref. [12], and is extensively applied in the framework of the static-path approximation. In this paper we reformulate the quantum number projection at finite temperature by employing the thermo field dynamics (TFD) [13, 14]. The TFD has been shown to be a powerful tool [15] to handle the thermal fluctuations within the mean-field theories. We 2 shall show that this is also true for the quantum-number-projected HFB theory at finite temperature. While the resultant formulae are equivalent to those in Ref. [12], the present formalism is advantageous in the following respects. In the TFD, the thermal expectation value of an observable is expressed in terms of a vacuum expectation value in an enlarged TFD space. Therefore explicit formulae are derived in a straightforward manner, keeping complete parallel to the zero-temperature formalism [16]. We demonstrate it for the simultaneous projection both of the particle number and the angular momentum. This leads to another advantage that variational parameters can be identified in an obvious manner and handled easily. While application of the present method to the VBP calculations is straightforward, further approximation is desired in order to carry out the variation-after-projection (VAP) calculations. We shall also discuss an approximation of the entropy, so as for the Peierls inequality [17] to hold which guarantees the variational principle for free energy. It is expected that this approximation scheme will make the VAP calculations practical. In Sec. II, application of the thermo field dynamics (TFD) to the HFB theory at finite temperature is presented. In Sec. III, we formulate the TFD version of the quantum-number projection in the HFB at finite temperature. As well as general arguments, explicit formulae for the particle number and angular momentum projection are presented by taking specific basis-sets. Formulae in the BCS approximation are also given. In Sec. IV, we propose an approximation scheme for the entropy, which keeps the Peierls inequality. Finally, the paper is summarized in Sec. V. II. FINITE-TEMPERATURE HFB THEORY AND THERMO FIELD DYNAM- ICS A. HFB equation at finite temperature In the variational derivation of the Hartree-Fock-Bogoliubov (HFB) equation at finite temperature [10, 18], a set of variational parameters U ,V ,U∗ ,V∗ is introduced { kµ kµ kµ kµ} through the generalized Bogoliubov transformation (GBT), which relates the original single- particle (s.p.) operators c ,c† to those in the quasiparticle (q.p.) picture α ,α† , { k k} { µ µ} c α U V∗ k = W µ , W kµ kµ . (1) c† kµ α† kµ ≡ V U∗ k ! µ µ ! kµ kµ ! X The transformation matrix W obeys the unitarity relation W†W = WW† = 1. We express † the original s.p. state as i = c vac. , with the vacuum vac. satisfying c vac. = 0. The | i i| i | i i| i q.p. vacuum 0 Π α vac. is annihilated by α , i.e. α 0 = 0. Note that 0 0 = 1. ν ν µ µ | i ∝ | i | i h | i An additional set of variational parameters E comes into the problem through the trial µ { } 3 statistical operator e−Hˆ0/T wˆ = ; Hˆ = E α†α , (2) 0 Tr(e−Hˆ0/T) 0 µ µ µ µ X where we put theBoltzmann constant k = 1. We denotethe thermalaverage ofanoperator B ˆ ˆ ˆ O by O = Tr(wˆ O), where the trace is taken over the grand canonical ensemble. 0 h i We consider the Hamiltonian comprised of up to two-body interactions, 1 Hˆ = ε c†c + v c†c†c c , (3) i i i 4 ijkl i j l k i ijkl X X where two kinds of nucleons are not discriminated to avoid unnecessary complication. The hermiticity of the Hamiltonian implies ε = ε∗ and v = v = v = v∗ . In the i i ijkl − jikl − ijlk klij constrained HFB (CHFB) approximation at finite temperature, the grand potential is given by Ω = Hˆ′ TS; Hˆ′ Hˆ λ Zˆ λ Nˆ ω Jˆ , (4) p n rot x h i− ≡ − − − where λ (λ ) and ω are Lagrange multipliers, which are interpreted as proton (neu- p n rot tron) chemical potential and rotational frequency of the system, respectively. The proton- (neutron-) number operator Zˆ (Nˆ) and the x-component of angular momentum operator Jˆ x are expressed in terms of the s.p. operators as Zˆ = c†c , Nˆ = c†c , Jˆ = i Jˆ j c†c , (5) i i i i x h | x| i i j i∈p i∈n ij(all) X X X where the sum ( ) extends over the proton (neutron) states, and over i∈p i∈n ij(all) both proton and neutron states. In the CHFB the conservations of proton number, neutron P P P number and angular momentum are taken into account in their averages, Zˆ = Z, Nˆ = N, Jˆ = J(J +1), (6) x h i h i h i p which indicate constraints. In calculations without some of the constraints, the associated Lagrange multipliers are set to be zero in Eq. (4). The auxiliary Hamiltonian in Eq. (4) is rewritten in terms of the q.p. operators by the GBT (1), Hˆ′ = U +Hˆ +Hˆ +Hˆ +Hˆ +Hˆ . (7) 0 11 20 22 31 40 For later convenience, explicit forms in the above expression are presented in Appendix A. Ensemble averages of the bilinear forms in the q.p. operators are given by 1 α†α = f δ ; f = , (8a) h µ νi µ µν µ eEµ/T +1 α α = α†α† = 0, (8b) h µ νi h µ νi 4 where f is the occupation number of the µ-th q.p. level. Thus, the variation of the µ parameter δE is converted to δf = f (1 f )δE /T. The approximate entropy S in µ µ µ µ µ − − Eq. (4) is expressed as S = Tr(wˆ lnwˆ ) = [f lnf +(1 f )ln(1 f )]. (9) 0 0 µ µ µ µ − − − − µ X The s.p. density ρ and the pair tensor κ are defined by ρ = c†c = [V∗(1 f)Vtr +UfU†] , (10a) ij h j ii − ij κ = c c = [V∗(1 f)Utr +UfV†] . (10b) ij j i ij h i − The ensemble average of Hˆ′ becomes 1 1 Hˆ′ = Tr ξρ+ Γρ+ ∆κ† = U + (H ) f +2 (H ) f f , (11) s.p. 0 11 µµ µ 22 µνµν µ ν h i 2 2 (cid:18) (cid:19) µ µν X X where Tr stands for the trace over the s.p. space, s.p. ξ = (ε λ )δ ω i Jˆ j (12) ij i τ ij rot x − − h | | i with τ = p (n) for proton (neutron), while the HF potential matrix Γ and the pair potential matrix ∆ are defined by Γ = (Γ†) = v ρ , (13a) ij ij ikjl lk kl X 1 ∆ = ∆ = v κ . (13b) ij ji ijkl kl − 2 kl X The variational principle δΩ = δ( Hˆ′ TS) = 0 yields [10] h i− (Heff) U†(ξ +Γ)U V†(ξ +Γ)∗V +U†∆V V†∆∗U 11 µν ≡ − − µν = (cid:2)(H ) +4 (H ) f (cid:3) 11 µν 22 µρνρ ρ ρ X ∂S = T δ = E δ , (14a) µν µ µν ∂f µ (Heff) U†(ξ +Γ)V∗ V†(ξ +Γ)∗U∗ +U†∆U∗ V†∆∗V∗ 20 µν ≡ − − µν = (cid:2)(H ) +6 (H ) f = 0. (cid:3) (14b) 20 µν 31 µνρρ ρ ρ X These equations are summarized in the form of the HFB coupled equation at finite temperature, ξ +Γ ∆ U U = E . (15) ∆∗ (ξ +Γ)∗ V V µ ! ! ! Xl − − kl lµ kµ In the CHFB case this equation should be solved together with the constraints in Eq. (6), by which the Lagrange multipliers λ , λ and ω , as well as self-consistent q.p. energies p n rot E , are determined as functions of quantum numbers and temperature. µ 5 B. Bogoliubov transformation extended by TFD In the thermo field dynamics (TFD) [13, 14], the original q.p. operator space α ,α† is { µ µ} enlarged by including newly introduced tilded operators; α ,α˜ ,α†,α˜† . Correspondingly, { µ µ µ µ} the q.p. vacuum in the enlarged space is defined by 0 ˜0 , where ˜0 is the tilded vacuum | i⊗| i | i satisfying α˜ ˜0 = 0. The TFD vacuum 0 ˜0 is hereafter denoted by 0 for the sake µ | i | i ⊗ | i | i of simplicity, i.e. α 0 = α˜ 0 = 0. In order to handle ensemble averages in a thermal µ µ | i | i equilibrium at temperature T, the TFD vacuum 0 is transformed to the temperature- | i dependent vacuum by the following unitary transformation, 0 = e−Gˆ 0 , Gˆ = Gˆ† = ϑ (α†α˜† α˜ α ), (16) | Ti | i − µ µ µ − µ µ µ X where sinϑ = f1/2 g , cosϑ = (1 f )1/2 g¯ . (17) µ µ ≡ µ µ − µ ≡ µ This unitary transformation relates the operator set α ,α˜ ,α†,α˜† to new set of the { µ µ µ µ} temperature-dependent operators β ,β˜ ,β†,β˜† , { µ µ µ µ} α = eGˆβ e−Gˆ = g¯ β +g β˜†, µ µ µ µ µ µ α˜ = eGˆβ˜ e−Gˆ = g¯ β˜ g β†, (18) µ µ µ µ − µ µ ˜ so that the temperature-dependent vacuum should fulfill β 0 = β 0 = 0. The µ T µ T | i | i temperature-dependent quasiparticles created by β† or β˜† will hereafter be called TFD quasiparticles. The thermal average of an observable Oˆ is then expressed by the vacuum expectation value at 0 , T | i Oˆ = Tr(wˆ Oˆ) = 0 Oˆ 0 . (19) 0 T T h i h | | i Now we unify two unitary transformations (1) and (18) to compose an extended form of the GBT [15], c β c˜ β˜ U¯ V¯∗   = W¯   , W¯ = , (20) c† kµ β† kµ V¯ U¯∗ ! c˜†  Xµ β˜†  kµ  k  µ     where Ug¯ V∗g Vg¯ U∗g U¯ = , V¯ = . (21) kµ V∗g Ug¯ kµ U∗g Vg¯ ! ! − kµ − kµ Because of the unitarity relation W¯ W¯ † = W¯ †W¯ = 1, the matrix inverse to W¯ is given by U¯† V¯† W¯ −1 = W¯ † = . (22) V¯tr U¯tr ! 6 III. QUANTUM NUMBER PROJECTION IN FINITE-TEMPERATURE HFB THEORY A. Projection operators 1. General form Since quantum numbers are usually associated with a certain group structure of the system, projection with respect to the quantum numbers is also introduced in connection to the group. In general, a projection operator Pˆ(α) is defined as a subring basis corresponding µν to an irreducible representation (irrep.) ̺(α) of a group [19, 20], G dim(̺(α)) Pˆ(α) = ̺(α)(x−1)x, (23) µν g νµ x∈G X where x stands for an element of , dim(̺(α)) is the dimension of the representation matrix G ̺(α), and g the order of . The property of group representation yields G Pˆµ(αν)Pˆµ(β′ν)′ = δαβδνµ′Pˆµ(αν′), (24) justifying a convenient bracket representation of the projection operator, Pˆ(α) = αµ αν . µν | ih | The projection on the subspace corresponding to the irrep. ̺(α), without referencing µ, is obtained by Pˆ = Pˆ(α) = αµ αµ . (25) α µµ | ih | µ µ X X It is obvious that Pˆ is idempotent, i.e. Pˆ2 = Pˆ . α α α When is not simple and is decomposed into a direct product of an invariant subgroup G and the complementary quotient group as , its irrep. ̺(α) is also a product such as 1 2 G ⊗G ̺(α1) ̺(α2), where ̺(αi) is an irrep. of (i = 1,2). Denoting the projection operator on 1 ⊗ 2 i Gi ̺(αi) by Pˆ , the projection operator on ̺(α) is Pˆ = Pˆ Pˆ . Therefore, projection operators i αi α α1 α2 of simple groups are essential. In the following we assume to be a compact simple group or a product of such groups, G whoseelement xisrepresented byanappropriateunitaryoperatorQˆ = e−iSˆ. HereSˆ = Sˆ(Θ) is a hermitian operator, which belongs to a Lie algebra associated with and is dependent G on a set of parameters Θ. The projection operator Pˆ is represented in the integral form α Pˆ = dΘζ (Θ)Qˆ, (26) α α Z where ζ (Θ) is an appropriate function of Θ and is derived from Eq. (23). α 7 2. Angular momentum projection operator For the angular momentum projection, the relevant group is SU(2), which is denoted by SU(2) in this paper. The group element is the rotation operator Rˆ, whose parameters are J represented by Φ. In practice, we consider two alternative parameterizations. One is the Euler angles Φ = (α,β,γ). The rotation is also represented by an angle ω (0 ω < 2π) ≤ around an axis indicated by a unit vector ~n. Using the representation n = sinθcosφ,n = x y sinθsinφ,n = cosθ, we arrive at the other parameterization Φ = (ω,θ,φ). Corresponding z to these parameterizations, the rotation operator is expressed in two ways as Rˆ(Φ) = e−iαJˆze−iβJˆye−iγJˆz = e−iω(nxJˆx+nyJˆy+nzJˆz), (27) where the angular momentum operators are defined in terms of the s.p. operators as in Eq. (5), Jˆ = i Jˆ j c†c (k = x,y,z). (28) k h | k| i i j ij(all) X The angles (ω,θ,φ) are related to the Euler angles by ω β α+γ β α+γ π +α γ cos = cos cos , tanθ = tan sin , φ = − . (29) 2 2 2 2 2 2 (cid:30) The order of the group is determined by the volume of the parameter space, 2π π 4π 2π ω π 2π g = dα sinβdβ dγ = 4 sin2 dω sinθdθ dφ = 16π2, (30) 2 Z0 Z0 Z0 Z0 Z0 Z0 where the range of γ is taken to be 0 γ < 4π for the double-valued representation ≤ corresponding to half-odd-integer spin. The irrep. of SU(2) is given by the Wigner D- J function [21, 22], DJ (Φ) JM Rˆ(Φ) JK = e−iαMdJ (β)e−iγK , (31) MK ≡ h | | i MK whichisaunitarymatrixofdimension(2J+1). ThenEq.(23)yields thefollowingprojection operator, 2J +1 2π π 4π PˆJ = PˆJ† = dα sinβdβ dγDJ∗ (Φ)Rˆ(Φ). (32) MK MK 16π2 MK Z0 Z0 Z0 4π 2π For integer spin J, the integral dγ may be replaced by 2 dγ. 0 0 We express the character of the representation matrix DJ (Φ) as R MRK J sin[(J +1/2)ω] χ (ω) = DJ (Φ) = . (33) J MM sin(ω/2) M=−J X The real function χ (ω) can be represented in many different ways as listed in Ref. [22]. If J the magnitude of angular momentum J is subject to the projection while its z-component in the laboratory frame M is not referenced, the projection operator is given by J 2J +1 2π ω π 2π Pˆ = Pˆ† PˆJ = sin2 dω sinθdθ dφχ (ω)Rˆ(Φ). (34) J J ≡ MM 4π2 2 J M=−J Z0 Z0 Z0 X 8 3. Particle number projection operators The relevant group to the particle number projection is the gauge group U(1), whose group elements are parameterized by a single variable ϕ for the particle number operator ˆ ˆ N; exp( iϕN). For neutrons, the group is denoted by U(1) , and the projection operator, N − which projects out states having the exact neutron number N, is given by, 1 2π Pˆ = Pˆ† = e−iϕn(Nˆ−N)dϕ . (35) N N 2π n Z0 Likewise, the proton number projection is implemented by the operator, 1 2π Pˆ = Pˆ† = e−iϕp(Zˆ−Z)dϕ , (36) Z Z 2π p Z0 and the relevant group to Pˆ is denoted by U(1) . The product operator Pˆ Pˆ is employed Z Z Z N for the U(1) U(1) projection, by which the canonical trace is calculable from the grand- Z N × canonical trace. 4. SU(2) U(1) U(1) projection operator J Z N × × The projector of simultaneous projection of angular momentum J, proton number Z and neutron number N is Pˆ = Pˆ Pˆ Pˆ , which satisfies Pˆ2 = Pˆ . This Pˆ (J,Z,N) J Z N (J,Z,N) (J,Z,N) (J,Z,N) can be expressed in the form of Eq. (26), with Θ = (Φ,ϕ ,ϕ ). The operator Qˆ stands for p n Qˆ = Rˆ(Φ)e−iϕpZê−iϕnNˆ = e−iSˆ, (37) where ~ ~ Sˆ = ϕ Zˆ +ϕ Nˆ +ω~n Jˆ; ~n Jˆ n Jˆ +n Jˆ +n Jˆ . (38) p n x x y y z z · · ≡ ˆ ˆ Obviously P commutes with the totalnuclear Hamiltonian H, which conserves angular (J,Z,N) momentum, nucleon number and charge. However, note that [Pˆ ,Hˆ ] = 0 [12]. (J,Z,N) 0 6 5. Parity and number-parity projection operators The space reflection forms the discrete group isomorphic to S . The projection operator 2 with respect to the space parity can be expressed in the form similar to Eq. (26), with the integral converted to a discrete sum, − 1 Pˆ = 1+ξexp iπ c†c . (39) π 2 k k (cid:20) k (cid:21) (cid:0) X (cid:1) 9 − Here ξ = +1 (ξ = 1) for the projection of positive (negative) parity state, and denotes − k the sum extending only over the s.p. states of negative parity. As far as the parity is not P mixed in the q.p. state µ, the projection operator is also written as − 1 Pˆ = 1+ξexp iπ α†α , (40) π 2 µ µ (cid:20) µ (cid:21) (cid:0) X (cid:1) as given in Ref. [10]. The number-parity is relevant to another S group, which is a discrete subgroup of U(1) 2 Z or of U(1) . Respective to protons and neutrons, the number-parity projection is carried N out by the operator 1 Pˆ = 1+ηexp iπ c†c , (41) q 2 k k (cid:20) k (cid:21) (cid:0) X (cid:1) where η = +1 (η = 1) for the projection of the state of even (odd) number-parity. Apart − from the denominator, this is obtained by restricting ϕ in Pˆ (or ϕ in Pˆ ) only to 0 and π. n N p Z ˆ Since the GBT (1) does not mix the number-parity, P is equivalently represented in terms q of the q.p. operators [10], 1 Pˆ = 1+ηexp iπ α†α . (42) q 2 µ µ (cid:20) µ (cid:21) (cid:0) X (cid:1) B. Extended form of linear transformation We now consider the projected statistics. The projected ensemble average of an operator Oˆ is newly defined by Tr(Pê−βHˆ0PÔˆPˆ) Tr(e−βHˆ0PÔˆPˆ) Oˆ Tr(wˆ Oˆ) = = , (43) h iP ≡ P Tr(Pê−βHˆ0Pˆ) Tr(e−βHˆ0Pˆ) where the statistical operator with projection is introduced as Pê−βHˆ0Pˆ wˆ . (44) P ≡ Tr(e−βHˆ0Pˆ) For an operator commutable with Pˆ, Oˆ is calculated in the TFD by P h i Tr(wˆ OˆPˆ) 0 OˆPˆ 0 Oˆ = 0 = h T| | Ti. (45) h iP Tr(wˆ Pˆ) 0 Pˆ 0 0 T T h | | i Substituting the projector by the form of Eq. (26), we obtain ˆ ˆ ζ(Θ) 0 OQ 0 dΘ Oˆ = h T| | Ti . (46) h iP ζ(Θ) 0 Qˆ 0 dΘ R T T h | | i R 10