Evaluation of overlaps between arbitrary Fermionic quasiparticle vacua
EEvaluation of overlaps between arbitrary Fermionic quasiparticle vacua
B. Avez and M. Bender
Universit´e Bordeaux 1, CNRS/IN2P3, Centre d’ ´Etudes Nucl´eaires de Bordeaux Gradignan,Chemin du Solarium, BP120, 33175 Gradignan, France (Dated: November 25, 2018)We derive an expression that allows for the unambiguous evaluation of the overlap between twoarbitrary quasiparticle vacua, including its sign. Our expression is based on the Pfaffian of a skew-symmetric matrix, extending the formula recently proposed by [L. M. Robledo, Phys. Rev. C ,021302(R) (2009)] to the most general case, including the one of the overlap between two differentblocked n -quasiparticle states for either even or odd systems. The powerfulness of the method isillustrated for a few typical matrix elements that appear in realistic angular-momentum-restoredGenerator-Coordinate Method calculations when breaking time-reversal invariance and using thefull model space of occupied single-particle states. I. INTRODUCTION
The evaluation of kernels in projection and more gen-eral applications of the Generator Coordinate Method(GCM) based on quasiparticle vacua requires the calcu-lation of the overlap between two different quasiparticlevacua. Its evaluation presents a long-standing technicalchallenge: the standard expression for this overlap, theso-called ”Onishi formula” [1] provides the square of the(complex) overlap only. As a consequence, the overallsign of the overlap is not determined, or, equivalently, itsphase is determined up to integer multiples of π only.One possible solution to the problem was proposed byNeerg˚ard and W¨ust [2], but the practical application oftheir technique becomes cumbersome in realistic appli-cations and has been rarely used in practice. Notableexceptions are Refs. [3, 4]. Many groups have resided todetermine the phase through a kind of Taylor expansionthat allows ”to follow the overlap” when the kernel canbe connected in small steps to a known reference over-lap (see, for example, Refs. [5, 6]). In practice this canbecome very cumbersome or even impossible when thephase is rapidly changing or when there is no symmetrythat establishes a reference phase.It was pointed out by Robledo in Ref. [7] that tech-niques for the manipulation of matrix elements betweenFermionic coherent states that are well-known in fieldtheory allow to express the overlap between two quasipar-ticle vacua, including its sign, as the so-called Pfaffian ofa skew-symmetric matrix. In a more recent paper, Rob-ledo [9] has also worked out the practical implementationof this idea for the unambiguous evaluation of the over-lap between fully-paired quasiparticle vacua, includingthe limit where some of the ”pairs” consist of fully occu-pied single-particle states. Even more recently, Bertschand Robledo [10] also investigated the case of systemswith an odd number of constituants, providing the un- The possibility to use Pfaffians for this purpose was alreadyconjectured much earlier by Balian and Br´ezin in Ref. [8], butnever worked out. ambiguous evalution of the overlap for the special case oftwo quasiparticle vacua linked by a symmetry transfor-mation.In the present paper, we present the extension of thisscheme to the calculation of the overlap between two pos-sibly different arbitrary quasiparticle vacua, generalizingthe treatment of completely filled single-particle states tothe most general case. This extension thus allows to han-dle quasiparticle vacua obtained from blocked 1-, 2-, . . . n arbitrary quasiparticle states. To this aim, we present analternative derivation of the overlap that makes use of anextension of the standard Thouless parameterization ofquasiparticle vacua [11] that is advantageous in the pres-ence of completely filled single-particle states and allowsto avoid many of the matrix manipulations elaborated inRef. [9]. Also, our final expression Eq. (63) allows forthe calculation of the overlap of two quasiparticle vacuathat are expressed in different single-particle bases thatdo not span the same sub-space of the Hilbert space ofsingle-particle states, a situation frequently encounteredin symmetry-restored GCM codes that use a coordinatespace representation of the quasiparticle vacua in termsof their canonical single-particle bases [12–16].The article is organized as follows: Section II intro-duces a generalization of the Thouless parameterizationof quasiparticle vacua for blocked states that will turnout to be useful for the purpose of our paper. Sec-tion III reviews key properties of Fermionic coherentstates and Grassmann calculus that will be needed lat-eron and thereby introduces our notation. Section IVdescribes the calculation of the overlap, and Section Vpresents some illustrative examples of overlaps from re-alistic calculations. Finally, Section VI summarizes ourfindings. An appendix gives the representation of de-terminants and Pfaffians of skew-symmetric matrices interms of integrals over Grassmann variables. II. PARAMETERIZATION OFQUASIPARTICLE VACUA
Let { ˆ a † , ˆ a } and { ˆ b † , ˆ b } be two not necessarily equiva-lent single-particle bases of dimension n ( n even ), with a r X i v : . [ nu c l - t h ] S e p which are defined two (not normalized) quasiparticlevacua | φ a (cid:105) and | φ b (cid:105) through the parameterizations | φ c (cid:105) = e (cid:80) kl M ( c ) kl ˆ c † k ˆ c † l (cid:46) (cid:89) i ∈{ I c } ˆ c † i |−(cid:105) , (1)where c is either a or b , |−(cid:105) is the bare vacuum ofsingle-particle operators, and where the M ( c ) are skew-symmetric matrices. Their elements with indices belong-ing to the ensemble of fully occupied states { I c } of car-dinality I c = r c are (can be) put to zero. This consti-tutes a natural way to regularize the matrix M ( c ) in thepresence of fully-occupied states.The triangle pointing to the right on top of the productsign means that it is a ”direct” product, (cid:46) (cid:89) i ∈{ I c } ˆ c † i = ˆ c † µ · · · ˆ c † ν ˆ c † δ , with µ > · · · > ν > δ , (2)as opposed to a ”reverse” product obtained, for example,by taking the adjoint of Eq. (2), i.e. (cid:47) (cid:89) i ∈{ I c } ˆ c i = ˆ c δ ˆ c ν · · · ˆ c µ , with µ > · · · > ν > δ . (3)The parameterization Eq. (1) of quasiparticle vacua isnot very different from the one by Thouless [11]. How-ever, it has two advantages important for our purpose.First, this parameterization is well-defined when dealingwith fully occupied states. And second, it allows to pa-rameterize a quasiparticle vacuum for systems with oddparticle number in terms of single-particle states, some-thing that cannot be achieved with the standard Thou-less formula. However, this parameterization can be set-up only in a specific single-particle basis that separatesthe fully-occupied single-particle states from the others.Such a single-particle basis is, for example, the canonicalsingle-particle basis of a quasiparticle vacuum. The useof these bases does not impose a serious restriction, sincethey provide the most compact representation of a quasi-particle vacuum, such that their use is often desirable innumerical applications.Finally, the convention for the ordering of single-particle levels in Eq. (1) (matrix elements of M ( c ) andproduct ordering) is to be kept unchanged for each calcu-lation involving a given state | φ c (cid:105) . In fact, Eq. (1) is justan alternative convention that circumvents the conven-tion of Ref. [7] to connect the phase of (cid:104) φ a | φ b (cid:105) to (cid:104) φ a |−(cid:105) and (cid:104)−| φ b (cid:105) , which cannot be achieved when either (cid:104) φ a |−(cid:105) or (cid:104)−| φ b (cid:105) (or both) is (are) zero. For time being, n will be the minimal number of single-particlestates that allows to represent both | φ a (cid:105) and | φ b (cid:105) in their respec-tive single-particle basis. The cardinality is the number of elements of an ensemble.
III. A REMINDER ON FERMIONICCOHERENT STATES AND GRASSMANNCALCULUS
In order to evaluate the overlap (cid:104) φ a | φ b (cid:105) between thetwo states, we introduce, following closely Ref. [7], twosets of Fermionic coherent states | z c (cid:105) = e ˆ c † . z c |−(cid:105) (4) (cid:104) z c | = (cid:104)−| e ¯ z c . ˆ c (5)for c = a, b , parameterized in terms of anticommuting z c k and ¯ z c k elements of a Grassmann algebra G , wherethe notations ˆ c † . z c and ¯ z c . ˆ c used in Eq. (4) and Eq. (5)stand for ˆ c † . z c = n (cid:88) i =1 ˆ c † i z c i , (6) ¯ z c . ˆ c = n (cid:88) i =1 ¯ z c i ˆ c i . (7)In particular, we notice that the coherent states | z c (cid:105) arenot normalized. Instead, one has (cid:104)−| z c (cid:105) = (cid:104) z c |−(cid:105) = 1.In what follows, we recall some useful propertiesof Grassmann algebra, its associated calculus, and ofFermionic coherent states that will be needed for the for-mal derivations outlined below. Concerning Grassmannalgebra and calculus [17–19], we recall that • The adjoint operator performs a one-to-one map-ping within G ( z c k ) † = ¯ z c k , (8)(¯ z c k ) † = z c k , (9)( z c k z c l ) † = ¯ z c l ¯ z c k . (10) • Grassmann variables anticommute z c k z c l = − z c l z c k , (11)¯ z c k ¯ z c l = − ¯ z c l ¯ z c k , (12) z c k ¯ z c l = − ¯ z c l z c k , (13) z c k z c k = ¯ z c k ¯ z c k = 0 . (14)In the following, a product of p Grassmann vari-ables ( G -variables) will be called a monomial of de-gree p . When p is even (odd), such a product will becalled an even (odd) monomial. We notice that aneven monomial of G -variables commutes with evenand odd monomials of G -variables. We also remarkthat exponentials of pairs of G -variables also com-mute with even and odd monomials of G -variables,such an exponential being a sum of even monomialsof G -variables. • G -variables commute with complex numbers andanticommute with Fermionic operators. • The fundamental Grassmann calculus rules are (cid:90) dz c k z c k = ∂∂z c k z c k = 1 , (15) (cid:90) dz c k = 0 . (16) • As a consequence of Eq. (14), the G -variables z c k play the role of their own δ -functions, e.g., for ananalytic function f of G -variables (c.f. [19], p. 91),we have (cid:90) dz c k z c k f ( z c k ) = f (0) . (17) • The adjoint variables ¯ z c k and z c k are independentintegration variables (c.f. [17], p. 28).Concerning Fermionic coherent states, the propertiesto be used in what follows are: • They are eigenstates of second quantized operatorsˆ c k | z c (cid:105) = z c k | z c (cid:105) , (cid:104) z c | ˆ c † k = (cid:104) z c | ¯ z c k , (18)with the eigenvalues being Grassmann variables. • They resolve the identity through the closure rela-tion c = (cid:90) d n(cid:46) ¯ z c d n(cid:47) z c | z c (cid:105) e − ¯ z c . z c (cid:104) z c | , (19)where c means that the resolution of the iden-tity is built for Fock spaces generated by { ˆ c † , ˆ c } .We use the short-hand notation ¯ z c . z c ≡ ¯ z tc z c = (cid:80) ni =1 ¯ z c i z c i . Finally, d n(cid:46) and d n(cid:47) represent prod-ucts of differential elements that are ordered suchthat d n(cid:46) x = dx n · · · dx dx , (20)d n(cid:47) x = dx dx · · · dx n . (21)As compared to [7] and many textbooks, we change theordering of the products of differential elements in theintegral in order to anticommute them in a more trans-parent way, i.e. we used n(cid:46) ¯ z c d n(cid:47) z c , (22)which is equivalent to the more commonly used ordering (cid:89) i d ¯ z c i dz c i . (23) In the commonly used ordering, there is no need to define aparticular product ordering as d ¯ z c i dz c i are even monomials of G -variables, and the overall order convention is carried only by d ¯ z c i dz c i . IV. EVALUATION OF THE OVERLAPSA. Preliminary considerations
To evaluate the expression for the overlap, we start byinserting two closure relations, the left (right) one beingbased on the single-particle basis of the left (right) state,e.g. (cid:104) φ a | φ b (cid:105) = (cid:104) φ a | a b | φ b (cid:105)(cid:104) φ a | φ b (cid:105) = (cid:90) d n(cid:46) ¯ z a d n(cid:47) z a d n(cid:46) ¯ z b d n(cid:47) z b ×(cid:104) φ a | z a (cid:105) e − ¯ z a . z a (cid:104) z a | z b (cid:105) e − ¯ z b . z b (cid:104) z b | φ b (cid:105) , (24)where we implicitely use that d n(cid:46) ¯ z c d n(cid:47) z c ( c = a , b ) areeven (2 n ) monomials of Grassmann differential elements,and thus commuting with Fermionic operators, in orderto move all differential elements to the very left.When the single-particle bases of a and b do not spanthe same subspace of the Hilbert space of single-particlestates, e.g. when they are not linked through a unitarytransformation, the resolution of the identity a ( b )works for the left (right) state alone, and the two clo-sure relations are not equivalent. However, there is noloss of generality for the following, their non-equivalencebeing carried by the overlap kernel (cid:104) z a | z b (cid:105) .Given that by definition (cid:104)−| z c (cid:105) = (cid:104) z c |−(cid:105) = 1, we firstevaluate the three overlaps (cid:104) φ a | z a (cid:105) = (cid:47) (cid:89) i ∈{ I a } z a i e − (cid:80) kl M ( a ) ∗ kl z ak z al , (25) (cid:104) z b | φ b (cid:105) = e + (cid:80) kl M ( b ) kl ¯ z bk ¯ z bl (cid:46) (cid:89) j ∈{ I b } ¯ z b j , (26) (cid:104) z a | z b (cid:105) = e (cid:80) kl ¯ z ak R kl z bl . (27)The two first expressions use the properties Eq. (18) ofcoherent states. The last one uses the Baker-Campbell-Hausdorff formula, such that (cid:104) z a | z b (cid:105) = (cid:104)−| e ¯ z a . ˆ a e ˆ b † . z b |−(cid:105) = (cid:104)−| e ˆ b † . z b e ¯ z a . ˆ a e (cid:104) ¯ z a . ˆ a , ˆ b † . z b (cid:105) |−(cid:105) . (28)Indeed, the latter is applicable because the commutator (cid:104) ¯ z a . ˆ a , ˆ b † . z b (cid:105) = (cid:88) ij ¯ z a i (cid:110) ˆ a i , ˆ b † j (cid:111) z b j (29)= (cid:88) ij ¯ z a i R ij z b j = ¯ z ta R z b , (30)commutes with ¯ z a . ˆ a and ˆ b † . z b , where R ij ≡ { ˆ a i , ˆ b † j } de-notes the matrix of overlaps of the single-particle states The Baker-Campbell-Hausdorff formula states that, if[ X, [ X, Y ]] = [ Y, [ Y, X ]] = 0, then e X e Y = e Y e X e [ X,Y ] . corresponding to ˆ a i and ˆ b † j . We finally obtain Eq. (27)by considering that |−(cid:105) is a common vacuum for the op-erators ˆ a and ˆ b . B. Integration of the reproducing kernel
As the next step, we integrate the expression for thereproducing kernel e − ¯ z a . z a (cid:104) z a | z b (cid:105) e − ¯ z b . z b = e − ¯ z a . z a + ¯ z ta R z b − ¯ z b . z b , (31)where we use that exponentials of pairs of G -variablescommute, thereby allowing to merge the three exponen-tial factors.Noticing that, in Eq. (24), ¯ z a and z b only appear in thereproducing kernel Eq. (31), we want to integrate these variables separately. In order to do so, we first remarkthat the expression Eq. (31) can be moved to the veryright of Eq. (24) because it commutes with G -variables.We can as well move the product of differential elementsd n(cid:46) ¯ z a d n(cid:47) z b in front of it by virtue of (cid:122) (cid:125)(cid:124) (cid:123) d n(cid:46) ¯ z a d n(cid:47) z a d n(cid:46) ¯ z b d n(cid:47) z b = ( − n d n(cid:47) z a d n(cid:46) ¯ z b d n(cid:46) ¯ z a d n(cid:47) z b = d n(cid:47) z a d n(cid:46) ¯ z b d n(cid:46) ¯ z a d n(cid:47) z b (cid:124) (cid:123)(cid:122) (cid:125) even product . (32)With d n(cid:46) ¯ z a d n(cid:47) z b being an even product of Grassmanndifferential elements, it commutes with (cid:104) φ a | z a (cid:105) and (cid:104) z b | φ b (cid:105) , quantities containing neither the variables ¯ z a nor z b , c.f. Eqns. (25) and (26). Rewriting Eq. (24) in a moresuitable way now gives (cid:104) φ a | φ b (cid:105) = (cid:90) d n(cid:47) z a d n(cid:46) ¯ z b (cid:104) φ a | z a (cid:105)(cid:104) z b | φ b (cid:105) (cid:90) d n(cid:46) ¯ z a d n(cid:47) z b e − ¯ z a . z a + ¯ z ta R z b − ¯ z b . z b (cid:124) (cid:123)(cid:122) (cid:125) reproducing kernel integral . (33)We now evaluate the reproducing kernel integral. Provided that R is non-singular, , we make the change of variables(c.f. Ref. [19] p. 14) ¯ η t = ¯ z ta − ¯ z tb R − η = z b − R − z a (34)such that ¯ η t R η = − ¯ z a . z a + ¯ z ta R z b − ¯ z b . z b + ¯ z tb R − z a . (35)The reproducing kernel can now be written e − ¯ z a . z a + ¯ z ta R z b − ¯ z b . z b = e − ¯ z tb R − z a e ¯ η t R η . (36)The Jacobian of the transformation being one, i.e. d n(cid:46) ¯ z a d n(cid:47) z b ≡ d n(cid:46) ¯ η d n(cid:47) η , the integration gives (cid:90) d n(cid:46) ¯ z a d n(cid:47) z b e − ¯ z a . z a + ¯ z ta R z b − ¯ z b . z b = e − ¯ z tb R − z a (cid:90) d n(cid:46) ¯ η d n(cid:47) η e ¯ η t R η = ( − n det ( R ) e − ¯ z tb R − z a (37)where we have used the determinant formula outlined in Eq. (A2) of Appendix A. The case of singular R is not equivalent of having zero overlap.As an example, consider the case of a partially or completelyempty single-particle state of the left vacuum which is orthogonalto all single-particle states of the right vacuum. In this case, R is singular, whereas the overlap is not necessarily zero. In case of singular R , one has to complete the single-particlebases of | φ a (cid:105) and | φ b (cid:105) in order to get an invertible matrix, forexample using a Gram-Schmidt orthonormalization procedure.Still, having a non-singular matrix R is not equivalent to havingequivalent bases a and b . C. Re-expression of the overlap
Using Eqns. (25), (26) and (37), we are now able to rewrite Eq. (33) as (cid:104) φ a | φ b (cid:105) = ( − n det ( R ) (cid:90) d n(cid:47) z a d n(cid:46) ¯ z b (cid:16) (cid:104) φ a | z a (cid:105)(cid:104) z b | φ b (cid:105) e − ¯ z tb R − z a (cid:17) (38)= ( − n det ( R ) (cid:90) d n(cid:47) z a d n(cid:46) ¯ z b (cid:47) (cid:89) i ∈{ I a } z a i (cid:46) (cid:89) j ∈{ I b } ¯ z b j e (cid:16) − (cid:80) kl M ( a ) ∗ kl z ak z al + (cid:80) kl M ( b ) kl ¯ z bk ¯ z bl − ¯ z tb R − z a (cid:17) , (39)where we used that exponentials of pairs of G -variables commute with G -variables. Closely following the notation ofRef. [7], we introduce the matrix M and the vector ζ M ≡ (cid:18) M ( b ) − R − (cid:0) R − (cid:1) t − M ( a ) ∗ (cid:19) , ζ ≡ (cid:18) ¯ z b z a (cid:19) , (40)such that 12 ζ t M ζ = 12 (cid:16) ¯ z tb M ( b ) z b + z ta R − t ¯ z b − ¯ z tb R − z a − z ta M ( a ) ∗ z a (cid:17) , (41)= 12 (cid:88) ij (cid:16) M ( b ) ij ¯ z b i ¯ z b j − M ( a ) ∗ ij z a i z a j (cid:17) − (cid:88) ij R − ij ¯ z b i z a j . (42)Inserting these definitions into Eq. (39), the overlap kernel becomes (cid:104) φ a | φ b (cid:105) = ( − n det ( R ) (cid:90) d n(cid:47) z a d n(cid:46) ¯ z b (cid:47) (cid:89) i ∈{ I a } z a i (cid:46) (cid:89) j ∈{ I b } ¯ z b j e ζ t M ζ . (43) D. Integration over fully-occupied states
We will now integrate variables corresponding to fully occupied states in Eq. (43) by virtue of Eq. (17). In orderto do so, we need to move the corresponding differential elements to the very right, which gives a sign factor becauseof the anticommutation of Grassmann differential elements. For example, such a rearrangement for a single variablein d n(cid:47) z c and d n(cid:46) ¯ z c gives d n(cid:47) z c = dz c · · · (cid:122)(cid:125)(cid:124)(cid:123) dz c k · · · dz c n = (cid:47) (cid:89) i (cid:54) = k dz c i ( − n − k dz c k , (44)d n(cid:46) ¯ z c = d ¯ z c n · · · (cid:122)(cid:125)(cid:124)(cid:123) d ¯ z c k · · · d ¯ z c = (cid:46) (cid:89) i (cid:54) = k d ¯ z c i ( − k − d ¯ z c k . (45)When there is more than one fully occupied state, the repeated application of this procedure givesd n(cid:47) z c = (cid:47) (cid:89) i/ ∈{ I c } dz c i (cid:46) (cid:89) k ∈{ I c } ( − n − k dz c k , (46)d n(cid:46) ¯ z c = (cid:46) (cid:89) i/ ∈{ I c } d ¯ z c i (cid:47) (cid:89) k ∈{ I c } ( − k − d ¯ z c k , (47)where in each equation we notice the opposite order for the products over indices corresponding to fully occupiedstates as compared to other states. Combining Eqns. (46) and (47) givesd n(cid:47) z a d n(cid:46) ¯ z b = (cid:47) (cid:89) i/ ∈{ I a } dz a i (cid:122) (cid:125)(cid:124) (cid:123) (cid:46) (cid:89) k ∈{ I a } ( − n − k dz a k (cid:46) (cid:89) j / ∈{ I b } d ¯ z b j (cid:47) (cid:89) l ∈{ I b } ( − l − d ¯ z b l (48)= ( − nr a (cid:47) (cid:89) i/ ∈{ I a } dz a i (cid:46) (cid:89) j / ∈{ I b } d ¯ z b j (cid:47) (cid:89) l ∈{ I b } ( − l − d ¯ z b l (cid:46) (cid:89) k ∈{ I a } ( − n − k dz a k . (49)The additional sign in Eq. (49) comes from the commutation of variables k ∈ { I a } to the very right, as indicated inEq. (48). Defining the sign factor σ ≡ ( − (cid:80) k ∈{ Ia } ( n + k )+ (cid:80) k ∈{ Ib } k , (50)Eq. (49) finally gives, after a suitable rearrangement of the sign factorsd n(cid:47) z a d n(cid:46) ¯ z b = ( − nr a + r b σ (cid:47) (cid:89) i/ ∈{ I a } dz a i (cid:46) (cid:89) j / ∈{ I b } d ¯ z b j (cid:47) (cid:89) l ∈{ I b } d ¯ z b l (cid:46) (cid:89) k ∈{ I a } dz a k . (51)The ordering of the differential elements corresponding to indices of fully occupied states are now in the appropriateorder with respect to their associated products in Eq. (43) to perform their integration.By virtue of Eq. (17), the integration over fully occupied levels of the integrand in Eq. (43) gives (cid:90) (cid:47) (cid:89) k ∈{ I b } d ¯ z b k (cid:46) (cid:89) k ∈{ I a } dz a k (cid:47) (cid:89) i ∈{ I a } z a i (cid:46) (cid:89) j ∈{ I b } ¯ z b j e ζ t M ζ = (cid:16) e ζ t M ζ (cid:17) ζ i =0 ∀ i ∈{ I b } ,n − i ∈{ I a } . (52)With this, Eq. (43) can be rewritten as (cid:104) φ a | φ b (cid:105) = ( − n ( − nr a + r b σ det ( R ) (cid:90) (cid:47) (cid:89) i/ ∈{ I a } dz a i (cid:46) (cid:89) j / ∈{ I b } d ¯ z b j (cid:16) e ζ t M ζ (cid:17) ζ i =0 ∀ i ∈{ I b } ,n − i ∈{ I a } , (53)where the integration only runs over variables of indices associated to not fully-occupied states. E. Integration over the remaining variables
We now define a new matrix M r , sub-matrix of M where rows and columns of indices i ∈ { I b } of fully occupied statesin | φ b (cid:105) and of incices j + n , where j ∈ { I a } are fully occupied levels in | φ a (cid:105) , have been removed. The correspondingappropriate vector ζ r is built from ζ in the same manner, removing components with indices i ∈ { I b } and j + n suchthat j ∈ { I a } . The matrix M r is skew-symmetric with dimension N r × N r , and the vector ζ r has N r elements, with N r = 2 n − ( r a + r b ).For an example where there are two fully occupied states in each quasiparticle vacuum, with indices i , k for thestate | φ b (cid:105) and indices j , l for the state | φ a (cid:105) , respectively, the matrix M r and the vector ζ r can be schematicallyrepresented as (54)(55) M r = M ( b ) − R − (cid:0) R − (cid:1) t − (cid:0) M ( a ) (cid:1) ∗ ζ r = ¯ z b z a i kki kilj j l lj (56)where labeled rows and columns have been removed from the original objects M and ζ . Using the submatrix M r andsubvector ζ r , Eq. (53) can be rewritten as (cid:104) φ a | φ b (cid:105) = ( − n ( − nr a + r b σ det ( R ) (cid:90) (cid:47) (cid:89) i/ ∈{ I a } dz a i (cid:46) (cid:89) j / ∈{ I b } d ¯ z b j (cid:16) e ζ tr M r ζ r (cid:17) . (57)In order to apply the Pfaffian formula Eq. (A4), the order of (cid:81) (cid:47)i/ ∈{ I a } dz a i has to be reversed, such that the differentialelements in Eq. (57) are in the appropriate order with respect to the matrix M r . This is achieved by reversing theorder of (cid:81) (cid:47)i/ ∈{ I a } dz a i (cid:47) (cid:89) i/ ∈{ I a } dz a i = ( − ( n − r a )( n − r a − / (cid:46) (cid:89) i/ ∈{ I a } dz a i . (58)The sign factor from the reversal of the product z · · · z n = ( − n ( n − / z n · · · z can be obtained by induction. Thedifferential elements are now in the appropriate orderd N r (cid:46) ζ r = (cid:46) (cid:89) i/ ∈{ I a } dz a i (cid:46) (cid:89) j / ∈{ I b } d ¯ z b j (59)= ( − ( n − r a )( n − r a − / (cid:47) (cid:89) i/ ∈{ I a } dz a i (cid:46) (cid:89) j / ∈{ I b } d ¯ z b j , (60)such that we can now integrate the remaining variables using the Pfaffian formula, Eq. (A4), (cid:104) φ a | φ b (cid:105) = ( − n ( − nr a + r b σ det ( R ) ( − ( n − r a )( n − r a − / (cid:90) d N r (cid:46) ζ r e ζ tr M r ζ r (61)= ( − n ( n +1) / ( − r a ( r a − / ( − r a + r b σ det ( R ) pf ( M r ) . (62)This formula, however, assumes N r to be even, see ap-pendix A. When N r is odd, | φ a (cid:105) and | φ b (cid:105) have in factdifferent number parity [20]. In that case, the overlap (cid:104) φ a | φ b (cid:105) is automatically zero. From the definition of thePfaffian, which by definition is zero for skew-symmetricmatrices of odd rank, we can thus notice that formulaEq. (57) can still be applied. In particular, ( − r a + r b will always be one except when multiplied by zero, al-lowing to drop this sign factor in the final formula. Wethus summarize the final expression for the overlap as (cid:104) φ a | φ b (cid:105) = s n s r a σ det ( R ) pf ( M r ) , (63)where s n = ( − n ( n +1) / , (64) s r a = ( − r a ( r a − / , (65) σ = ( − (cid:80) k =1 ..ra ( n + i ka )+ (cid:80) k =1 ..rb i kb . (66)The sign factors depend on the number of states inthe single-particle bases n , the number of fully occupiedstates r a in a , and the indices i k c of fully occupied statesin the bases c = a , b .Equation (63) provides the generalization of Eq. (7)of Ref. [7] to the overlap between different quasiparticle vacua with an arbitrary number of fully occupied single-particle states. In particular, it can be applied to over-laps that involve an odd number of blocked quasiparticlestates, a case not considered at all in Refs. [7, 9], and forthe special case of symmetry restoration only in Ref. [10].Moreover, when an even number of particles is fully oc-cupied (either for blocked 2 n quasiparticle states, or as aresult of the minimization, or both), Eq. (63) provides aformally justified alternative to the regularization of thematrix M performed in [9].Besides this important generalization, there is anothernoteworthy difference to previous work by Robledo [7, 9].Indeed, Eq. (63) is directly expressed in the single-particle bases of | φ a (cid:105) and | φ b (cid:105) , respectively, that allowfor the most compact representation of these quasipar-ticle vacua. In particular, Eq. (63) can also be appliedwithout invoking a complete single-particle basis span-ning the single particle subspace a ∪ b , cf. the discussionsabove. However, as explained there, if the matrix R issingular, one is forced to complete each single-particlebasis until det ( R ) (cid:54) = 0 is achieved.For quasiparticle vacua for which there are no fullyoccupied states in their respective single-particle basis,it is easy to show that Eq. (7) of Ref. [7] is recovered. I m ( h φ a | φ b i ) Re ( h φ a | φ b i ) FIG. 1: Real and imaginary parts of the overlap withoutparticle-number projection for the lowest one-quasiparticlestate in Mg obtained with SIII. The Euler angles α and β are held fixed at values of α = 1 . ◦ and β = 7 . ◦ , wheras γ is varied in the interval [0 , ◦ ] with a discretization of288 points. Filled circles on the curve represent a discretiza-tion of 48 points in the interval [0 , ◦ ], which is sufficient toconverge observables. Note the difference in scale of real andimaginary parts. I m ( h φ a | φ b i ) Re ( h φ a | φ b i ) FIG. 2: Same as Fig. 1, but for α = 43 . ◦ and β = 71 . ◦ . V. SOME ILLUSTRATIVE EXAMPLES
The determination of the phase of the overlap by thewidely used techniques that rely on a Taylor expansionof the overlap around a matrix element of known phase[5, 6, 14] works very well when restricting the calcula-tions to time-reversal invariant HFB states. However, itbecomes increasingly difficult when time-reversal is bro-ken, as it happens for odd- A or odd-odd nuclei, or forstates obtained with cranked HFB, cf. the examples dis-cussed in Refs. [5, 6, 21].We have implemented Eq. (63) into our numericalcodes for particle-number and angular-momentum re-stored GCM calculations based on triaxial HFB states[15, 16], using the routines for the calculation of the Pfaf- fian of Ref. [22]. We now present three examples wheretechniques to follow the overlap through Taylor expan-sion might fail and the direct calculation of the overlapbecomes a necessity. These illustrations will show trajec-tories in the complex plane of overlaps of quasi-particlevacua as obtained during angular-momentum projection (cid:104) φ a | φ b (cid:105) = (cid:104) ϕ | ˆ R ( α, β, γ ) | ϕ (cid:105) , (67)where ˆ R is the rotation operator that depends on thethree Euler angles α , β , and γ .The first two examples are presented in Figs. 1 and 2.They illustrate the trajectory of the overlap in the com-plex plane when varying the Euler angles γ when pro-jecting the lowest self-consistent one-quasiparticle stateof Mg, for two different combinations of α and β . Thefirst one, Fig. 1, illustrates that real and imaginary partsof the overlap can vary on quite different scales. In thisparticular case, most of the modulus of the overlap iscarried by the real part, and the phase of the overlapis most of the time either close to zero or close to ± π .Unless the discretization of the Euler angles is carefullyadapted, the phase of the overlap might change by al-most π when crossing the imaginary axis, which is verydifficult to distinguish from a discontinuity by π encoun-tered when having lost the phase. The second example,Fig. 2, obtained for a different combination of Euler an-gles α and β , shows that the trajectory of the overlap inthe complex plane may exhibit cusps, which might againbe difficult to resolve when discretizing Euler angles.In Fig. 3, we present the trajectory of the overlapof a high-spin state in Mg, obtained from crankedHFB+Lipkin Nogami [23]. The two inserts illustrate thatvariations may occur on very different scales with quiteinvolved structures.These three examples demonstrate that a Taylor-expansion-based algorithm to determine unambiguouslythe sign of the overlap may become difficult in applica-tions that use time-reversal invariance breaking quasi-particle vacua. Indeed, it should be able to resolve dis-continuities or cusps, or other involved structures. Thelatter might happen at very different scales, and the dis-cretization must be chosen in order to account for allthese details. Furthermore, we expect the complexityof such trajectories to increase with increasing intrinsicangular momentum. A direct determination of the over-lap is thus a considerable improvement not only from aformal point of view, but also from the perspective ofthe complexity of reliable algorithms for the computa-tion of the overlap on the one hand and of computingtime on the other hand, as it will often allow for the useof a smaller number of combinations of Euler angles inangular-momentum projection.
VI. DISCUSSION AND OUTLOOK
To summarize our main findings I m ( h φ a | φ b i ) Re ( h φ a | φ b i ) FIG. 3: Real and imaginary parts of the overlap withoutparticle-number projection for the cranked I = 8 (cid:126) state in Mg obtained with SIII. The Euler angles α and β are heldfixed at values of α = 23 . ◦ and β = 66 . ◦ , respectively,wheras γ is varied in the interval [0 , ◦ ] with a discretiza-tion of 144 points. The inserts amplify the zone at very smalloverlaps, whereas filled circles on the curves represent a dis-cretization of 24 points in the interval [0 , ◦ ], which is suf-ficient to converge observables.
1. An extension of the Thouless parameterization ofquasiparticle vacua with completely filled single-particle states allows to calculate the overlap di-rectly in a formalism based on Grassmann algebraand coherent states similar to the one outlined inRef. [7], but in such a manner that fully occupied orempty single-particle levels are automatically takencare of without any need for the manipulation (orregularization) of matrices elaborated in Ref. [9].2. This extension of the Thouless expression allowsto handle all possible quasiparticle vacua thathave completely filled states, i.e. also 1-, 2-, . . . n -quasiparticle states, not just quasiparticle vacuathat can be expressed as limits of fully-paired quasi-particle vacua as in Ref. [9].3. The handling of blocked states is not restricted topure symmetry restoration as the one proposed inRef. [10], and therefore can be also applied whenthe non-rotated left and right states are different,which is necessary for GCM calculations.4. Our final expression for the overlap allows for thecalculation of the overlap of two quasiparticle vacuathat are expressed in two different single-particlebases that do not span the same sub-space of theHilbert space of single-particle states. The knowl-edge of a complete basis spanning both single-particle bases is not needed, as compared to Ref. [9].In this way, the technique can be directly imple-mented in codes that use a coordinate space rep-resentation of the quasiparticle vacua in terms oftheir canonical single-particle bases [13–16]. The expression has been implemented into our numer-ical codes for particle-number and angular-momentumrestored GCM calculations based on triaxial HFB statesusing the full space of occupied single-particle states[15, 16]. It has been extensively tested for symmetryrestoration and for the calculation of non-diagonal ma-trix elements for symmetry-restored GCM calculationswithout encountering cases where it fails.By contrast, the technique “to follow the phase” fromRef. [5, 14], or the one to follow the overlap in the com-plex plane from Ref. [6] require often to use a very finediscretization to resolve the sign ambiguity when pro-jecting on angular momentum as soon as time-reversalinvariance of the HFB states is broken. In particular,it may become necessary to use a discretization of theintegrals over Euler or gauge angles that are much finerthan what is actually needed to converge observables. Inaddition, the direct calculation of the overlap also hasthe advantage to avoid complicated coding for the set-upof a reliable Taylor-expansion-based algorithm, in par-ticular since reliable routines to compute pfaffians areavailable [22]. Finally, in the general case of configura-tion mixing where there might not be a symmetry thatestablishes a reference sign for the overlap, such a directcalculation of the overlap could become to be mandatory.In summary, we report an expression for the overlapbetween arbitrary quasiparticle vacua that is easy to cal-culate and that is very robust in realistic applications. Itis a key ingredient for the extension of symmetry restora-tion and Generator Coordinate Method-type calculationsto angular-momentum-optimized states, either by crank-ing, or by blocking. Acknowledgments
This work was supported by the Agence Nationale dela Recherche under Grant No. ANR 2010 BLANC 0407”NESQ”. The computations were performed using HPCresources from GENCI-IDRIS (Grant 2011-050707).
Appendix A: Some remarkable Gaussian integralsover Grassmann variables
For the readers’ convenience, we give the identities thatrepresent determinants and pfaffians as integrals overGrassmann variables using our convention for the orderof the differential elements, Eq. (22).0
1. Determinant
The first one is the determinant identity that is defined,for a given matrix M , asdet ( M ) = (cid:90) (cid:89) i ( dz i d ¯ z i ) exp n (cid:88) i,j =1 ¯ z i M ij z j (A1)= ( − n (cid:90) d n(cid:46) ¯ z d n(cid:47) z exp n (cid:88) i,j =1 ¯ z i M ij z j , (A2)where the first equation is the expression from Ref. [19],p. 13, Eq. (1.67), and the second uses an alternative con-vention in the ordering of differential elements. The lat-ter differs by a sign because of the anticommutation of G -variables.
2. Pfaffian
The second one is the Pfaffian identity that is definedfor a skew-symmetric matrix A of dimension 2 n × n , as (see [19], p.15, Eq. (1.80)):pf ( A ) = (cid:90) dz n · · · dz dz exp n (cid:88) ij =1 z i A ij z j (A3)= (cid:90) d n(cid:46) z exp n (cid:88) ij =1 z i A ij z j . (A4)To obtain the correct sign, it is crucial that the differen-tial elements are in the same order as the indices of thematrix A ij . This implies sometimes to make some ma-nipulations to bring the entire expression into the properform, as for example between Eq. (57) and Eq. (61).As can be easily seen, this definition is only valid formatrices A of even rank. The determinant and the Pfaf-fian of a skew-symmetric matrix A are related by[pf ( A )] = det ( A ) . (A5)As the determinant of a skew-symmetric matrix of odd rank is always zero, the pfaffian of such a matrix is de-fined to be zero as well. [1] N. Onishi and S. Yoshida, Nucl. Phys. , 367 (1966).[2] K. Neerg˚ard and E. W¨ust, Nucl. Phys. A , 311 (1983).[3] K. W. Schmid, Prog. Part. Nucl. Phys. , 565 (2004).[4] T. R. Rod´ıguez and J. L. Egido, Phys. Rev. C , 064323(2010).[5] K. Hara, A. Hayashi, and P. Ring, Nucl. Phys. A ,14 (1982).[6] K. Enami, K. Tanabe, and N. Yoshinaga, Phys. Rev. C , 135 (1999).[7] L. M. Robledo, Phys. Rev. C , 021302(R) (2009).[8] R. Balian and E. Brezin, Il Nuovo Cimento B, , 37(1969).[9] L. M. Robledo, Phys. Rev. C , 014307 (2011).[10] G. F. Bertsch and L. M. Robledo, arXiv[nucl-th]:1108.5479v1 (2011).[11] D. J. Thouless, Nucl. Phys. 21, 225 (1960).[12] H. Flocard, th`ese, Orsay, S´erie A, 1543, Universit´e ParisSud (1975).[13] P. Bonche, J. Dobaczewski, H. Flocard, P.-H. Heenen,and J. Meyer, Nucl. Phys. A , 466 (1990). [14] A. Valor, P.-H. Heenen, and P. Bonche, Nucl. Phys. A , 145 (2000).[15] M. Bender and P.-H. Heenen, Phys. Rev. C , 024309(2008).[16] M. Bender, B. Avez, B. Bally, and P.-H. Heenen (unpub-lished).[17] J.-P. Blaizot and G. Ripka, Quantum Theory of FiniteSystems (MIT Press, 1985).[18] F. A. Berezin,
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