The sign of the overlap of HFB wave functions

0.17 MB·English

by  Luis M. Robledo

The sign of the overlap of HFB wave fun
tions
∗
L.M. Robledo
Dep. Físi
a Teóri
a C-XI, Fa
ultad de Cien
ias,
Universidad Autónoma de Madrid, 28049 Madrid, Spain
The problem of how to
ompute a

urately and e(cid:30)
iently the sign of the overlap between two
general HFB wave fun
tions is addressed. The results obtained
an easily be extrapolated to the
evaluation of thesign of thetra
e of a densityoperator exponential of one body operators.
I. INTRODUCTION statisti
al density operators [11, 12℄. In this
ase, how-
ever,Neergard'smethodhasnotbeenimplementedupto
9 date, leavingastheonly
hoi
ethe
ontinuitymethodin
Beyondmean(cid:28)eld
al
ulationsarebe
omingverypop-
0 su
h(cid:28)nite temperature
al
ulations. The samedi(cid:30)
ulty
0 ular [1℄ as they allow a fairly good des
ription of many alsoappliestothere
entlyproposedmethodto
ompute
2 nu
learstatepropertiesofboththegroundstateandsev-
multiquasiparti
le overlaps that relays on the statisti
al
eralkinds of ex
ited states all overthe Nu
lide
hart. In
n Wi
k's theorem [13℄. Re
ently, [14℄ the group stru
ture
a these
al
ulations,overlapsofHartree-Fo
k-Bogoliubov of the unitary Bogoliubov transformation has been dis-
J (HFB) wave fun
tions have to be
omputed. Standard

ussed, as well as its impli
ations in the relative phase

1 formulas[2℄involvethesquarerootofadeterminantleav-
betweentwoHFBwavefun
tions. However,itspra
ti
al
2 ingthesignoftheoverlapunde(cid:28)ned. However,whenthe
impli
ations are still un
lear.
HFBstatespreservesomekindofdis
retesymmetrylike
In this paper, I will introdu
e a new way to
ompute
]
time reversal or simplex, the blo
k stru
ture of the ma-
h theoverlapoftwoHFBwavefun
tionsbasedonthe
on-
t tri
es involved (cid:28)xes the sign. This has been dis
ussed,
eptoffermion
oherentstates[15, 16, 17℄. Thenewfor-
-
l for instan
e, in some re
ent appli
ations of angular mo- mulainvolvesaquantitysimilartothedeterminant
alled
c mentum proje
tion (AMP) using axially symmetri
and
u pfa(cid:30)an of a skew-symmetri
matrix. The advantage of
time reversal preserving intrinsi
wave fun
tions [3, 4℄.
n the proposed method is that the numeri
alevaluation of
To move forward, HFB wave fun
tions that do not have
[ the pfa(cid:30)an is simple and la
ks the problems previously
any spatial symmetry (triaxial) and also breaking time
mentioned about pairwise degenerate eigenvalues. An-
1
reversal symmetry have to be
onsidered in order to in-
v K 6= 0 otheradvantageofthepresentformulationisitsappli
a-

orporate
on(cid:28)gurations. This is the
ase to de-
3 bility to the evaluation of the tra
e of density operators
s
ribe, for instan
e, the ground state of odd-A nu
lei.
1 like the ones found in appli
ations of the auxiliary-(cid:28)eld
2 For the usual time reversal breaking (TRB) mean (cid:28)eld shell model Monte Carlo[11℄ or symmetry restoration at
3 wave fun
tions, the simplex symmetry endows the HFB
(cid:28)nite temperature[12℄. A reliable determination of the
. amplitudes U and V with a
ommon bipartite stru
ture
1 sign of the norm
an also be useful in to order to pin
0 and the usual arguments used to extra
t out the sign down the lo
ation of the zeros of the HFB overlaps [10℄.
9 of the overlap apply. However, when full triaxial an- This determination would eventuallybe useful to getrid
0 gular momentum proje
tion of HFB intrinsi
states [5℄
of the so
alled (cid:16)pole problem(cid:17) that plagues present be-
: is
onsidered, the simplex symmetry is no longer pre-
v yond mean (cid:28)eld
al
ulations.
i served in the evaluation of rotated overlaps and the de-
X
termination of the sign be
omes more di(cid:30)
ult. A gen-
r eral solution to the sign problem was given in Ref. [6℄,
II. OVERLAPS AND TRACES
a
where it was shown that the overlap, in
luding the sign,

an be
omputed from the pairwise degenerate eigen-
A. Preliminaries
values of a non-hermitian matrix. Handling the eigen-
values of non-hermitian matri
es is a di(cid:30)
ult task [7℄, |φ i |φ i
0 1
Let and be two HFB wave fun
tions de(cid:28)ned
that in
reases its
omplexity if the pairwise degenerate
eigenvalueshaveto beobtainednumeri
allywithoutany in terms of a set ao+f singlea parti
le
reation and anni-
hilation operators k and k that are assumed to be re-
symmetryenfor
ingdegenera
y,asis the
ase with HFB
latedbyhermitian
onjugationandalsotosatisfyfermion
wavefun
tionsbreakingsimplex. Neergard'smethodhas

ommutation relations. The HFB wave fun
tions, in the
been used along with small
on(cid:28)guration spa
es [8℄ but
Thouless representation[2, 16℄, are given by
in the majority of the
al
ulations
ontinuity arguments
areused(seeRefs. [5, 9,10℄forre
entexamples)inspite 1
|φ i=exp M(i)a+a+ |0i
opfrotbhleemdi(cid:30)is
aulsltoiepsrewsietnhttihnatthepreov
aelduuarteio.nTohfethseamtrea
seigonf i 2Xkk′ kk′ k k′! (1)
where the skew-symmetri
matri
es
∗ M(i) =(V U−1)∗
Ele
troni
address: luis.robledouam.es i i
2
U V
i i
are de(cid:28)ned in terms of the and
oe(cid:30)
ients of the B. Evaluation of the overlap
Bogoliubovtransformationsde(cid:28)ningtheHFBwavefun
-
|0i hφ |φ i
tions and is the true va
uum. The arbitrary phase To
omputetheoverlap 0 1 , the
losurerelationof
that
analwaysbeasso
iatedwithave
torstateinquan- Eq. (4) is inserted to obtain
tumme
hani
shasbeen impli
itly (cid:28)xedin thede(cid:28)nition
of Eq. (1) by requiring h0|φii = 1. Ways to enfor
e hφ0|φ1i = dµ(z)h0|e12Pkk′Mk(0k)′∗ak′ak|zi
this normalization for general HFB wave fun
tions are Z
dis
ussed,h0fo|φr iin=sta0n
e in Refs. [2, 16℄. In the eMve(nit) × hz|e12Pkk′Mk(1k)′a+ka+k′|0i
i
of having (as a
onsequen
e of divergent
and/orzeroo

upan
ies)thebestpra
ti
alstrategyisto
Using now Eqs. (2) and (3) one arrivesto
uvase
uaunmoth|0eir. rTefheerenne
we wreafveerefnu
ne
tHioFnBinwstaevaedfuonf
tthioentr|uφ¯ei hφ0|φ1i= dµ(z)e12Pkk′Mk(0k)′∗zk′zke21Pkk′Mk(1k)′zk∗zk∗′
h|φasito be
onveniently
hosen as to stay
lose to both Z
i
(for instan
e by taking a wave fu|φn
ition with similar |h0|zi|2 = 1 (5)
i
dM¯ef(oi)rmation para|mφ¯ieters as those of ). The matri
es where the property is used. The integral is
referred to
an be straightforwardly
omputed of the Gaussian type but for Grassmann variables. The
in terms of the previous quantities and the Bogoliubov te
hniquestoevaluatethiskindofintegrals
anbefound
transformation amplitudes of the referen
e state. In the in many textbooks [15, 16, 17℄ but its evaluation will be
raree|φv¯eintofnot(cid:28)ndinga
onvenientreferen
ewavefun
-
arried out expli
itly here. The reason is that in order
tion M¯(ia) regularization pro
edure to handle the diver- to determine the sign of the norm we have to be
areful
gent matrix elements (or the zero o

upan
ies) is with some intermediate steps. The above integral
an
in order. In this
ase, the expressions get more involved be written in a more
ompa
t way by introdu
ing the
and a detailed a

ountis deferred to a forth
omingpub- bipartite skew-symmetri
matrix
islnie
vnaottelivodenid.n.ARnefo.th[6e℄rbwuatythteordeesaulltwinigthexthparetspsiroonbsleamreirsatphreer- Mµ′µ = M11kk(′1′kk) −−M11kk(0′′kk)∗ !
|zi,
Let me now introdu
e fermion
oherent states and the ve
tor of Grassmann variables zµ =(zk∗′,zk′) as
whi
harzeparamz∗etrizedintermsoftheanti
ommutingel-
aemndenfutsl(cid:28)lklinagndthekeoqfuaatGiornasssmannalgebra[15, 16, 17,18℄ hφ0|φ1i= (dzk∗dzk)e12Pµµ′zµ′Mµ′µzµ (6)
Z k
Y
M
a |zi=z |zi
k k The skew-symmetri
matrix
an always be trans-
(2)
formed [19℄ to
anoni
al form by means of a unitary
U
transformation
0 ··· 0 β 0 0
and 1
 .. .. .. 0 .. 0 
hz|a+ =z∗hz| 0. ···. 0. 0 0. β
k k (3) M=U N UT =UM UT
−β1 0 0 0 ··· 0  c
 
 
sFtraotme othfethhazeb|oavnendihei(cid:28)lantiitoionnoipse
rleaatorrthaaktaw|+ziithiseaigreignhvtaleuigeeznk-  00 .0.. −β0N 0... ·.·.·. 0... 
wz∗hereas is a left eigenve
tor of k with eigenvalue β1,...,βN
k(Thenotationusedforthemembersofthe Grassmann and the M
oe(cid:30)
ients of the (cid:16)
anoni
al form(cid:17)
zakl∗geisbrnaotis
tohneneu
stueadl otonezkbubty

aonmbpeleaxl
itotnlejumgaistlieoand)i.nTghaes othfethneewmaGtrriaxssmaanrne rveaarliaabnledspηoµsit=ive. µI′n(tUroTd)uµ
µi′nzµg′ntohwe

oherent states satisfy a
losure relation exponent in the integrand of Eq. (5)Pbe
omes
N
1
= dµ(z)|zihz| 2 µµ′ ηµMcµµ′ηµ′ =k=1βkηk∗ηk
11 (4) X X
Z
whi
h is straightforward to integrate. The Jdae
to(bUiaTn)o=f
dµ(z) = thetransformation
anbeshowntobesimply
ew−hze∗rze thkedzmk∗deztkri.
oTfhethsee ainntdegortahlerisrgeilveveanntbyde(cid:28)nitions det(U). The remainingintegrals
anbeperformedeasily
being the result
and pQroperties of fermion
oherent states
an be found
inmanytextbookorintheoriginalliterature[15, 16,17, dη∗dηeβη∗η =−β.
18℄. Z
3
The (cid:28)nal expression for the overlap is then By ex
hanging blo
ks 2 and 4 we obtain
N (1)
0 0 − M
hφ0|φ1i=(−1)Ndet(U) βk. (011)+
kY=1 pf(M)=(−1)Npf 0 −M0(0)∗ M0 101 
 
This expression
an be
ast in terms of the pfa(cid:30)an  −M11(1)T − 0 0 
of a skew-symmetri
matrix. The pfa(cid:30)an of a skew-  11 
 
symmetri
matrix(see,forinstan
e,[20℄)isanumberob-
that
an be evaluated using Eq. (A2) to give
tainedout ofthematrix elementsofthe skew-symmetri
matrix in a way quite similar to the one used to de(cid:28)ne − M(1) (0)+ (1)
the determinant (see Appendix A for details and prop- hφ0|φ1i=det M(011)+ !=det(11+M M ).
ertiβes used below). The
onne
tion between the produ
t 11
i
of 's and tNheβpfa=(cid:30)a(−n1i)sNa(N
−o1n)/s2equ(eMn
e) of Eq. ((MA2)) (10)
andreads k=1 k Mpf c wherepf c
c
obviously Qde(nMo)te=s th(eUpMfa(cid:30)UaTn)=ofdet(.U)Us(inMg )the prop-
erty(A1)pf pf c pf c the(cid:28)nal C. Evaluation of statisti
al tra
es
result is obtained,
M(1) − Now I turn to the evaluation of the tra
e of density
hφ0|φ1i=sNpf(M)=sNpf −M1(10)∗ (7) operators. In theDˆstatisti
al HFB theory the statisti
al
(cid:18) 11 (cid:19)
density operatoDrˆ i=s geixvpe[n1by thγe Rexpoγne]ntial ofγa one
where sN = (−1)N(N+1)/2. To make the
onne
tion body operator (β2,..µ.ν,βµ ,βµν+,ν...w,βh+er)e µisRa
shorthand notation for 1P N 1 N and
with th(eA)s2ta=nddaertdAformula for the overlap [2℄ the rela- 2N
is a skew-symmetri
matrix of dimension
hara
ter-
tion pf is used (and this is here where the
izing the density operator (see Ref [12℄ for details). An-
sign is lost) to write
other way to
hara
terize the density operator is to de-
hφ0|φ1i= det M(1) −M−1(10)∗ 1/2 (8) l(cid:28)anteiohnoowpeitrattroarnssfDoˆr−m1γsµqDˆua=siparνtiT
µleνγ
νrewathioenreatnhdeamnantirhiix-
(cid:18) (cid:18) 11 (cid:19)(cid:19) T = exp(σR) σµν = {γµ,γν}
and P . Introdu
ing the bi-
T
This expression redu
es, by using standard formulas for partite stru
ture of
the determinant of a bipartite matrix (see below), to
hφ0|φ1i= det(11−M(0)∗M(1)) 1/2 (9) T =(cid:18)TT1211 TT1222 (cid:19)
(cid:16) (cid:17)
Dˆ
whi
h is the usual expression for the norm (Onishi for- the Balian and Brezin's de
omposition [21℄ of is given
mula). Please noti
e that in going from Eq. (7) to Eq. by
s(8eq)utehne
seigonf pthreesewnrtitiinngthoef(cid:28)trhset esqquuaarteionofisthloestpfaas(cid:30)aan
oans- Dˆ =e21Pijβi+Xijβj+e−12Tr[Y℄ePijβi+Yijβje21PijβiZijβj
a determinant. Also signs appearing in the manipula- X = T T−1 Z = T−1T (11)
tions needed to obtain Eq. (9) have been negle
ted. We with 12 22 and T22TσT21=skσew-symTmetri
(as

learlyseethatthesignproblemappearsinthestandard a
onesxepq(u−enY
)e=oftThTerelation that sDaˆtis(cid:28)es)
formulas be
ause of the wrong impli
it use of the above and 22. To evaluate the tra
e of using
relation between the pfa(cid:30)an and the determinant.
|φ i |φ i fermion
oherent states we have to use the formula [17℄
0 1
IfbothHFBwavefun
tions and sharea
om-
mondis
retesMym(im) etrylikesimplexortimereversal,then (Dˆ)= dµ(z)h−z|Dˆ|zi
Tr (12)
the matri
es de(cid:28)ning them
an a
quire a
ommon Z
blo
k stru
ture |zi
0 M(i) where are again a set of fermion
oherent states but
M(i) = −M(i)T 0 !
hosen this time asβeiigenstβati|ezsio=f tzhie|zqiuasiparti
le anni-
hilation operators , i.e. . Using Eq. (11)
the evaluationof the overlapbetween the fermion
oher-
that
anbeusedtosimplifytheresultofEq. (7). Inthis ent states gives

ase (1) h−z|Dˆ|zi = e−12Tr[Y℄e12Pijzi∗Xijzj∗e21PijziZijzj
0 M − 0
(1)T 11 × h−z|ePijβi+Yijβj|zi
M=−M 0 0 −1(10)∗ 
0 0 −M
 101 11 M(0)+ 0  Tsuolteevxapluateijtβhi+eYrijeβmjain|zinig=o|veeYrzlaipustehde tsotgaentdhaerrdwirteh-
  (cid:16) (cid:17)
P
4
h−z|z′i=exp(−z∗z′)
(see Refs. [15, 16, 17℄ ) gives HFB wavefun
tions. The result given in terms of pfa(cid:30)-
h−z|Dˆ|zi = e−21Tr[Y℄e21Pijzi∗Xijzj∗e−Pijzi∗(eY)ijzj ansissimplertoimplementthanprevious
onsiderations
[6℄ based on pairwisedegenerate eigenvaluesof a general
× e21PijziZijzj matrixanditisfreefromtheun
ertaintiesofothermeth-
ods based on
ontinuity arguments. Indi
ations on how
Combining this result with Eq. (12), the following inte- to evaluate e(cid:30)
iently the pfa(cid:30)an are also given. Hope-
gral is obtained fully,thisnewmethodwillhelptosimplifytheimplemen-
tationofambitiousproje
tsliketriaxialangularmomen-
Tr(Dˆ)=e−21Tr[Y℄ (dzk∗dzk)e12Pµµ′zµ′Mµ′µzµ tum proje
tion. On the other hand, the method used is
Z Yk straightforwardly extended to the evaluation of the sign
ofthetra
eofstatisti
aldensityoperatorswhi
hisanew
wherethesamenotationasinEq(6)isused. Inthis
ase result not
onsidered previously in the literature.
X −(eY + )
M= 11
(eY + )T Z
(cid:18) 11 (cid:19) A
knowledgments
Applying the same
onsiderationsasin the evaluationof
Work supported in part by MEC (FPA2007-66069)
the overlapwe (cid:28)nally arrive to
and by the Consolider-Ingenio 2010 program CPAN
1
(Dˆ)=s exp − (M) (CSD2007-00042)
N
Tr 2Tr[Y℄ pf
(cid:18) (cid:19)
s = (−1)N(N+1)/2
where N . Taking into a

ount the re- Appendix A: DEFINITION, BASIC PROPERTIES
X Z Y
lationship between , and and the blo
ks of the AND NUMERICAL EVALUATION OF THE
T
PFAFFIAN
matrix the above result
an be expressed as
(Dˆ) = s (detT )1/2 R
N 22
Tr The pfa(cid:30)an of a skew-symmetri
matrix of dimen-
× pf ((TT221)2−T12−2+1 11) −((cid:0)TT2T2−22(cid:1)1−T121+11)! sionp2fN(Ra)n=d 2w1nitnh1!matriǫx(Pel)ermi1ie2nrtis3ir4irji5iis6.d.e.(cid:28)rn2end−1a,2sn[20℄
(detT22)1/2 PXerm
Texhpe −i1ntrodu
tion of in pla
e of
2Tr[Y℄
an lead to the (right)
on
lusion where the sum extends to all possible permutations of
i ,...,i ǫ(P)
1 2n
that(cid:0)a sign in(cid:1)determina
y has been introduDˆ
ed in the and is the parity of the permutation.
expression of the tra
e. The dTe(cid:28)nition of in terms For matri
es of odd dimension the pfa(cid:30)an is by def-
of the transformation matrix leaves a phase open inition equal to zero. As an example, the pfa(cid:30)an of
2 × 2 R (R) = r 4 × 4
in the de(cid:28)nition of the density operator whi
h is also a matrix is pf 12 and for a one
(R) = r r −r r +r r
12 34 13 24 14 23
present in the expression of Eq. (11). A way to (cid:28)x the pf . Similarly to the
ase
i j
phase is to require some
ondihtφion|Dˆli|kφe,ifo=r in(sdteatnT
e,)t1h/e2 of determinantis, ex
hjanging rows and and the same
realness and positiveness of 0 0 22 time
olumns and , multiplies the pfa(cid:30)an by minus
|φ i
wβhere 0 is the va
uum of thDˆe quasiparti
le operators one. Other useful properties of the pfa(cid:30)an are
i
entering in the de((cid:28)dneittiTon)o1f/2 . This
onditionimplies
the repla
ement of 22 by its modulus. Using (PTRP)= (P) (R),
pf det pf (A1)
property (A1) of the pfa(cid:30)an the (cid:28)nal result is obtained
|detT |1/2
(Dˆ) = s 22
Tr N detT22 (13) 0 R =(−1)N(N−1)/2det(R)
× T12T2−21 −(T2T2+11) pf(cid:18)−RT 0 (cid:19) (A2)
pf (cid:18)(T22+11) T21T2T2 (cid:19) (14)
(R)=(detR)1/2
Thisresultisapparentlyquitedi(cid:27)erentfromthestandard pf (A3)
oneof[11,12℄,butaftersometediousmanipulations(see
A useful formula to
ompute pfa(cid:30)ans of small or simple
appendix B) one
an obtain the usual result.
matri
es is
(R)= (−1)i+j−1r (R )
ij ij
pf pf (A4)
III. CONCLUSIONS j
X
R
ij
I have used the te
hnique of fermion
oherent states where is the pfa(cid:30)an-minor obtained by eliminating
R i j
to
omputeunambiguouslythesignoftheoverlapoftwo from the two rows and two
olumns and .
5
M˜
The pfa(cid:30)an of a
omplex skew-symmetri
matrix UsingEq. (A3)thepfa(cid:30)anof iswrittenasthesquare
R 1/2
of dimension 2N is evaluated numeri
ally by (cid:28)rst M˜ = detM˜
RT rootofits determinant, pf . Byex
hang-
redu
ing the matrix to tridiagonal form . This
(cid:16) (cid:17)
redu
tion is a

omplish by means of a set of 2(N- ing rows and
olumns
onveniently
P
i
1) su

essive Householder transformations exa
tly
(T + ) T TT
in the same way as in the standard redu
tion of a detM˜ =(−)N T22T−111 −(T2T1 +22 )
sPy2m(Nm−e1t)r.i.
.Pm2aPt1rRixP1tToP2Ttr.id.i.aPg2To(nNa−l1)fo=rmRT[7w℄.ithWe have (cid:18) 12 22 22 11 (cid:19)
0 r 0 0 ... 0 and applying the formula of a bipartite determinant
1 A B
−r 0 r 0 ... 0 det =detAdet(D−CA−1B)
 1 2  C D the followingre-
(cid:18) (cid:19)
R = 0 −r2 0 ... ... ...  sult is obtained
T  
 0 0 ... 0 ... 0  detM˜ = detT22det( +T22)
  11
 0... 0... ······ .0.. −r20N−1 r2N0−1  × det(cid:16)11+(cid:0)T2T2(cid:1)−1+T12T2−21(11+T22)−1T21(cid:17).
wtrhideiradegeottn(hPael1)msp.a.et.
rdiiaxelti(ssPte2rv(uNi
d−teu1nr)t)e. Uo(fsRina)g=snkoeww(R-EsTyqm.)(mAe1t)riw
eaonbd- BT2u−2t1 (11T+11T22)−1=−11 =(cid:0)T−2T2((cid:1)11−+1 T2+2)−1T1s2oTt2−h21aTt21 and
tain pf pf . As House-
(cid:2) (cid:3)
holderdemt(aPtri)i
e=sa−r1ehermitian,unitaryand(Rha)v=edet(eRrmT)i- detM˜ = detT22det( +T22)
nant we (cid:28)nally arrive to pf pf . 11
× det +T −T ( +T )−1T
To evaluate the pfa(cid:30)an of the tridiagonal matrix we use 11 12 22 21
(R ) = 11 11
T
trh1re3m..i.nro2rNe−x1p=ansioNi=n1orf2iE−1q.. (A4) that gives pf = detT(cid:0)22det (T1T12+1 11) (T2T21+2 )(cid:1)
(cid:18) 11 (cid:19)
Q
When the pfa(cid:30)an is written as the square of the deter-
minantthesignislostandthereforephasesareirrelevant
Appendix B: DERIVATION OF THE STANDARD
FORMULA FOR THE TRACE in the derivation. Taking all this into a

ount the result
1/2
(T + ) T
Inthisappendixthestandardresultof[12℄forthetra
e (Dˆ)= det 11 11 12
Tr T21 (T22+ )
of a density operator is dedu
ed from Eq. (13). I start (cid:20) (cid:18) 11 (cid:19)(cid:21)

onsidering
M˜ = T12T2−21 −(T2T2+11) is obtained up to a sign, whi
h is the sought formula of
(cid:18)(T22+11) T21T2T2 (cid:19) Ref [12℄.
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