Sum-rules and Goldstone modes from extended RPA theories in Fermi systems with spontaneously broken symmetries

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by  D. S. Delion

EPJ manuscript No.
(will be inserted by the editor)
Sum-rules and Goldstone modes from extended RPA theories in
Fermi systems with spontaneously broken symmetries
D.S. Delion1,2,3, P. Schuck4,5, and M. Tohyama6
6 1 ”Horia Hulubei”National Instituteof Physics and Nuclear Engineering,
1
407 Atomi¸stilor, POB MG-6, Bucharest-Ma˘gurele, RO-077125, Romˆania
0
2 2 Academy of Romanian Scientists, 54 Splaiul Independen¸tei, Bucharest, RO-050094, Romˆania
3 Bioterra University,81 Gˆarlei str., Bucharest, RO-013724, Romˆania
n 4 Institut dePhysiqueNucl´eaire, F-91406 Orsay CEDEX, France
a 5LaboratoiredePhysiqueetMod´elisation desMilieuxCondens´es,CNRSandUniversit´eJosephFourier,25AvenuedesMartyrs
J
BP166, F-38042 Grenoble C´edex 9, France
6 6 Kyorin University School of Medicine, Mitaka, Tokyo181-8611, Japan
1

] the dateof receipt and acceptance should beinserted later
l
e
- Abstract. TheSelf-ConsistentRPA(SCRPA)approachiselaboratedforcaseswithacontinuouslybroken
r
t symmetry,thisbeingthemainfocusofthepresentarticle.CorrelationsbeyondstandardRPAaresummed
s up correcting for the quasi-boson approximation in standard RPA. Desirable properties of standard RPA
.
t such as fullfillment of energy weighted sum rule and appearance of Goldstone (zero) modes are kept. We
a
show theoretically and, for a model case, numerically that, indeed, SCRPA maintains all properties of
m
standard RPA for practically all situations of spontaneously broken symmetries. A simpler approximate
- formofSCRPA,theso-calledrenormalisedRPA,alsohastheseproperties.TheSCRPAequationsarefirst
d
outlined as an eigenvalue problem, but it is also shown how an equivalent many body Green’s function
n
approach can beformulated.
o
c
[ PACS. 21.60.Jz , 31.15.Ne, 71.10-w

2
v
1 Introduction and,thus,theequationswillnotonlybeintegralequations
5
in coordinate or momentum spaces but in addition one
9
4 The present trend in many body physics mostly goes in will have to deal with integrals in energy space what ren-
3 the direction of a direct numerical treatment of the prob- dersmostofthe problemsverycomplicated.Inthis work,
0 lemathand.Asexamples,onecouldciteQuantumMonte we want to advocate a different route where we extend
1. Carlo (QMC) [1,2] and Density Matrix Renormalisation the standard HF-RPA (BCS-QRPA) approach including
0 Group (DMRG) techiques [3,4] which are very success- highercorrelationswithoutdestroyingtheaforementioned
6 full.Ontheotherhanditmayalsobe interestingtomake desirableproperties.Asmentioned,ourapproachisbased
1 advances in many body theory proper. We will try to do on the equation of motion method and leads to a self-
: thisinthisworkemployingtheequationofmotion(EOM) consistent version of RPA (SCRPA) which largely over-
v
i technique (see e.g. [5]). We think that this formalism has comes a well known defect of standard RPA, the quasi-
X not been exploited to its full power in the past. For ex- bosonapproximation.Thereisonepointwherethisdefect
r ample, as is well known, the fullfillment of sum-rules and is easy to trace: RPA, besides excitations, also describes
a
appearance of Goldstone modes in systems with sponta- correlations in the ground state. However, fermion occu-
neously broken symmetries are corner stones of any valu- pationnumbers used in RPA are the uncorrelatedones,a
able many body theory. We will dwell on this specific as- clearcontradictionwhichisduetotheneglectofthePauli
pect in this work. Unfortunately, going beyond basic ap- principle.TheSCRPAcuresthispointinusingcorrelated
proacheslikeHartree-Fock(HF)-RPAand/orBCS-QRPA occupation numbers which couple back to the RPA. Ad-
(quasi-particle RPA) [6], it becomes immediately a non- ditional Pauli corrections as vertex screening are also in-
trivialproblemtosatisfythoseproperties.Thereexiststhe cluded. We willshow applications to various exactly solv-
approachofΦderivablefunctionalspromotedbyKadanoff able models and demonstrate the strong improvement of
and Baym [7]. However, it is mostly very difficult to im- SCRPA over standard RPA. We will treat Fermi systems
plement this numerically beyond HF-RPA (BCS-QRPA), but we think that the approach can be applied to Bose
thelowestorderapproach,becausethistechniqueinvolves systems as well.
ingeneralverticeswhichdependonmorethanoneenergy
2 D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length
Thepaperisorganizedasfollows.InSectionIIwegive where E and E are supposed to be exact eigenvalues of
µ 0
a shortoutline of SCRPA. A very simplifying approxima- the Hamiltonien H corresponding to the eigenstates 0
| i
tion thereof, the renormalised RPA (r-RPA) is presented and µ . Therefore Ω can be considered as some average
ν
| i
in Sction III. In Section IV the Goldstone mode is ana- excitationenergyanditis thisquantitywhichwewantto
lyzedandinSectionVitisshownthattheenergyweighted minimise. As we will see, the summation in (2) must ex-
sum-rule is fulfilled within SCRPA. In Section VI, we ap- clude the diagonal terms with indices α=β. In addition,
ply SCRPA to the Hubbard and pairing models and we we want the states ν to be normalised. Accordingly we
| i
demonstratenumericallywithathreelevelmodelthat,in- write for (2) more explicitly
deed, the Goldstone (zero)mode appears.In Section VII,
weshowhowSCRPAcanequivalentlybeformulatedwith Q† = [Xν δQ† Yν δQ ] , (5)
manybodyGreen’sfunctions.Inthe lastSectionwedraw ν αβ αβ − αβ αβ
α>β
X
our conclusions.
where
A
2 Self Consistent Random Phase δQ† = αβ , A a†a (6)
αβ M αβ ≡ α β
Approximation (SCRPA) αβ
are the normalised papir creation operators
Asmentioned,ourtheorywillbebasedontheEquationof with
Motion(EOM)approach.Wewillberathershortwiththe
presentation of the formalism, since it has been exposed αMˆ β M =n n , (7)
αβ β α
several times before [5,8,9,10,11,12]. For finite systems h | | i≡ −
with discrete levels, the EOM mostly is applied so that where
an eigenvalue problem results. We will follow first this
routebutlateralsooutlinetheequivalentGreen’sfunction n = 0a†a 0 , (8)
α h | α α| i
approachwhichismostlyappliedincondensedmatterfor
arethesingleparticleoccupationnumbers.Withthischoice
infinite homogeneous systems. Then for finite systems, if
and with ν ν = 0[Q ,Q†]0 , one immediately verifies
we wantto stay within RPA, one may makethe following h | i h | ν ν | i
that with
ansatz for an excited state of the system [5]
|νi=Q˜†ν|0i , (1) |Xανβ|2−|Yανβ|2 =1 , (9)
where in general Q˜† = ν 0 is a complicated many body αX>β(cid:18) (cid:19)
ν | ih |
operator, 0 and ν beingexactgroundandexcitedstates the excited states ν are normalised under the assump-
of the syst|eim, res|piectively. To lowest order, we may con- tion that we work i|nithe canonical basis where the single
sider the one body operator particledensitymatrixonlyhasdiagonalelements,thatis
ρ = 0a†a 0 =n δ . From (7) and (6), we see that
Q˜† = χ˜ν a†a , (2) αβ h | β α| i α αβ
ν αβ α β the configuration α=β must be excluded in (5).
Xαβ With these definitions, we obtain from the minimisa-
where a†,a are single particle creation and annihilation tion of the sum rule (4) the following eigenvalue equation
α β
operators in a general basis but which, to fix the ideas,
may be chosen to correspond to a diagonalisation of the
single particle density matrix, the so-called canonical ba- A B Xν =Ω 1 0 Xν , (10)
sis. In EOM, one also always supposes that there exists a B∗ A∗ Yν ν 0 1 Yν
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19)
ground state which is the vacuum to the destructors Q˜ ,
ν
i.e., where
Q˜ν|0i=0 . (3) Aαβα′β′ =h0| δQαβ, H,δQ†α′β′ |0i , (11)
TherearemanywaystoderivestandardHF-RPA.One (cid:20) (cid:20) (cid:21)(cid:21)
of the best known is the linearisationof the Time Depen- and
dent HF (TDHF) equations, see, e.g., [13]. Here, we want
to go a slightly different way. Let us consider the nor- Bαβα′β′ =−h0| δQαβ, H,δQα′β′ |0i . (12)
(cid:20) (cid:20) (cid:21)(cid:21)
malised energy weighted sum rule
For the following it is useful to introduce the quantities
1 0[Q˜ ,[H,Q˜†]]0
Ων = 2h |h0|[νQ˜ν,Q˜†ν]ν|0i| i S = BA∗ AB∗ , X = XY XY∗∗ ,
(E E ) 0Q˜ µ 2 (cid:18) (cid:19) (cid:18) (cid:19)
= Pµ µµ−h0|Q0˜νh|µ|i|2ν| i| , (4) N0 = (cid:18)10 −01(cid:19) . (13)
P
D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length 3
We realise that (10) has exactly the same mathematical Minimisingthegroundstateenergywithrespecttothe
structure as standard RPA [13]. For instance, we see that basis,let us firstwrite the hamiltonianin ourgeneralsin-
the eigen vectors χν with components Xν and Yν form a gle particle basis with greek indices more explicitly. Sup-
complete orthonormal set. posingthatthehamiltonianisoriginallygiveninthebasis
Equation(10)canthenbewritteninamorecompactform of plane waves which are written in the sought for basis
as as c† = R∗ a† where k includes momenta, spin, and
k α α,k α
isospin,thehamiltonianwithtwobodyinteractionsinthe
= 0 Ω , (14) new basPis reads
SX N X
whereΩisadiagonalmatrixwithrealeigenvaluesΩ , Ω ,
ν ν
−
if is positive definite.
SThe closure relation is H = ekRα∗,kRβ,ka†αaβ
kαβ
X
(XανβXαν′β′ −YανβYαν′β′)=δαα′δββ′ . (15) + 1 v¯ R∗ R∗ R R a†a†a a ,
ν 4 k1k2k3k4 α,k1 β,k2 γ,k3 δ,k4 α β δ γ
X
The orthonormality relations allow us to invert the oper- X (19)
ator (5). For α>β we have
with e = k2/(2m) the kinetic energy and v¯ the
k k1k2k3k4
antisymmetrised matrix element of the two body interac-
a†a = M (Xν∗Q† +Yν∗Q )
α β αβ αβ ν αβ ν tion.Thecorrespondingvariationalequationareobtained
ν
p X from
a†a = M (Xν Q +Yν Q†) . (16)
β α αβ αβ ν αβ ν
ν
p X ∂
With(3),itthenfollowsthatthedensitymatrix 0a†a 0 0H 0 E R∗ R =0 , (20)
h | α β| i ∂R∗ h | | i− β β,k β,k
only has diagonal elements, as it was introduced already α,k(cid:18) β k (cid:19)
X X
after (9).
It can immediately be verified that, if all expecta- ∂
0H 0 E R∗ R =0 , (21)
tion values in (10) are evaluated with the HF ground ∂R h | | i− β β,k β,k
state,thenthestandardRPAequations[13]arerecovered α,k(cid:18) Xβ Xk (cid:19)
with,inparticular,onlyXν andYν amplitudessurviving
ph ph where,asusual,weensuredwithLagrangemultipliersthat
where the indices p(h) stand for ’particle (hole)’, i.e., in-
the transformation is unitary. For the common situation
dices above (below) the Fermi energy.The equations (10)
where the transformationis real, this yields the following
are, however, much more general and it is obvious that,
eigenvalue problem
if the expectation values in (10) are evaluated with the
RPA ground state obeying (3), then the matrices A and
B will depend in a complicatednonlinear wayon the am- Hkk′Rα,k′ =EαRα,k , (22)
plitudes X and Y. This we will call the Self-Consistent Xk′
RPA (SCRPA). with
Before we come to the explicit evaluation of the ma-
trixelementsA,B in(10)andtothediscussionofsponta-
1
neouslybrokensymmetriestogetherwithsum-rules,Gold- =hMF + v¯ C , (23)
stone modes, etc., we first shall deal with the so far open Hk1k2 k1k2 2 k1k3k4k5 k4k5k2k3
butveryimportantquestionofthe optimalsingleparticle k3Xk4k5
basis. As usual, we will obtain this from the minimisa- and the mean field (MF) hamiltonian given by
tion of the ground state energy with respect to the basis.
Wfolelowwiilnlgshaodwdittihoantaltheqisumatiinoinmoisfamtiootnioins equivalent to the hMkkF′ =ekδkk′ + v¯kk1k′k2ρk2k1 . (24)
kX1k2
0[H,Q†]0 = 0[H,Q ]0 =0 , (17) The single and two particle density matrices correspond-
h | ν | i h | ν | i ing to the operators ρˆand Cˆ, respectively, are
which obviously is correct if 0 is an eigenstate of H.
Because there are as many ope|raitors Q†,Q as there are
ν ν
components a†αaβ,a†βaα, we also can write for (17) ρ = 0a† a 0
k1k2 h | k2 k1| i
h0|[H,a†αaβ]|0i=h0|[H,a†βaα]|0i=0 , (18) Ck1k2k3k4 =h0|a†k1a†k2ak4ak3|0i−(ρk1k3ρk2k4
where we again recall our convention α > β. Equations ρ ρ ) , (25)
− k1k4 k2k3
(18) are of the one body type and one can directly ver-
ify that with a Slater determinant as ground state, they It is now easy to show that (18) and (22) are equivalent.
reduce to the HF equations.
4 D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length
Supposing that we work in this optimised single par-
ticle basis with greek indices, we obtain for the SCRPA
1 1
matrix in (14) [14] M =1 M Yν 2 M Yν 2 . (30)
ph − 2 ph| ph| − 2 ph| ph|
p,ν h,ν
X X
˜αα′,ββ′ = Mαα′(eαβδα′β′ eα′β′δαβ) Mββ′
S − Letusremindthatp(h)standforgreekindicesabove(be-
1
+ pδα′β′ v¯αγγ′γ′′Cγ′γ′′γβ p low) the Fermi surface.
2
γγ′γ′′
X Thecorrelationfunctions a†a a†a needextracare.
δαβ1 v¯γγ′α′γ′′Cβ′γ′′γγ′ If α = β, there is no problehmαsαincβe βthie expression re-
− 2
γγ′γ′′ duces to a single particle occupation number which we
X
+ Mαα′v¯αβ′α′βMββ′ justtreatedabove.Inthecaseα=β,wecanwrite(please
6
be aware that all greek quantum numbers represent just
+ (v¯αγβγ′Cβ′γ′α′γ +v¯β′γα′γ′Cαγ′βγ) a single quantum state)
γγ′
X
1
− 2 (v¯αβ′γγ′Cγγ′α′β +v¯γγ′α′βCαβ′γγ′ ), a†a a†a =n a†a a†a , α=β . (31)
Xγγ′ h α α β βi α−h α β β αi 6
(26)
Again,wecannowexpressthe twobodycorrelationfunc-
whereweintroducedthenon-diagonalkineticenergiese tionin(31)bytheX,Y amplitudesandallsingleandtwo
αβ
and ˜=Mˆ1/2 Mˆ1/2 and supposed that the single parti- body density matrices can be fully included in a selfcon-
S S sistent solution of the SCRPA equations (10).
cle density matrix is also diagonal,as statedafter (16). It
Theefficiencyofthemethodhasbeendemonstratedin
seems, however, clear that density matrix and single par-
several earlier publications [8,9,10,11,12]. Examples will
ticleHamiltonian(24)cannotbediagonalsimultaneously.
Thiscontradictionstemsfromthefactthatthekillingcon- begivenbelowincludingonewithabrokensymmetrywith
dition (3) cannot be satisfied exactly. With (5) it is only the appearence of a Goldstone (zero) mode which is the
main subject of this paper.
satisfiedtogoodapproximationchoosing,e.g.,theground
statewavefunctiontobetheoneofCoupledClusterThe-
ory (CCT) at SUB2 level, as this is discussed in detail in
ref. [15]. 3 Renormalised RPA
In order to establish selfconsistency, we must express
the matrices ofMˆ andCˆ by the amplitudes X,Y.To this
A very much simplified version of SCRPA consists in the
end, we write
so-calledrenormalisedRPA(r-RPA).Itsimplyisobtained
in discarding in (26) and (27) the two-body correlations
δαβ′δβα′nβ(1−nα)+Cαβα′β′ =ha†α′aαa†β′aβi , (27) Cˆ, keeping, however, the correlated occupation numbers
(28-30).
withabreviation ... = 0...0 .With (16)andthekilling
h i h | | i
condition (3), we can express this correlationfunction for (r) =hMF (32)
α = α′ and β = β′ by X and Y together with single Hk1k2 k1k2
6 6
particle occupation numbers n .
α
Concerning the s.p. occupation numbers, we refer to
esox-pcraelslesidonnsumdebreivreodpeearraltioerrimnetthheodli,tesreaetuer.ge.,em[1p1l]o.yWinegatlhsoe S˜α(rα)′,ββ′ =(Eα−Eα′)Mαα′δαβδα′β′
have derived these expressions from the Coupled Cluster +Mαα′v¯αβ′α′βMββ′
(33)
(CC) wave function at level SUB2 [15]. In model cases it
hasbeen shownthatthis wavefunction fullfills the killing
relation (3) to very good accuracy [10] as was mentioned Since the correlated occupation numbers are rounded
already above. The equations read neartheFermisurface,weagaincanconsideranRPAop-
eratorwithallconfigurationsbesides diagonalgenerators.
This feature preserves the property of fullfillment of sum
1 1
n =1 M Yν 2 =1 a+a a+a , (28) rule and appearance of Goldstone (zero) modes to be dis-
h −2 ph| ph| −2 h p h h pi cussedbelow.Inaway,r-RPAisthemostdirectextension
p,ν p
X X of standard RPA: we know that standard RPA accounts
forcorrelations.However,as alreadymentioned,the RPA
n = 1 M Yν 2 = 1 a+a a+a , (29) equations are set up, e.g., with uncorrelated occupation
p 2 ph| ph| 2 h p h h pi numbers. In r-RPA this contradiction is lifted. The per-
h,ν h
X X formance of r-RPA is somewhat between standard and
or SCRPA [10,11,8]
D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length 5
4 Goldstone modes operatorhas no diagonalelements either because it is not
time-reversalinvariant.
More subtle is the question of pairing which is one of
It is well established that the standard HF-RPA (BCS-
the broken symmetries often encountered in Fermi sys-
QRPA) approachexhibits a Goldstone (zero) mode if the
tems. Indeed, in this case, the symmetry operator is the
HFsolutioncorrespondstoacontinuouslybrokensymme-
try. For finite systems, with discrete quantum numbers, particle number operator Nˆ = αa†αaα. In quasiparticle
one mostly talks about a zero or spurious mode. For ex- representationa†α =uαqα† −vαqP−α this becomes
ample, for nuclei and other selfbound Fermi systems like
3He droplets, HF always breaks translational invariance
andthe correspondingRPAshowsazeromode[17,16,13] Nˆ = [vα2 +u2αqα†qα−vα2q−†αq−α
whichcorrespondstoacoherentdisplacementofthewhole α
X
system. In trapped cold atom gases the corresponding u v (q†q† +q q )]. (34)
mode is the Kohn mode where the atom cloud oscillates − α α α −α −α α
coherentlyinanexternalharmonictrappingpotential[18, whichcontainsahermitiandiagonalpiecewhichcannotbe
19].WithinBCS-QRPA,oneobtainsininfinitematterbe- included into the (quasi-particle) RPA operator Q† as we
cause of broken particle number U(1) symmetry a Gold- discussed already above. In this case the standard BCS-
stonemode,theso-calledBogoliubov-Andersonmode[20, QRPA scheme [13], analogous to HF-RPA in the non-
21,22] Also in finite superfluid systems, like many super- superfluidcase,shows,aswellknown[23,24],azeromode
fluid nuclei, zero modes appear [23,24]. because the so-called scattering terms (q†q) drop out of
As mentioned, these Goldstone (zero) modes reflect the QRPA equations. However, with the Self-Consistent
basic principles of quantum mechanics and it is very im- QRPA(SCQRPA) this is not the case and in general the
portant not to destroy this property in theories which go zero mode will be absent. As it turns out, this is only a
beyond the HF-RPA scheme. In this respect, what is the problemforfinite systems.Forhomogeneousinfinite mat-
situation with the SCRPA approach outlined above? As ter we have as QRPA operator
we see from (4), the crucial point is that the Q† opera-
tor canrepresentthe symmetry operator(let us callit Pˆ)
in question as a particular solution of SCRPA equations Q†q = [Xkqqq†−kqk† −Ykqqq−kqk
and that the relation [H,Pˆ] = 0 is not destroyed in the kX>0
course of applying the formalism. In standard RPA, only + Z(+)qq† q Z(−)qq† q ] (35)
the ph and hp components of the symmetry operator en- q+k k− k −k q−k
ter the equations. One can ask the question why the zero whereqisthec.o.m.momentumofthepairexcitation(we
mode appears nonetheless. The answer comes from the suppressed spin indices). So, as long as q is not strictly
fact that even if one included the pp and hh matrix ele- zero, we can include the scattering terms and one will
ments of Pˆ into the RPA, these elements become, in the approachthezeromodeinthelimitasq 0.Onerealises
HF basis, completely decoupled from the rest of the RPA that in finite systems with discrete level→s, the zero mode
equations. Therefore, in standard RPA, it is as if the to- is absent in MF-SCQRPA and one will have to extend
tality of the symmetry operator were included and, thus, the theory. This can be done in including to the QRPA
the Goldstone mode is presentnonetheless.In SCRPA all operator,besidestheonebodysector,thetwobodysector
components of the symmetry operator,besides the diago- what will allow the inclusion of diagonal hermitian one
nal ones, are present and they all play a role. Therefore, body pieces in the QRPA operator and the zero mode is
onemaythinkthatifPˆ =0,theninanycasethe Gold- saved. Some works on including the two-body sector can
αα
stone mode will come in the solution. In this respect, we be found, e.g., in [25] and references in there. However,
must remember that SCRPA joins HF-RPA in the weak it shall not be the task of this work to enter into the
coupling limit. Therefore, the inclusion of the generalised more complicated structure of this approach. It may be
mean field (MF) equation (22) must also be assured for investigated in future work. For the moment, let us be
theappearanceofazeromode.InanalogytotheHF-RPA satisfiedthattheBogoliubov-Andersonmode[20]appears
scheme,onemay talk ofthe MF-SCRPAscheme to imply in the infinite matter case.
a zero mode in the broken symmetry case. In the next Another particular case occurs, if we write the RPA
section, we will demonstrate explicitly the appearance of operatorwith collective ph operatorsas this may be intu-
the Goldstone mode with an application to a model case. itivelythe casewhens.p.states aredegenerateandwhich
oftenreducesthedimensionoftheSCRPAequationsdras-
All this holds true under the condition that Pˆ = 0. tically.The price to be paidare some extra complications
αα
Many symmetry operators have this property. This is the withcertaintwobodycorrelationfunctionsaswewilldis-
case for the linear momentum because of its odd parity. cusswiththe applicationtoamodeljustbelow.However,
So the zero mode corresponding to translational motion in general, the so-called m-scheme where one considers
willcomeinselfboundsystemslikenucleior3Hedroplets. quantumstatebyquantumstateispreferablebecausethe
Equally the abovementioned Kohnmode in trappedcold complications with the evaluation of the density matrices
atom systems. Some systems may be deformed and then are absent. For example with the ansatz (35) the latter is
rotational symmetry is broken. The angular momentum the case.
6 D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length
5 Sum-rules space, the Hamiltonian is given by (see, e.g., [26])
We show that the energyweightedsum rule, givenby the H = (ǫ µ)nˆ (41)
k k,σ
following identity −
k,σ
X
U
+ a† a a† a
(E E ) ν F 0 2 = 1 0[F,[H,F]]0 (36) 2N k,σ k+q,σ p,−σ p−q,−σ
ν − 0 |h | | i| 2h | | i k,Xp,q,σ
ν
X
where nˆ = a† a is the occupation number oper-
is fullfilled within SCRPA. Here ν is a complete set of k,σ k,σ k,σ
| i ator and the single particle energies are given by ǫ =
eigenstates and F is a one body operator k
2t D cos(k ) with the lattice spacing set to unity. It
− d=1 d
F = f A (37) is convenientto transformthe fermion creation and anni-
αβ αβ hilatPion operatorsa†,a to HF quasi-particle operators.In
αβ
X 1D, we have
where A is defined in (6). One can show that the iden-
αβ
tity (36) is automatically fullfilled if one considers that a =b† , a =b (42)
ν is the set of SCRPA or r-RPA eigenstates. By using h,σ h,σ p,σ p,σ
| i
the inversetransformationof the basisoperatorsA one where h and p are momenta below and above the Fermi
αβ
obtains momentum, respectively, so that b HF = 0 for all k
k,σ
| i
where HF is the Hartree-Fock ground state in the plane
(Eν E0) ν F 0 2 wave b|asisi. Introducing the operators
− |h | | i|
ν
X
= (Eν E0) 0[Qν,F]0 2 n˜k,σ =b†k,σbk,σ , (43)
− |h | | i|
ν
X
= (Eν −E0)| fαβMα1/β2(Xανβ +Yανβ)|2 . Jp−h,σ =bh,σbp,σ , Jp+h,σ =(Jp−h,σ)† (44)
ν αβ
X X we write for the RPA operator
(38)
Q+ = [X¯ν J+ Y¯ν J− ] (45)
UsingthegeneralsystemofRPAequationswithexcitation q,ν ph,σ ph,σ− ph,−σ ph,−σ
energy Ων =Eν E0 one gets the following identity pXh,σ
−
With our Eqs. (28, 29) for the evaluation of the occu-
(Eν E0) ν F 0 2 pation numbers, we also verify straightforwardlythat
− |h | | i|
ν
X
= f M1/2 f M1/2(A B ) .
αβ αβ γδ γδ αβ,γδ− αβ,γδ n = n˜ = J+ J−
αβ γδ h pσi h pσi h phσ phσi
X X
h
(39) X
= (1 M )Yν 2,
−h phσi | phσ|
Ontheotherhandbysplittingthesummationsintoα>β
h,ν
X
and α < β parts and using the definitions of the RPA
n =1 n˜ =1 J+ J−
matrices one obtains that the right-hand side of the Eq. h hσi −h hσi − h phσ phσi
(39) has the same form Xp
=1 (1 M )Yν 2 (46)
1 − −h phσi | phσ|
0[F,[H,F]]0 p,ν
2h | | i X
1 where M = n˜ + n˜ , andmore complicatedex-
= fαβfγδ 0[Aαβ,[H,Aγδ]]0 . (40) h phσi h hσi h pσi
2h | | i pressions for the quadratic terms. Those expressions are
αβ γδ
XX the same as derived in our earlier publication [26].
We would like to emphasise that this sum rule also
remains fulfilled in the case of broken symmetry, in spite The Hubbard molecule. In the Hubbardmodel,the results
ofthefactthatforazeroenergyeigenstatetheamplitudes are again quite promising. For example the half filled 2-
X,Y diverge. sites problem, the so-called Hubbard molecule, is solved
exactly.Thisisnotatotallytrivialresult.Forexamplethe
well known GW approximation fails in this respect [26].
6 Applications to models
Thehalf-filled6-siteschain.WeshowinFig.1,forachoice,
6.1 The Hubbard model the excitation spectrum for the momentum transfer q =
| |
π. The abreviations ’ch’ and ’sp’ stand for ’charge’, i.e.
Weshortlywanttooutlinehowourformalismworksinthe spin S = 0 and for ’spin’, that is S,M = 1,0 excitations,
Hubbard model and present some results. In momentum respectively. The results for q =2π/3,π/3 are of similar
| |
D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length 7
|q|=π 0.1
3.5
0.08
3 Exat
SCRPA
sp s−RPA
2.5
0.06
n
pσ
2
ε/t
ch 0.04
1.5 |k|=2π/3
sp
0.02 k=−π
1
ph −RPA standard
Exact
ph −SCRPA
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5
0 U/t
0 0.5 1 1.5 2 2.5 3 3.5
U/t Fig. 3. Particle occupation numbers for momenta k = π/3
and k = −π in SCRPA and standard RPA (s-RPA) for the
Fig. 1. Excitationspectrumforthetransfer|q|=π asafunc- 6-sites half filled Hubbard chain as a function of U/t.
tion of U/t of the 6-sites half-filled Hubbard chain. ’sp’ and
’ch’ stand for spin and charge response, respectively. 1
−4
Exact 0.95
ph −RPA Standard
ph −SCRPA
−5 HF
nhσ 0.9 |k|=π/3
EGS/t −6
Exact
k=0
s−RPA
0.85
SCRPA
−7
0.8
0 0.5 1 1.5 2 2.5 3 3.5
U/t
−8
0 0.5 1 1.5 2 2.5 3
Fig. 4. Sameas Fig. 3 but for hole occupations.
U/t
Fig. 2. Ground state energy as a function of U/t for the 6-
i.e., what is depleted below the Fermi surface is exactly
sites half-filled Hubbard chain with SCRPA, standard RPA,
replaced by non-zero values above the Fermi surface. In
HF, and exact solution.
themacroscopiclimit,thisimpliesthattheLuttingerthe-
orem(see,e.g.,[27])isrespected.Letusalsomentionthat
the1DHubbardmodelhasbeensolvedwithrenormalised
quality [26]. In Fig.2, we show the ground state energy.
RPA in the infinite system limit with interesting results
There is good agreement with the exact solution and it
particularly in the strong coupling limit, [28].
presents a maximum error of about 0.8 percent at U/t=
3.5. In that figure, we also display the results of HF and
standard RPA. We can appreciate the gain in precision
6.2 Pairing model
with SCRPA.
InFigs.3and4,weshowtheoccupationnumbers,see The pairing or picket fence model (PFM) is the only one
Eq.(46).WeseethatwithSCRPAtheycompareverywell whereinthepasttheSCRPAschemecouldbeappliedfor
with the exact values and are very much improved over thefirsttimewithoutanyapproximationandwithoutthe
the corresponding values from standard RPA (s-RPA). It explicitknowledgeofthevacuum[10].Thisstemmedfrom
is worth mentioning that particle number is conserved, thefactthatinthisparticularmodelwiththeHamiltonian
8 D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length
H = ε N +V P+P (47)
i i i k
i ik
X X
and N two fold degenerate equidistant levels labeled with
theindex’i’,theoccupationnumberoperatorscanexactly
beexpressedbytheproductoftwofermionpairoperators,
that is
N =2P†P (48)
i i i
with N = a† a +a† a and P† = a† a† . It is seen
i i+ i+ i− i− i i+ i−
that the pair operators are the ones which enter the Bo-
goliubovtransformationoffermionpairsinthepp-SCRPA
Q+ = X¯αP++ Y¯αP+ (49)
α p p h h
p h
X X
and,therefore,with (3)itwaspossiblein[10]to calculate
N and N N completely selfconsistently and without
i i j
h i h i
the use of any procedures external to the SCRPA ones.
Wealsoremarkthattheevaluationof N N necessitates
i j
h i
the knowledge of four particle correlation functions what
makes the approach rather heavy. However, factorisation
N N N N turned out to work quite well, thus
i j i j
h i ∼ h ih i
strongly simplifying the expressions [29].
Fig.5.FirstexcitedstateinSCppRPAforN=10asafunction
InFigs.5and6,weshowresultsobtainedwiththisfac- of the pairing coupling strength G. The strong improvement
torised approximation for the first excitation energy with over standard ppRPA should be observed. The exact result is
N =10 particles and the correlation energy, respectively, presented bythe full line
as a function of the pairing coupling strength G. The ex-
act results were obtained from the equations established
by Richardson almost half a century ago [30].
Commenting on the SCRPA results, we see that they
are very much improved over standard ppRPA, [13]. On
the other hand, we also see from the sum rule relation
given in [10], Eq. (69), i.e. pp′hNpNp′i = phh(2 −
N )N andTablesVIIandXIin[10]thatthePauliprin-
h p
i P P
ciple,isstillslightlyviolated,oftheorderof4 5percent,
−
what stems from the fact that the killing condition (3)
is not exactly fulfilled. Let us mention that SCRPA was
solvedin[10]and[29]amongothersforthecaseof100lev-
els where it is even difficult to solvethe problemwith the
Richardson equations. A very instructive example is the
N = 2, i.e., the single Cooper pair case. Though already
presented in [10], let us discuss it here again. In standard
RPA the excitation energy is given by
E √1 G
∝ −
whereas in SCRPA the result is
E √1+G .
∝
Thelattercoincides,asalreadymentioned,withtheexact
result.TheRPAresultshowstheusualBCSinstabilityat
Fig. 6. Correlation energy for 10 particles from SCppRPA
G=1. With SCRPA the vertex renormalisationfrom the
compared with theexact result as function of coupling
self consistency, i.e. screening, has effectively turned the
sign of G around and with screening the effective inter-
action is now repulsive ! This stems from the fact that
for N = 2 the constraint from the Pauli principle is, as
D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length 9
one easily realises,of maximumimportance. It is testified
again with this example that the Pauli principle is very
well respected with SCRPA. 1
0.9
Let us alsomentionthat the SCRPA scheme has been
0.8
generalisedtofinitetemperaturesusinganequivalentGreen’s 0.7
function formalism in an application to the PFM in [12] 0.6
0.5
withthesamequalityofresultsasatzerotemperature.In
0.4
particular it could be shown that also in the PFM, there 0.3
opens a pseudo gap in the level density approaching the 0.2
0.1
critical temperature from above. 0
-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1
6.3 Goldstone mode within a 3-level spin model and 1
SCRPA 0.95
0.9
The SCRPA scheme is a selfconsistent in-medium two 0.85
0.8
body equation of the Schroedinger type. We think it is
0.75
amenablefornumericalsolutionforrealisticsystems.How-
0.7
ever,thisneedsmajorinvestmentsandthisisnotavailable
0.65
at this moment. We, therefore, have chosen a simplified
0.6
model to demonstrate numerically the fullfillment of the -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1
Goldstone mode. We have chosen the three level Lipkin
model which also can be seen as a three sites spin model,
corresponding to a SU(3) algebra [31]. This model has
Fig. 7. (a) First SCRPA excitation energy for N = 20 and
been used in order to test different many body approxi-
ǫ1 = 0, ǫ2 = 1, ǫ3 = 1+∆ǫ versus log10∆ǫ (in arbitrary
mations[32,33].Wealsohavetreateditalreadyin[8]with
units).(b) The ratio Y/X versuslog ∆ǫ.
10
results of similar quality as in the two preceding models
of sections VI.1 and VI.2. Here we want to dwell specifi-
cally on the zero mode. It is so far the only model where This means that we only have the single parameter φ to
the appearance of the Goldstone (zero) mode has been be varied to obtain the ’deformed’ solution. First, let us
demonstrated in a numerical application with SCRPA. write down the RPA operator in the deformed basis
By labeling the levels with 0, 1, 2, we consider the
following Hamiltonian written in some ’original’ basis
Q† =Xν A +Xν A +Xν A
ν 10 10 20 20 21 21
YνA YνA YνA , (54)
2 V 2 − 10 01− 20 02− 21 12
H = ǫ S (S S +S S ) , (50)
k kk k0 k0 0k 0k where the A operators correspond to the S ones of (51)
− 2
kX=0 Xk=1 but in the deformed basis.
where ǫ =0; ǫ =1; ǫ =1+∆ǫ and We also write the Hamiltonian in the deformed basis
0 1 2
andthenconstructtheSCRPAequationsfromthedouble
N commutator relations (11, 12). While solving the SCRPA
Skk′ = a†kµak′µ (51) equations we have to minimise the ground state energy
µ=1 with respect to φ in order to fullfill the generalised mean
X
fieldequation(18).Thelattercanbecalculatedasafunc-
WesupposethatthethreelevelshaveequaldegeneracyN
tion of X,Y,φ in expressing its expectation values from
andthatinthenon-interactingcasethelowestlevelisfull
the inversion of (54) and then using (3). Our procedure
so that N corresponds also to the particle number. The
works with collective generators what seems natural in
operators (51) satisfy simple commutation relations
this model, since we can suppose that the non-collective
states decouple to a large extent from the rest of states.
[S ,S ]=δ S δ S . (52)
k1k2 k3k4 k2k3 k1k4 − k1k4 k3k2 Working with the individual quantum states (m-scheme)
Standard HF-RPA shows a zero mode in the so-called would considerably complicate the solution of the model
deformed region where HF in the original basis is un- withthe SCRPA scheme.Employingthe collectiveopera-
stable and when the two upper level become degenerate tors has, however, the disadvantage that expectation val-
(∆ǫ = 0). This because the hamiltonian commutes with ues of products of diagonal operators as A , A A ,
00 00 22
the ’angular momentum’ operator Lˆ =i(S S ). Ac- etc. cannot directly be expressed with thhe Xi,hY amplii-
0 21 12
−
cordingto[31],thetransformationmatrixtothedeformed tudes. We, therefore, in [8] found expressions via the uni-
basis in (19) can be written as follows tary operator method what yields
cosφ sinφ 0
R† = sinφ cosφ 0 . (53)
αk − 0 0 1! nα ≈ yαα+y11y22/N
(cid:20) (cid:21)
10 D.S.Delion1,2,3, P. Schuck4,5, and M. Tohyama6:Title Suppressed Dueto Excessive Length
−1
employs propagatorsor many body Green’s functions. Of
1+2(y +y )/N +3y y /N2
× 11 22 11 22 course, it is clear that every eigenvalue problem has a
(cid:20) (cid:21) corresponding formulation with Green’s functions but it
α=1,2 , (55)
may be useful to give some more details on the ingredi-
wherey = Yν Yν .Here,evidentlyn =N n n ents of the presentformalism.However,the Green’s func-
αβ ν α0 β0 0 − 1− 2 tionequivalent to the eigenvalue equationofSCRPA (10)
and n = A . For the quadratic terms we obtain [8]
α h αPαi is, in a way, somewhat particular. As one may immedi-
ately realise, it cannot come from the familiar many time
N 1
A00Aαα − Aα0A0α Green’sfunctionapproachwhere,e.g.,thetwobodyprop-
h i≈ N h i
agator(andalsoitsintegralkernel)dependsonfourtimes
1
( A2 A2 + A A A A ) onceonegoesbeyondthestandardHF-RPAscheme.This
− N2(N 1) h α0 0αi h 10 20 02 01i
stems from the fact that in an eigenvalue problem only
−
1 one energy (the eigenvalue) is involved and then the cor-
A A A A A A .
h 11 22i≈ N(N 1)h 10 20 02 01i respondingintegralequationforthe Green’sfunctionalso
−
(56) can involve only one energy, even in the integral kernel.
Though the formalism has been described in earlier pub-
Evaluating the four body correlation functions with lications, see, for instance, refs. [12] and [14], we feel that
the inversion (16) and the killing condition (3), one ob- it may be helpful for the reader to give a short outline of
tainsasetoflinearequationsforthetwobodycorrelation the procedure. To this purpose, we write down the corre-
functions with diagonal A opertors which can be solved. spondingintegralequationformof(14),thatistheBethe-
With this, we get the solution of the SCRPA equations Salpeter equation
in the deformed regime. In Fig. 7 (a) we show the energy
(oifntharebfiitrrsatryexucnitietds).stAasteaalrseaadyfunmcetniotnionoefd∆, ǫin=thǫe2l−imǫi1t ( ¯hω−Ek1 +Ek2)G˜kω1k2k3k4
∆ǫ 0azeromodeshouldappear.Weseethatthereisa = [δ δ + ]˜ω (57)
→ N0,k1k2 k1k3 k2k4 Sk1k2k3′k4′ Gk3′k4′k3k4
rapiddecreaseofthefirstexcitationenergyΩ tozero.Of
1 kX3′k4′
course the whole system of SCRPA equations is very sen-
Inserting the spectral representation for the Green’s
sitivetonumericalaccuracy.Forinstancetheminimumof
function
the ground state energy at φ = φ is not easy to deter-
0
mine withhighaccuracywhichweestimateto beoforder χµN χµ∗
10−3. With this, for ∆ǫ = 10−5, we obtain an excitation ˜ω = µ (58)
energy of order 10−2. It also is interesting to follow the G ¯hω Ωµ+iηNµ
µ −
X
values of the X,Y amplitudes. In Fig. 7 (b) we show the
where the sum goes over positive and negative values of
evolution of their ratio Y/X as it approaches one. At ex-
µ and N = N ,Ω = Ω , and taking the limit
µ −µ µ −µ
actlyzeroexcitationenergytheamplitudeswoulddiverge. − −
¯hω Ω ,weobtainincomparingthesingularitiesonleft
ν
Here, they are still of reasonable value, i.e. X Y 20. →
∼ ∼ and right hand sides, the eigenvalue equation (14).
It is worth mentioning that even a very tiny inaccuracy
In order to see how this scheme with the equation of
in φ destabilises the zero mode showing that it is ab-
0 motion technique can go on and lead to an ω-dependent
solutely neccessary to work in the basis which fullfills eq.
termintheintegralkerneloftheBethe-Salpeterequation,
(18).Thescenariostaysmoreorlessthesame,ifinsteadof
we extend the opertor (2) to include a two body term as
SCRPAthesimplerr-RPAisapplied(seeSectionIII).We
a first extension, eventually higher order terms
think thatthis isaveryinstructiveexamplewhichclearly
demonstratesthatSCRPAintheformpresentedherewith
allcomponentsincludedconservesallappreciatedproper- Q˜† = [χ˜ a†a + ˜ a†a†a a +...] (59)
ν αβ α β Xαβγδ α β δ γ
ties of standard HF-RPA. To the best of our knowledge,
we do not know of any other method which with more or This leaXds to an extended eigenvalue problem involv-
less equal performances obtains the zero mode in a simi- ing also the two body amplitudes . Eliminating the lat-
X
larly easy way. Even for this simple model the realisation ter fromthe coupledequationsof onebody andtwobody
ofthezeromodewiththeΦderivablefunctional[7]would amplitudes, one obtains an effective equation for the χ
be very combersome. amplitudes with an effective, energy dependent potential
containing implicitly the two body amplitudes. This ef-
fective potential can be qualified to corresponds to the
ω dependent part of a two body self energy. This proce-
7 A short outline of the equivalent Green’s
durecanformallybepusheduptotheN-bodyamplitudes
function description of the equation of
leadingthustoanexacttwobodyDysonequationform,in
motion method analogyto whatisknownfromthe singleparticleGreen’s
function.
In condensed matter physics dealing with homogeneous Let us shortly show how the same scheme can be ob-
infinite systems,one usually does notformulate the prob- tained beginning directly with the Green’s function. We
lems in the form of an eigenvalue equation. One rather start with the following chronologicalpropagator